Adaptive multiconfigurational wave functions

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.

Compact Jastrow-Slater-type correlated wave functions for three- and four-electron atoms and ions (Li, ${\mathrm{Be}}^{+}$, ${\mathrm{B}}^{2+}$, ${\mathrm{C}}^{3+}$, Be, ${\mathrm{B}}^{+}$, and ${\mathrm{C}}^{2+}$) are proposed. The Jastrow factor we employed consists of one-body and two-body functions including only two variational parameters in total. We found that a one-body Jastrow function with one variational parameter, which is responsible for the screening effect around the nucleus, is effective for reproducing good total energies of the three- and four-electron atomic systems if hydrogenoid orbitals are adopted in the Slater determinant. On the other hand, a determinant composed of Hartree-Fock orbitals multiplied by the same Jastrow factor was found to give rather worse results than that composed of hydrogenoid orbitals. This result clearly indicates that analytic hydrogenoid orbitals coupled with the one-body Jastrow function are useful for describing simple wave functions and understanding the physical properties of these systems.

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.]

Although the concept of a potential energy curve (PEC) originates from the outgrowth of the Born-Oppenheimer (BO) approximation, we propose the application of analysis methods for the physical PEC with non-Born-Oppenheimer (non-BO) wave functions. A numerical examination was performed with the highly accurate non-BO vibronic wave functions of hydrogen molecular ion, which were obtained in our previous studies with the free complement method. The reduced density function integrated over the electron coordinates plays an important role in understanding nuclear motion dynamics, since it corresponds to the wave function density of the vibrational and rotational motions. The maximum positions of this density indicate the high existence probability of nuclei and can be considered as a discrete representation of the PEC. Whereas an ordinary PEC with the BO approximation is obtained as a numeric curve after multiple electronic state calculations at fixed nuclear coordinates, we propose a new analytical expression of the PEC from a non-BO wave function.

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.

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).

In the previous sections I have discussed how to visualize spaces adapted to various operators and spatial point group symmetries. All eigenfunctions of the desired operators and of the desired symmetry that can be built from a set of given one-particle states were taken as the bases of model spaces. Symmetry was thus the only selection criterium employed so far. For practical purposes this is usually not sufficient: full symmetry-adapted many-particle space grows very rapidly with the number of primitive states and although the graphs can always be drawn and their general properties analyzed for practical calculations the dimension of the full space becomes quickly too large to handle even for the best computers. The traditional way of reducing the size of a model space is based on the concept of reference states. Coulomb forces are rather weak, therefore in atomic and molecular problems the independent-particle approximation works rather well and it is possible to choose the one-particle states in such a way, that among all many-particle symmetry-adapted states there is one state or a combination of a few states clearly dominating the exact solution. In such cases one may argue that the model space with a basis composed of these few states (called ‘the reference states’ or simply ‘references’) plus the states that are ‘not far’ from them allow to form a very good approximation to the exact state (wave function) of our system. ‘Not far’ usually means configurations built from almost the same orbitals as the references are, with at most one or two orbitals changed. Matrix elements of a two-particle operator between the reference and the singly or doubly substituted states are in general different from zero, justifying the choice of these states to our model space by their contributions in the second-order perturbation theory.

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.

Taking the ground state of helium as an example, evidence is presented in support of the theory that when an approximate wave function satisfies a hypervirial theorem, then the expectation values of certain operators calculated using that wave function will be quite accurate. Firstly, it is discovered which hypervirial theorems are satisfied by various simple approximate wave functions when their disposable parameters are chosen to optimize the energy. Appropriate expectation values are tested, and found to be reasonably accurate. Secondly, the optimum energy criterion is abandoned in certain cases, and parameters are chosen to satisfy alternative hypervirial theorems. This leads to improvement in expectation values related to the hypervirial operators. Some advantages and drawbacks of the hypervirial method of fixing parameters in approximate wave functions are briefly discussed.

Variational calculations and the nature of the phase transition in Z(2) gauge theory