Session J25: Focus Session: Explicitly correlated Methods and Quantum Few-Body Systems

Ultracold physics with 3, 4, or 5 atoms

Recent studies will be reviewed [1-3], which utilize hyperspherical coordinates to treat few-body systems, concentrating on processes such as recombination, in which the initial state has 3 or more free particles in the continuum. Of particular interest are ultracold species with large two-body scattering lengths, for which universal behavior has been seen experimentally [4] that goes beyond the ordinary universality associated with the Efimov effect. The so-called three-body parameter, now understood to be universal for systems having van der Waals interactions, is readily interpreted using this theoretical framework, and predictions are made concerning A$+$A$+$B collisions as well as the homonuclear case A$+$A$+$A. Various aspects of the work presented have been carried out in collaboration with Jia Wang, Yujun Wang, Jose D'Incao, Javier von Stecher, and Brett Esry. \\[4pt] [1] J. Wang et al., Phys. Rev. Lett. \textbf{108}, 263001 (2012)\\[0pt] [2] J. Wang et al., Phys. Rev. A \textbf{84}, 052721 (2011)\\[0pt] [3] Y. Wang et al. arXiv:1207.6439 (2012).\\[0pt] [4] M. Berninger et al., Phys. Rev. Lett. \textbf{107}, 120401 (2011).   

A continued fraction approach for calculating Auger electron sprectra

In 'core valence valence' Auger spectroscopy (AES), X-ray absorption leads to the appearance of a core hole, which then decays into two valence holes and an Auger electron. The Auger electron carries information about the spectrum of the two additional holes thus introduced in the system. This is straightforward to compute if the two holes move in an otherwise full band, but accurate results for partially filled bands are still missing. Here we present a novel approach to calculating few-body lattice Green's functions that allows us to obtain the AES spectrum for systems with both filled and open bands, such as CuO and NiO. For full bands, comparison against exact results allows us to propose efficient variational schemes, which can then be used to study partially filled bands.   

Progress in calculating the PES of H$_3^+$

The most accurate electronic structure calculations are performed using wave-function expansions in terms of basis functions explicitly dependent on the interelectron distances. In our recent work we use such basis functions to calculate a highly accurate potential energy surface (PES) for the H$_3^+$ ion. The functions are explicitly correlated Gaussians which include inter-electron distances in the exponent. Key to obtaining the high accuracy in the calculations has been the use of the analytical energy gradient determined with respect to the Gaussian exponential parameters in the minimization of the Rayleigh-Ritz variational energy functional. The effective elimination of linear dependencies between the basis functions, as well as the automatic adjustment of the positions of the Gaussian centers to the changing molecular geometry of the system, are key to the success of the computational procedure. After adiabatic and relativistic corrections are added to the PES and with an effective accounting of the non-adiabatic effects in the calculation of the rotational/vibrational states, the experimental H$_3^+$ rovibrational spectrum is reproduced at the 0.1~cm$^{-1}$ accuracy level up to 16,600~cm$^{-1}$ above the ground state.   

Variational methods with all-particle explicitly correlated Gaussians

Accurate treatment of electron correlation in quantum systems of various nature remains an important challenge for modern theoretical and computational approaches. The variational method in conjunction with explicitly correlated Gaussian (ECG) basis sets is one of the most capable, accurate, and conceptually simple methods for calculating the ground, excited, and even scattering state properties of small quantum systems. I will review the basic theoretical foundations, recent advances, and the applications of the ECG method to Coulomb systems such as atoms, molecules, and systems containing positrons. I will also discuss some of the most important challenges that need to be overcome in order to extend the current range of applicability of the method.   

Hylleraas coordinates in few-body atomic and molecular systems

In this talk, we will present recent progress on the calculations of few-body Coulombic systems, such as atomic lithium and hydrogen molecular ions, using variational method in Hylleraas coordinates, including relativistic and quantum electrodynamic corrections. We will also discuss the applications of these calculations in the determination of nuclear charge radii and the proton-electron mass ratio.   

Development of explicitly correlated congruent transformed Hamiltonian

The central idea of the explicitly correlated congruent transformed Hamiltonian (CTH) method is the treatment of the Coulomb singularity in the Hamiltonian by performing congruent transformation using an explicitly correlated wave function. However, unlike the transcorrelated methods, the CTH is Hermitian and amenable to standard variational methods. The variational solution of the CTH was obtained using FCI and the comparison between the transformed and untransformed calculation will be discussed. We found that the CTH dramatically improves the convergence of the FCI expansion. The CTH can also be represented in the occupation number (ON) space, however this representation is approximate due to the finite size of the underlying basis. Analogous to the diagrammatic summation in MBPT, we have developed partial infinite order summation (PIOS) for improving the CTH calculation in ON space and analysis of the real space, ON and ON-PIOS calculation of CTH will be discussed. The CTH has been applied to a series of 10 electron systems and comparison of the results with other methods will be presented. Preliminary results on the excited state of water will be compared with R12-MP2 and MRCI methods. The size-consistency of the CTH method was numerically analyzed and will be discussed.   

Stability of high and low spin states

Octahedral CoL$_6$ complexes exhibit high or low spin states, depending on ligand L. We present an explicitly correlated first principles calculation of CoL$_6$ with five different ligands, and show that the total energy difference $\Delta E$ between the high and low spin states is variationally determined in an intricate interplay of the interelectron repulsion $V_{ee}$, internuclear repulsion $V_{nn}$, and electronuclear attraction $V_{ne}$. This is in stark contrast to ``ligand field theory'' [1,2], where $\Delta E$ is approximated as $\Delta E \approx \Delta V_{ee}$ in a first order perturbation theory. Moreover, we show that $\Delta V_{ee}$ exhibits the opposite trend to $\Delta E$ and is three or four orders of magnitude greater than $\Delta E$, which demonstrates the failure of ligand field theory both quantitatively and qualitatively. Correctly, the crossover of high and low spin states is a consequence of different Co--L bondings, ionic or covalent, which is found by an accurate treatment of Coulomb correlation between ligand $p$ and cobalt $d$ electrons in the present calculation. [1] J. H. Van Vleck, J. Chem Phys {\bf 3}, 807 (1935). [2] Y. Tanabe and S. Sugano, J. Phys. Soc. Jpn. {\bf 9}, 766 (1954).   

Computation of Low-Energy Positronium-Hydrogen Collisions using the Kohn Variational Method

The Kohn variational method is an established method that can provide benchmark calculations for quantum few-body systems. We consider the four-body Coulomb process of positronium-hydrogen (Ps-H) scattering. We improve upon the numerics of prior accurate S- and P-wave Kohn variational calculations of Ps-H elastic scattering [1,2]. For instance, we use a procedure that removes Hylleraas-type terms that lead to linear dependence [3]. In addition to using the Kohn and inverse Kohn variational methods as previously used, we use the generalized and complex Kohn variational methods [4]. We are extending the calculations of Ps-H to include the D-wave.\\[4pt] [1] P. Van Reeth and J. W. Humberston, J. Phys. B \textbf{36}, 1923 (2003).\\[0pt] [2] P. Van Reeth and J. W. Humberston, Nucl. Instrum. Methods B \textbf{221}, 140 (2004).\\[0pt] [3] A. Todd, Ph.D. thesis, The University of Nottingham, (2007), \emph{unpublished}.\\[0pt] [4] J.N. Cooper, M. Plummer, and E.A.G. Armour, J. Phys. A \textbf{43}, 175302 (2010).   

Influence of Angular and Spin-dependent Terms on Variational Energies of Lithium

Improved nonrelativistic energy bounds for the low-lying states of lithium are presented using the variational method in Hylleraas coordinates [1]. For example, the nonrelativistic energies for the infinite nuclear mass case are $-7.478\,060\,323\,910\,147(1)$ a.u. for $1s^22s\;^2{\rm S}$, $-7.354\,098\,421\,444\,37(1)$ a.u. for $1s^23s\;^2{\rm S}$, $-7.318\,530\,845\,998\,91(1)$ a.u. for $1s^24s\;^2{\rm S}$, $-7.410\,156\,532\,652\,4(1)$ a.u. for $1s^22p\;^2{\rm P}$, and $-7.335\,523\,543\,524\,688(3)$ a.u. for $1s^23d\;^2{\rm D}$. These results represent the most accurate nonrelativistic energies in the literature. The completeness of the angular momentum and spin configurations is investigated and examples presented for the 2P and 3D states to demonstrate the effect of different coupling schemes. In particular, the so-called second spin function (i.e.\ coupled to form an intermediate triplet state) is shown to have no effect on the final converged results, even for the expectation values of spin-dependent operators. This resolves a long-standing controversy concerning the completeness of the spin-coupling terms.\\ \mbox{}[1] L.M. Wang, Z.-C. Yan, H.X. Qiao, and G.W.F. Drake, Phys.\ Rev.\ A {\bf 85}, 052513 (2012).   

Multi-determinant electron-nuclear quantum Monte Carlo method for ground state solution of molecular Hamiltonian

The focus of this work is to obtain the ground state energy of the non-relativistic spin-independent molecular Hamiltonian without making the Born-Oppenheimer (BO) approximation. In addition to avoiding the BO approximation, this approach avoids imposing separable-rotor and harmonic oscillator approximations. The ground state solution is obtained variationally using multi-determinant variational Monte Carlo method where all nuclei and electrons in the molecule are treated quantum mechanically. The multi-determinant VMC provides the right framework for including explicit correlation in a multi-determinant expansion. This talk will discuss the construction of the basis functions and optimization of the variational coefficient. The electron-nuclear VMC method will be applied to H$_2$, He$_2$ and H$_2$O and comparison of the VMC results with other methods will be presented. The results from these calculations will provide the necessary benchmark values that are needed in development of other multicomponent method such as electron-nuclear DFT and electron-nuclear FCIQMC.   

Low energy model estimation from detailed quantum Monte Carlo calculations for transition metal systems

Systems of strongly correlated electrons have incredible potential for new devices and new quantum states. However, it is very challenging to a priori predict the quantum state of a system of correlated electrons. Detailed calculations using quantum Monte Carlo methods on the first principles Hamiltonian have in recent years shown to be quite reliable for some example transition metal oxide systems, such as FeO, ZnO, among others. These calculations, although they are accurate, have not provided much information in terms of the correct approximate low-energy model that should describe the systems in question. In this talk, I'll summarize the results of matching the two-body correlations from first principles quantum Monte Carlo on transition metal systems to models and discuss the implications for the commonly used models.