Determination of simple correlated wave functions for few-electron systems using a Jastrow factor

A variational approach is developed for bound state calculations in three-\nand four-electron atomic systems. This approach can be applied to determine, in\nprinciple, an arbitrary bound state in three- and four-electron ions and atoms.\nOur variational wave functions are constructed from four- and five-body\ngaussoids which depend upon the six ($r_{12}, r_{13}, r_{14}, r_{23}, r_{24},\nr_{34}$) and ten ($r_{12}, r_{13}, r_{14}, r_{15}, r_{23}, r_{24}, r_{25},\nr_{34}, r_{35}$ and $r_{45}$) relative coordinates, respectively. The approach\nallows one to operate with the different number of electron spin functions. In\nparticular, the trial wave functions for the ${}^1S$-states in four-electron\natomic systems include the two independent spin functions $\\chi_1 = \\alpha\n\\beta \\alpha \\beta + \\beta \\alpha \\beta \\alpha - \\beta \\alpha \\alpha \\beta -\n\\alpha \\beta \\beta \\alpha$ and $\\chi_2 = 2 \\alpha \\alpha \\beta \\beta + 2 \\beta\n\\beta \\alpha \\alpha - \\beta \\alpha \\alpha \\beta - \\alpha \\beta \\beta \\alpha -\n\\beta \\alpha \\beta \\alpha - \\alpha \\beta \\alpha \\beta$. We also discuss the\nconstruction of variational wave functions for the excited $2^3S$-states in\nfour-electron atomic systems.\n

We introduce a method for solving a self consistent electronic calculation within localized atomic orbitals, that allows us to converge to the complete basis set (CBS) limit in a stable, controlled, and systematic way. We compare our results with the ones obtained with a standard quantum chemistry package for the simple benzene molecule. We find perfect agreement for small basis set and show that, within our scheme, it is possible to work with a very large basis in an efficient and stable way. Therefore we can avoid to introduce any extrapolation to reach the CBS limit. In our study we have also carried out variational Monte Carlo (VMC) and lattice regularized diffusion Monte Carlo (LRDMC) with a standard many-body wave function (WF) defined by the product of a Slater determinant and a Jastrow factor. Once the Jastrow factor is optimized by keeping fixed the Slater determinant provided by our new scheme, we obtain a very good description of the atomization energy of the benzene molecule only when the basis of atomic orbitals is large enough and close to the CBS limit, yielding the lowest variational energies.

A procedure is developed to construct improved wave functions for the bound states in various four-electron atoms and ions. The bound state wave functions are improved in the sense that our trial wave functions are compact and accurate. We employ two independent four-electron spin functions ${\ensuremath{\chi}}_{1}=\ensuremath{\alpha}\ensuremath{\beta}\ensuremath{\alpha}\ensuremath{\beta}+\ensuremath{\beta}\ensuremath{\alpha}\ensuremath{\beta}\ensuremath{\alpha}\ensuremath{-}\ensuremath{\beta}\ensuremath{\alpha}\ensuremath{\alpha}\ensuremath{\beta}\ensuremath{-}\ensuremath{\alpha}\ensuremath{\beta}\ensuremath{\beta}\ensuremath{\alpha}$ and ${\ensuremath{\chi}}_{2}=2\ensuremath{\alpha}\ensuremath{\alpha}\ensuremath{\beta}\ensuremath{\beta}+2\ensuremath{\beta}\ensuremath{\beta}\ensuremath{\alpha}\ensuremath{\alpha}\ensuremath{-}\ensuremath{\beta}\ensuremath{\alpha}\ensuremath{\alpha}\ensuremath{\beta}\ensuremath{-}\ensuremath{\alpha}\ensuremath{\beta}\ensuremath{\beta}\ensuremath{\alpha}\ensuremath{-}\ensuremath{\beta}\ensuremath{\alpha}\ensuremath{\beta}\ensuremath{\alpha}\ensuremath{-}\ensuremath{\alpha}\ensuremath{\beta}\ensuremath{\alpha}\ensuremath{\beta}$ in each trial wave function. Optimization of the nonlinear parameters is performed for each radial function. Our procedure is applied to obtain the total energies of the ground $^{1}S^{e}$ states in the following four-electron berylliumlike systems: ${\mathrm{Li}}^{\ensuremath{-}}$, Be, ${\mathrm{B}}^{+}$, ${\mathrm{C}}^{2+}$, ${\mathrm{N}}^{3+}$, ${\mathrm{O}}^{4+}$, ${\mathrm{F}}^{5+}$, ${\mathrm{Ne}}^{6+}$, ${\mathrm{Na}}^{7+}$, and ${\mathrm{Mg}}^{8+}$. All our trial wave functions used in this study are very compact. In fact, each of these functions contains only 300 basis (radial) functions which are constructed from ten-dimensional gaussoids of the five-body relative coordinates ${r}_{12}$, ${r}_{13}$, ${r}_{14}$, ${r}_{15}$, ${r}_{23}$, ${r}_{24}$, ${r}_{25}$, ${r}_{34}$, ${r}_{35}$, and ${r}_{45}$.

The direct inversion in the iterative subspace (DIIS) method is applied to several simple SCF wave functions in an effective Fock matrix formulation. The following cases are treated: high-spin-restricted open shell, open-shell singlet, and two-configuration wave functions. Open-shell singlet states are described by a three-determinant 2×2 CAS expansion which is equivalent to Davidson’s nonorthogonal SCF method in the case of the first open-shell singlet. Very sharp convergence is usually obtained in less than 20 cycles. The method is applicable to slowly convergent or even inherently divergent cases, and able to enforce convergence to excited states not the lowest of their symmetry. For these simple wave functions, the present first order method is asymptotically more efficient than second-order methods. Examples are presented for H2O, H2O2, C2H4, F2, several states of NO2, C2H5, formaldehyde, and ketene.

Momentum distributions of a 9Li fragment arising from the ( 11Li, 9Li) reaction and neutron correlations

We have studied via a Green-function Monte Carlo (GFMC) method the S=(1/2 Heisenberg quantum antiferromagnet in two dimensions. We use a well-known transformation to map the spin problem onto a system of hard-core bosons that allows us to exploit interesting analogies between magnetism and superfluidity. The GFMC method is a zero-temperature stochastic method that projects out the component of the true ground state in a given variational wave function. This method is complementary to previously used finite-temperature Monte Carlo methods and is well suited to studying the ground state and low-lying excited states. Starting with even a simple wave function, e.g., the classical N\'eel state, the GFMC method can obtain the short-range correlations very accurately, and we find the ground-state energy per site ${\mathit{E}}_{0}$/J=-0.6692(2). We show that it is important to include the zero-point motion of the elementary excitations in the ground state and by a spin-wave analysis find that it produces long-range correlations in the wave function. Upon inclusion of such long-range correlations, we obtain a staggered magnetization ${\mathit{m}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$=0.31(2) and the structure factor scrS(q)\ensuremath{\sim}q at long wavelengths. Using the Feynman-Bijl relation, from the slope we deduce the renormalization of the spin-wave velocity by quantum fluctuations to be ${\mathit{Z}}_{\mathit{c}}$==c/${\mathit{c}}_{\mathit{s}}$=1.14(5).

A method is suggested to build simple multiconfigurational wave functions specified uniquely by an energy cutoff Λ. These are constructed from a model space containing determinants with energy relative to that of the most stable determinant no greater than Λ. The resulting Λ-CI wave function is adaptive, being able to represent both single-reference and multireference electronic states. We also consider a more compact wave function parameterization (Λ+SD-CI), which is based on a small Λ-CI reference and adds a selection of all the singly and doubly excited determinants generated from it. We report two heuristic algorithms to build Λ-CI wave functions. The first is based on an approximate prescreening of the full configuration interaction space, while the second performs a breadth-first search coupled with pruning. The Λ-CI and Λ+SD-CI approaches are used to compute the dissociation curve of N2 and the potential energy curves for the first three singlet states of C2. Special attention is paid to the issue of energy discontinuities caused by changes in the size of the Λ-CI wave function along the potential energy curve. This problem is shown to be solvable by smoothing the matrix elements of the Hamiltonian. Our last example, involving the Cu2O2(2+) core, illustrates an alternative use of the Λ-CI method: as a tool to both estimate the multireference character of a wave function and to create a compact model space to be used in subsequent high-level multireference coupled cluster computations.

我们在文献[1]中设计了一套五个参数的变分波函数用来计算了周期表中前面十个原子的能量,所得结果比过去一些作者用四参数波函数所算得的结果为好。我们在过去计算经验的基础上,另外找到了一套特别简单的解析波函数,其形式为1s电子:ψ1(r)=N1e-μar,2s电子:ψ2(r)=N2[(μr)e-μr-Ne-μar],2p电子:ψ3(r)=N3(μr)cosθe-μr,ψ4(r)=N4(μr)sinθeiφ-μr,ψ5(r)=N5(μr)sinθe-iφ-μr,式中的α与μ为变分参数;N1,N2,N3,N4,N5为归一化因子;N为正交化系数。μ可用解析法来决定,因而只有一个参数α要由数值法来决定。我们用这样的波函数算出了第二周期元素的正常态原子和离子(共有八十几个原子态)的各电子的各种能量积分值及总能量值,并确定了波函数的最佳参数值。其结果与五参数波函数的计算结果相比,一般相差在万分之一至千分之一的范围内,并比最近有些作者用一种三参数波函数所算的结果还好。根据这些结果,我们还讨论了Slater近似计算法的可靠程度和适用范围。

In a recent paper a general impulse approximation formalism for ($p,\ensuremath{\pi}$) reactions in nuclei was developed which incorporated distortion, spin, and antisymmetrization effects. Here we discuss two further considerations in connection with this model with particular reference to its application to the reactions $\mathrm{pd}\ensuremath{\rightarrow}t{\ensuremath{\pi}}^{+}$ and $\mathrm{pd}\ensuremath{\rightarrow}^{3}\mathrm{He}{\ensuremath{\pi}}^{0}$. First, realistic wave functions for $t$ and $^{3}\mathrm{He}$ which produce the dip in the electromagnetic form factor are incorporated. We find results qualitatively similar to those found previously when simple analytic wave functions were used. This is in contrast to the work of Locher and Weber who found that the dip in the electromagnetic form factor produced a similar dip in the ($p,\ensuremath{\pi}$) cross section. In the present model correct antisymmetrization of the wave functions and inclusion of spin components of the wave functions, in particular the $D$ state of the deuteron, fill the dip. Second, we examine the factors which affect the energy at which a bump due to the $\ensuremath{\Delta}(1232)$ resonance would appear in the ($p,\ensuremath{\pi}$) cross section. This is motivated by a recent experiment which found no evidence of a bump at 450 MeV where it would be expected from simple kinematic arguments. We find that distortion and form factor effects both tend to push the bump to lower energies and that the net result is that the model predicts a smooth falloff with energy of the fixed angle cross section above 350 MeV, in qualitative agreement with the experiment, and a bump below 350 MeV, where there are no data. The physical mechanisms responsible for the shift, and some possible approaches which might improve quantitative agreement, are then briefly discussed.[NUCLEAR REACTIONS ($p,\ensuremath{\pi}$) reactions in distorted-wave impulse approximation. Effects of realistic wave functions on cross section and of form factors and distortion on resonance position. Differential cross section and energy dependence of fixed angle cross section for $^{2}\mathrm{H}(p,{\ensuremath{\pi}}^{+})^{3}\mathrm{H}$ and $^{2}\mathrm{H}(p,{\ensuremath{\pi}}^{0})^{3}\mathrm{He}$ at 250-800 MeV.]

In a recent paper [Bressanini et al. Phys. Rev. Lett. 95, 110201 (2005)] it was pointed out that ``the nodes of even simple wave functions are largely unexplored.'' Here we show that for $N$-particle wave functions nodal surfaces arise from the spin and orbital entanglement of constituent two-particle wave functions and derive, for two-electron atoms, 11 exact nodal rules applicable in $LS$ coupling. In addition, the ``higher symmetry'' identified numerically in the above paper is shown to be an approximate dynamical symmetry described by a molecular model or a classical unstable periodic orbit. We show that the analysis is readily extended to four-particle wave functions and consider the case of the hydrogen molecule, both fixed in space and rotating.

Ionization of hydrogen atoms by the impact of fast charged particles can be accurately treated theoretically using simple two-center wave functions to describe the scattering system both initially and finally. For the final state, we use a continuum distorted wave (CDW) that contains the product of three Coulombic distortion factors (one for each two-body interaction), hence it is called the 3C wave function. This CDW (3C) wave function is ideal for studying fast collisions since it is asymptotically correct in all asymptotic domains of momentum space for all configurations of the three particles in coordinate space [S. Jones and D. H. Madison, Phys. Rev. A $62,$ 42 701 (2000)]. Coulomb distortion in the entrance channel is provided by an eikonal initial state (EIS), and this CDW-EIS approximation, introduced by Crothers and McCann [J. Phys. B $16,$ 3229 (1983)], has proven to be the most accurate perturbative method ever devised within a two-state approximation. The first fully differential cross sections for ion-atom ionization in the CDW-EIS approximation are presented. In addition, by considering projectiles of different mass and charge, it will be shown that although the charge of the projectile is important, the mass of the projectile plays virtually no role in the vast majority of fast ionizing collisions.

The ground-state hyperfine structure and other properties are calculated for muonic helium atoms (3 He +2μ-e- and 4 He +2μ-e-), with the use of some local properties of the wave functions in the domains where two particles are close to each other or far away. Simple wave functions incorporating these properties with one variational parameter is developed. Also, the electron-muon correlation function is considered in proposed wave functions. It has a correct behavior for r12 tending to zero and infinity. The calculated values for hyperfine structure, energy and expectation values of r2n in ground-state, are compared with the multibox variational approach and the correlation function hyperspherical harmonic method. Our obtained results are very close to the values calculated by mentioned methods and giving strong indications that the proposed wave functions in addition to being very simple, provide relatively accurate values for hyperfine structure, energy and expectation values of r2n, emphasizing on the importance of the local properties of the wave functions.

A simple five-parameter wave function is constructed and the positions and widths of the lowest two compound states of the ($e,\mathrm{H}$) system are calculated for both singlet and triplet spin multiplicities with zero total angular momentum. Very good agreement with previous calculations using more elaborate trial functions is obtained because particular attention is paid to the asymptotic form of the wave function. A method adopted from the double-perturbation theory for the systematic improvement of the calculation of widths and shifts of compound states is presented in an Appendix.

We demonstrate that electrons in quantum dots defined by electrostatic gates in semiconductor nanotubes freeze orderly in space realizing a `Wigner molecule'. Our exact diagonalisation calculations uncover the features of the electron molecule, which may be accessed by tunneling spectroscopy -indeed some of them have already been observed by Deshpande and Bockrath [Nature Phys. 4, 314 (2008)]. We show that numerical results are satisfactorily reproduced by a simple ansatz vibrational wave function: electrons have localized wave functions, like nuclei in an ordinary molecule, whereas low-energy excitations are collective vibrations of electrons around their equilibrium positions.

We use a simple Gutzwiller variational wave function to study the dispersion relation of one single hole in the t-J model. Such a wave function gives a minimum of energy for the hole at (\ensuremath{\pi}/2,\ensuremath{\pi}/2), in agreement with previous results, and both a band width and an energy that are linear in J for moderate values of J.