Session V20: Explicitly Correlated Methods and Quantum Few-Body Systems

Possibilities of the free-complement methodology for solving the Schrödinger equation of atoms and molecules

Chemistry is a science of complex subjects that occupy this universe and biological world and that are composed of atoms and molecules. Its essence is diversity. However, surprisingly, whole of this science is governed by simple quantum principles like the Schrödinger and the Dirac equations. Therefore, if we can find a useful general method of solving these quantum principles under the fermionic and/or bosonic constraints accurately in a reasonable speed, we can replace somewhat empirical methodologies of this science with purely quantum theoretical and computational logics. This is the purpose of our series of studies -- called “exact theory” in our laboratory. Some of our documents are cited below [1-8]. The key idea was expressed as the free complement (FC) theory (originally called ICI theory [3]) that was introduced to solve the Schrödinger and Dirac equations analytically. For extending this methodology to larger systems, order N methodologies are essential, but actually the antisymmetry constraints for electronic wave functions become big constraints. Recently [8], we have shown that the antisymmetry rule or `dogma' can be very much relaxed when our subjects are large molecular systems. In this talk, I want to present our recent progress in our FC methodology. The purpose is to construct “predictive quantum chemistry” that is useful in chemical and physical researches and developments in institutes and industries. [1] H. Nakatsuji, Acc. Chem. Res. 45, 1480 (2012). [2] H. Nakatsuji and H. Nakashima, TSUBAME e-Science J. 11, 8, 24 (2014). [3] H. Nakatsuji, Phys. Rev. Lett. 93, 030403 (2004). [4] H. Nakatsuji and H. Nakashima, Phys. Rev. Lett. 95, 050407 (2005). [5] H. Nakatsuji, et al, Phys. Rev. Lett. 99, 240402 (2007). [6] H. Nakatsuji and H. Nakashima, J. Chem. Phys.142, 084117 (2015). [7] H. Nakashima and H. Nakatsuji, J. Chem. Phys. 139, 044112 (2013). [8] H. Nakatsuji and H. Nakashima, J. Chem. Phys. 142, 194101 (2015).   

Natural generalization of Slater determinants to more than one dimension

The calculation of realistic $N$-body wave functions for identical fermions is still an open problem in physics, chemistry, and materials science, even for $N$ as small as two. Here a fundamental algebraic structure of many-body Hilbert space is described, enabling theoretically well-founded systematic investigation of wave-function space. The structure allows an arbitrary many-fermion wave function to be written in terms of a finite number of antisymmetric functions called shapes, which cannot be constructed by combining one-dimensional wave functions. Shapes naturally generalize the single-Slater-determinant form for the ground state to more than one dimension. Their number is exactly $N!^{d-1}$ in $d$ dimensions. A general algorithm is given to list them all in terms of standard Slater determinants. Conversely, excitations which can be induced from the one-dimensional case are bosonised into a system of distinguishable bosons, called Euler bosons, much like the electromagnetic field is quantized in terms of photons distinguishable by their wave numbers. Their wave functions are given explicitly in terms of elementary symmetric functions, reflecting the fact that the fermion sign problem is trivial in one dimension. The shapes are all possible vacua for the Euler bosons.   

Configuration space method for calculating binding energies of exciton complexes in quasi-1D/2D semiconductors.

A configuration space method, pioneered by Landau and Herring in studies of molecular binding and magnetism[1], is developed to obtain universal asymptotic relations for lowest energy exciton complexes (trion, biexciton) in confined semiconductor nanostructures such as nanowires and nanotubes[2], as well as coupled quantum wells. Trions are shown to be more stable (have greater binding energy) than biexcitons in strongly confined quasi-1D structures with small reduced electron-hole masses. Biexcitons are more stable in less confined quasi-1D structures with large reduced electron-hole masses. The theory predicts a crossover behavior, whereby trions become less stable than biexcitons as the transverse size of the quasi-1D nanostructure increases, which might be observed on semiconducting carbon nanotubes of increasing diameters. This method is also efficient in calculating binding energies for trion-type electron-hole complexes formed by indirect excitons in double coupled quantum wells, quasi-2D nanostructures that show new interesting electroabsorption/refraction phenomena. -- [1]Landau {\&} Lifshitz, Quantum Mechanics; C.Herring, RMP 34, 631 (1962). [2]I.V.Bondarev, PRB 90, 245430 (2014); PRB 83, 153409 (2011).   

Tests for Wavelets as a Basis Set

A wavelet transformation is a special type of filter usually reserved for image processing and other applications. We develop metrics to evaluate wavelets for general problems on test one-dimensional systems. The goal is to eventually use a wavelet basis in electronic structure calculations. We compare a variety of orthogonal wavelets such as coiflets, symlets, and daubechies wavelets. We also evaluate a new type of orthogonal wavelet with dilation factor three which is both symmetric and compact in real space.   

How competitive are expansions in orbital products with explicitly correlated expansions

Helium dimer potential is of great importance for metrology since several future measurement standards will be based on helium gas. Such potential can be used to predict all thermodynamic properties of diffuse helium gas (nonadditive three-body potential is needed for higher densities). The accuracy required by these standards is so high, that in the past work of our group we had to include nonadiabatic, relativistic, and quantum electrodynamics effects. The current state is that the largest contribution to the uncertainty of the helium dimer potential is due to the Born-Oppenheimer (BO) part of this potential. This uncertainty was reduced and became comparable to other uncertainties in the new calculations that will be presented. These calculations used explicitly correlated Gaussian (ECG) basis sets and represent nearly exact solutions of the Schr\"odinger equation in the BO approximation. Similar calculations were also performed in orbital basis sets using a multilevel approach up to the full configuration interactions level. Largest existing basis sets were used at each level so that our calculations represent the best results that can currently be obtained using orbitals. These results will be critically compared with those obtained using ECG bases.   

New Types of Explicitly Correlated Gaussian Functions for Non-Born-Oppenheimer Molecular? Calculations.

In this work we explore the possibility of using a new functional form of Explicitly Correlated Gaussian-type functions (ECGs) for performing non-Born-Oppenheimer calculations of diatomic molecular systems. Namely we focus our attention on ECGs with pre-exponential factors in the form of sin/cos functions of the square of the internuclear distance (sin/cos-ECGs). These ECGs can be generated as linear combinations of ECGs with complex exponential parameters (complex-ECGs) The complex-ECGs were previously used to calculate energy levels of He atom [1] and with the sin/cos-ECGs the vibrational energy levels for the H$_2$ molecule described by an effective Morse potential were calculated [2]. The focus of this study is ab-initio description of a real diatomic molecule, namely HD$^+$, within the framework where the BO approximation is not assumed using these basis sets. The aim is to compare their accuracy and efficiency with ECGs with pre-exponential factors in the form of even powers of the internuclear distance and to assess their potential usefulness in non-BO calculations of molecules with more than two nuclei. [1] S. Bubin and L. Adamowicz, J. Chem. Phys. 124, 224317 (2006) [2] M. Formanek, K.L. Sharkey, N. Kirnosov and L. Adamowicz, J. Chem. Phys. 141, 154103 (2014)   

Complex explicitly correlated Gaussians for non-Born--Oppenheimer calculations of small molecules

Non-Born--Oppenheimer calculations of molecular systems, where all particles are properly treated on an equal footing, represents a big challenge for the theory. Due to the huge difference in the masses of the electrons and nuclei the latter move more slowly and their correlation functions have distinct localization around the equilibrium internuclear separations. This feature is hard to implement in explicitly correlated variational approaches with Gaussian type basis functions while maintaining an analytic integrability of all necessary matrix elements. In this work we demonstrate that the difficulties can be overcome by using complex Gaussians. In our benchmark calculations on HD$^+$ molecular ion we have achieved excellent performance of this simple complex basis set that is on par or better than what was seen in previous Non-BO calculations of small diatomic molecules.   

How Large are Nonadiabatic Effects in Atomic and Diatomic Systems?

We have developed a fixed-node quantum Monte Carlo method to simulate atoms and molecules without the Born-Oppenheimer approximation with sub milli-Hartree accuracy [1]. For this purpose, we construct trial wave functions with nodes that depend on both the electronic and ionic positions. We report ground-state energies and the ionization energies for the first-row atoms and atomization energies for the first-row hydrides. The latter show effects of the nonadiabatic coupling between electrons and nuclei. We discuss how the method scales to larger systems. [1] Y. Yang, I. Kyl\"{a}np\"{a}\"{a}, N. M. Tubman, J. T. Krogel, S. Hammes-Schiffer, D. M. Ceperley, J. Chem. Phys. \textbf{143}(12), 2015.   

The structure of the second-order non-Born-Oppenheimer density matriz D2:

Properties of the non-Born-Oppenheimer 2-matrix are examined. Using a coordinate system formed by internal translationally invariant plus the total center-of-mass coordinates it is shown that regardless of the point of reference selected, the operator for the reduced second order density matrix, 2-RDM, solely depends upon the translationally invariant internal coordinates. We apply this result to examine the nature of the 2-RDM extracted from the exact analytical solutions for model non-Born-Oppenheimer four-particle systems of the Coulomb-Hooke and Moshinsky types. We obtain for both these models explicit closed-form analytic expressions for the electron and nuclear 2-RDM. An explicit expression is also obtained for the electron-nuclear 2-RDM in the Moshinsky case, which shows coupling between the electron and nuclear coordinates.   

Extracting g tensor values from experimental data with Markov Chain Monte Carlo methods

Quantum Dot Molecules(QDMs) have emerged as a new platform for optoelectronic and spintronic devices.QDMs consist of multiple Quantum Dots (QDs) arranged in close proximity such that interactions between them can tailor their optical and spin properties.These properties can be tuned during growth and in-situ by applying electric fields that vary the coupling between QDs,which controls the formation of delocalized molecular-like states.Engineering the formation of molecular states in QDMS can be used to achieve new functionalities unavailable with individual QDs. Using molecular engineering approaches to tailor QDMs require precise knowledge of parameters such as binding energies of charge complexes,magnitude of many body interactions or components of the g tensor.Precise values of these parameters are difficult to extract from either experimental measurements or theoretical calculations.We develop and demonstrate a Markov Chain Monte Carlo method for extracting elements of the g tensor for a single hole confined in a QDM from photoluminescence data obtained as a function of electric and magnetic fields.This method can be applied to extract precise quantitative values of other physical parameters from sparse experimental data on a variety of systems.   

QUANTUM MONTE CARLO METHOD FOR HEAVY ATOMIC AND MOLECULAR SYSTEMS WITH SPIN-ORBIT INTERACTIONS

We present a new quantum Monte Carlo (QMC) method that can treat spin-orbit and other types of spin-depentent interactions explicitly. It is based on generalization of the fixed-phase and projection of the nonlocal operators with spinor trial wave functions. For testing the method we calculate several atomic and molecular systems such as Bi, W, Pb, PbH and PbO, some of them with both large- and small-core pseudopotentials. We validate the quality of the results against other correlated methods such as configuration interaction in two-component formalism. We find excellent agreement with extrapolated values for the total energies and we are able to reliably reproduce experimental values of excitation energies, electron affinity and molecular binding. We show that in order to obtain the agreement with experimental values the explicit inclusion of the spin-orbit interactions is crucial.