Many-electron Hamiltonian Research Articles

Unified construction of relativistic Hamiltonians.

It is shown that the four-component (4C), quasi-four-component (Q4C), and exact two-component (X2C) relativistic Hartree-Fock equations can be implemented in a unified manner by making use of the atomic nature of the small components of molecular 4-spinors. A model density matrix approximation can first be invoked for the small-component charge/current density functions, which gives rise to a static, pre-molecular mean field to be combined with the one-electron term. As a result, only the nonrelativistic-like two-electron term of the 4C/Q4C/X2C Fock matrix needs to be updated during the iterations. A "one-center small-component" approximation can then be invoked in the evaluation of relativistic integrals, that is, all atom-centered small-component basis functions are regarded as extremely localized near the position of the atom to which they belong such that they have vanishing overlaps with all small- or large-component functions centered at other nuclei. Under these approximations, the 4C, Q4C, and X2C mean-field and many-electron Hamiltonians share precisely the same structure and accuracy. Beyond these is the effective quantum electrodynamics Hamiltonian that can be constructed in the same way. Such approximations lead to errors that are orders of magnitude smaller than other sources of errors (e.g., truncation errors in the one- and many-particle bases as well as uncertainties of experimental measurements) and are, hence, safe to use for whatever purposes. The quaternion forms of the 4C, Q4C, and X2C equations are also presented in the most general way, based on which the corresponding Kramers-restricted open-shell variants are formulated for "high-spin" open-shell systems.

Theoretical molecular spectroscopy of actinide compounds: the ThO molecule

The tiny-core generalised (Gatchina) relativistic pseudopotential (GRPP) model provides an accurate approximation for many-electron Hamiltonians of molecules containing heavy atoms, ensuring a proper description of the effects of non-Coulombian electron-electron interactions, electronic self-energy and vacuum polarisation. Combining this model with electron correlation treatment in the frames of the intermediate Hamiltonian Fock space coupled cluster theory employing incomplete main model spaces, one obtains a reliable and economical tool for excited state modelling. The performance of this method is assessed in applications to ab initio modelling of excited electronic states of the thorium monoxide molecule with term energies below 20,000 cm. Radiative lifetimes of excited states are estimated using truncated expansions of effective and metric operators in powers of cluster amplitudes.

The Valence-Bond (VB) Model and Its Intimate Relationship to the Symmetric or Permutation Group.

VB and molecular orbital (MO) models are normally distinguished by the fact the first looks at molecules as a collection of atoms held together by chemical bonds while the latter adopts the view that each molecule should be regarded as an independent entity built up of electrons and nuclei and characterized by its molecular structure. Nevertheless, there is a much more fundamental difference between these two models which is only revealed when the symmetries of the many-electron Hamiltonian are fully taken into account: while the VB and MO wave functions exhibit the point-group symmetry, whenever present in the many-electron Hamiltonian, only VB wave functions exhibit the permutation symmetry, which is always present in the many-electron Hamiltonian. Practically all the conflicts among the practitioners of the two models can be traced down to the lack of permutation symmetry in the MO wave functions. Moreover, when examined from the permutation group perspective, it becomes clear that the concepts introduced by Pauling to deal with molecules can be equally applied to the study of the atomic structure. In other words, as strange as it may sound, VB can be extended to the study of atoms and, therefore, is a much more general model than MO.

Universal formulation of second-order generalized Møller–Plesset perturbation theory for a spin-dependent two-component relativistic many-electron Hamiltonian

We have derived and implemented a universal formulation of the second-order generalized Møller–Plesset perturbation theory (GMP2) for spin-dependent (SD) two-component relativistic many-electron Hamiltonians, such as the infinite-order Douglas–Kroll–Hess Hamiltonian for many-electron systems, which is denoted as IODKH/IODKH. Numerical assessments for He- and Ne-like atoms and 16 diatomic molecules show that the MP2 correlation energies with IODKH/IODKH agree well with those calculated with the four-component Dirac–Coulomb (DC) Hamiltonian, indicating a systematic improvement on the inclusion of relativistic two-electron terms. The present MP2 scheme for IODKH/IODKH is demonstrated to be computationally more efficient than that for DC.

Isomorphism between the multi-state Hamiltonian and the second-quantized many-electron Hamiltonian with only 1-electron interactions.

We introduce the isomorphism between an multi-state Hamiltonian and the second-quantized many-electron Hamiltonian (with only 1-electron interactions). This suggests that all methods developed for the former can be employed for the latter, and vice versa. The resonant level (Landauer) model for nonequilibrium quantum transport is used as a proof-of-concept example. Such as the classical mapping models for the multi-state Hamiltonian proposed in our previous work [J. Liu, J. Chem. Phys. 145, 204105 (2016)] lead to exact results for this model problem. We further demonstrate how these methods can also be applied to the second-quantized many-electron Hamiltonian even when 2-electron interactions are included.

New quantum number for the many-electron Dirac-Coulomb Hamiltonian

By breaking the spin symmetry in the relativistic domain, a powerful tool in physical sciences was lost. In this work, we examine an alternative of spin symmetry for systems described by the many-electron Dirac-Coulomb Hamiltonian. We show that the square of many-electron operator $\mathcal{K}_+$, defined as a sum of individual single-electron time-reversal (TR) operators, is a linear Hermitian operator which commutes with the Dirac-Coulomb Hamiltonian in a finite Fock subspace. In contrast to the square of a standard unitary many-electron TR operator $\mathcal{K}$, the $\mathcal{K}^2_+$ has a rich eigenspectrum having potential to substitute spin symmetry in the relativistic domain. We demonstrate that $\mathcal{K}_+$ is connected to $\mathcal{K}$ through an exponential mapping, in the same way as spin operators are mapped to the spin rotational group. Consequently, we call $\mathcal{K}_+$ the generator of the many-electron TR symmetry. By diagonalizing the operator $\mathcal{K}^2_+$ in the basis of Kramers-restricted Slater determinants, we introduce the relativistic variant of configuration state functions (CSF), denoted as Kramers CSF. A new quantum number associated with $\mathcal{K}^2_+$ has potential to be used in many areas, for instance, (a) to design effective spin Hamiltonians for electron spin resonance spectroscopy of heavy-element containing systems; (b) to increase efficiency of methods for the solution of many-electron problems in relativistic computational chemistry and physics; (c) to define Kramers contamination in unrestricted density functional and Hartree--Fock theory as a relativistic analog of the spin contamination in the nonrelativistic domain.

Creation of 4fn-4fn-15d1 Transition Energy Diagram of Trivalent Lanthanide Ions in Fluorides Based on First-Principles Calculations

The theoretical prediction of the 4fn-4fn-15d1 transition energies of lanthanides in crystals is an important issue in the development of novel luminescent materials such as solid-state lasers or phosphors. In our previous work, the first-principles calculations of the entire multiplet structures of lanthanide (Ln) ions in LiYF4 (YLF) corresponding to 4fn and 4fn-15d1 configurations were performed using the relativistic discrete variational multi-electron (DVME) method [1]. However, the lowest 4fn-4fn-15d1transition energy was overestimated for light lanthanides while it was underestimated for heavy lanthanides. In this work, in order to improve the accuracy of the theoretical transition energy, a correction based on the molecular orbital (MO) calculation in the Slater’s transition state was introduced. The LnF8 clusters were constructed based on the crystal structures of YLF and CaF2. The fully-relativistic MO calculations were performed using the relativistic DV-Xα code. The multiplet calculations were performed using the relativistic DVME code. The Slater determinants corresponding to 4fn and 4fn-15d1 configurations were constructed from the four-component relativistic MOs and used as the basis functions to diagonalize the relativistic many-electron Hamiltonian. In order to correct the errors in the transition energy, MO calculations in the Slater’s transition state where a half electron was excited from the Ln 4f level to the Ln 5d level were also performed. A constant correction term was added to the diagonal matrix element of the many-electron Hamiltonian for 4fn-15d1 configuration so that the 4fn -4fn-15d1transition energy is consistent with the 4f-5d transition energy obtained by the MO calculation in the Slater’s transition state. The diagrams representing the lowest 4fn-4fn-15d1transition energies of trivalent lanthanide ions in YLF calculated with and without the above correction are shown in Fig. 1. The overestimation in the light lanthanides and the underestimation in the heavy lanthanides are significantly reduced by the correction and the theoretical transition energies approached to the experimental values in the literature [2-5].

Effective Hamiltonians for correlated narrow energy band systems and magnetic insulators: Role of spin-orbit interactions in metal-insulator transitions and magnetic phase transitions.

Using a second-quantized many-electron Hamiltonian, we obtain (a) an effective Hamiltonian suitable for materials whose electronic properties are governed by a set of strongly correlated bands in a narrow energy range and (b) an effective spin-only Hamiltonian for magnetic materials. The present Hamiltonians faithfully include phonon and spin-related interactions as well as the external fields to study the electromagnetic response properties of complex materials and they, in appropriate limits, reduce to the model Hamiltonians due to Hubbard and Heisenberg. With the Hamiltonian for narrow-band strongly correlated materials, we show that the spin-orbit interaction provides a mechanism for metal-insulator transition, which is distinct from the Mott-Hubbard (driven by the electron correlation) and the Anderson mechanism (driven by the disorder). Next, with the spin-only Hamiltonian, we demonstrate the spin-orbit interaction to be a reason for the existence of antiferromagnetic phase in materials which are characterized by a positive isotropic spin-exchange energy. This is distinct from the Néel-VanVleck-Anderson paradigm which posits a negative spin-exchange for the existence of antiferromagnetism. We also find that the Néel temperature increases as the absolute value of the spin-orbit coupling increases.

Big picture of relativistic molecular quantum mechanics

AbstractAny quantum mechanical calculation on electronic structure ought to choose first an appropriate Hamiltonian H and then an Ansatz for parameterizing the wave function Ψ, from which the desired energy/property E(λ) can finally be calculated. Therefore, the very first question is: what is the most accurate many-electron Hamiltonian H? It is shown that such a Hamiltonian i.e. effective quantum electrodynamics (eQED) Hamiltonian, can be obtained naturally by incorporating properly the charge conjugation symmetry when normal ordering the second quantized fermion operators. Taking this eQED Hamiltonian as the basis, various approximate relativistic many-electron Hamiltonians can be obtained based entirely on physical arguments. All these Hamiltonians together form a complete and continuous ‘Hamiltonian ladder’, from which one can pick up the right one according to the target physics and accuracy. As for the many-electron wave function Ψ, the most intriguing questions are as follows. (i) How to do relativistic explicit correlation? (ii) How to handle strong correlation? Both general principles and practical strategies are outlined here to handle these issues. Among the electronic properties E(λ) that sample the electronic wave function nearby the nuclear region, nuclear magnetic resonance (NMR) shielding and nuclear spin-rotation (NSR) coupling constant are especially challenging: they require body-fixed molecular Hamiltonians that treat both the electrons and nuclei as relativistic quantum particles. Nevertheless, they have been formulated rigorously. In particular, a very robust ‘relativistic mapping’ between the two properties has been established, which can translate experimentally measured NSR coupling constants to very accurate absolute NMR shielding scales that otherwise cannot be obtained experimentally. Since the most general and fundamental issues pertinent to all the three components of the quantum mechanical equation HΨ = EΨ (i.e. Hamiltonian H, wave function Ψ, and energy/property E(λ)) have fully been understood, the big picture of relativistic molecular quantum mechanics can now be regarded as established.

Efficient and accurate treatment of electron correlations with Correlation Matrix Renormalization theory.

We present an efficient method for calculating the electronic structure and total energy of strongly correlated electron systems. The method extends the traditional Gutzwiller approximation for one-particle operators to the evaluation of the expectation values of two particle operators in the many-electron Hamiltonian. The method is free of adjustable Coulomb parameters, and has no double counting issues in the calculation of total energy, and has the correct atomic limit. We demonstrate that the method describes well the bonding and dissociation behaviors of the hydrogen and nitrogen clusters, as well as the ammonia composed of hydrogen and nitrogen atoms. We also show that the method can satisfactorily tackle great challenging problems faced by the density functional theory recently discussed in the literature. The computational workload of our method is similar to the Hartree-Fock approach while the results are comparable to high-level quantum chemistry calculations.

Distributed Atomic Polarizabilities of Amino Acids and their Hydrogen-Bonded Aggregates.

With the purpose of rational design of optical materials, distributed atomic polarizabilities of amino acid molecules and their hydrogen-bonded aggregates are calculated in order to identify the most efficient functional groups, able to buildup larger electric susceptibilities in crystals. Moreover, we carefully analyze how the atomic polarizabilities depend on the one-electron basis set or the many-electron Hamiltonian, including both wave function and density functional theory methods. This is useful for selecting the level of theory that best combines high accuracy and low computational costs, very important in particular when using the cluster method to estimate susceptibilities of molecular-based materials.

Theoretical characterization of the collective resonance states underlying the xenon giant dipole resonance

We present a detailed theoretical characterization of the two fundamental collective resonances underlying the xenon giant dipole resonance (GDR). This is achieved consistently by two complementary methods implemented within the framework of the configuration-interaction singles (CIS) theory. The first method accesses the resonance states by diagonalizing the many-electron Hamiltonian using the smooth exterior complex scaling technique. The second method involves a new application of the Gabor analysis to wave-packet dynamics. We identify one resonance at an excitation energy of 74 eV with a lifetime of 27 as, and the second at 107 eV with a lifetime of 11 as. Our work provides a deeper understanding of the nature of the resonances associated with the GDR: a group of close-lying intrachannel resonances splits into two far-separated resonances through interchannel couplings involving the 4d electrons. The CIS approach allows a transparent interpretation of the two resonances as new collective modes. Due to the strong entanglement between the excited electron and the ionic core, the resonance wave functions are not dominated by any single particle-hole state. This gives rise to plasma-like collective oscillations of the 4d shell as a whole.

Infinite-order diagrammatic summation approach to the explicitly correlated congruent transformed Hamiltonian

A resolution of identity approach to explicitly correlated congruent transformed Hamiltonian (CTH) is presented. One of the principle challenges associated with the congruent transformation of the many-electron Hamiltonian is the generation of three, four, five, and six particle operators. Successful application of the congruent transformation requires efficient implementation of the many-particle operators. In this work, we present the resolution of identity congruent transformed Hamiltonian (RI-CTH) method to handle many-particle operators. The resolution of identity was used to project the explicitly correlated operator in a N-particle finite basis to avoid explicit computation of the many-particle operators. Single-particle states were obtained by performing Hartee-Fock calculations, which were then used for construction of many-particle states. The limitation of the finite nature of the resolution of identity was addressed by developing partial infinite order (PIOS) diagrammatic summation technique. In the PIOS method, the matrix elements of the projected congruent transformed Hamiltonian was expressed in terms of diagrammatic notation and a subset of diagrams were summed up to infinite order. The RI-CTH and RI-CTH-PIOS methods were applied to isoelectronic series of 10-electron systems (Ne,HF,H2O,NH3,CH4) and results were compared with CISD and CCSD(T) calculations. One of the key results from this work is that for identical basis set, the RI-CTH-PIOS energies are lower than CISD and CCSD(T) values.

A Cartesian quasi-classical model to nonequilibrium quantum transport: The Anderson impurity model

We apply the recently proposed quasi-classical approach for a second quantized many-electron Hamiltonian in Cartesian coordinates [B. Li and W. H. Miller, J. Chem. Phys. 137, 154107 (2012)] to correlated nonequilibrium quantum transport. The approach provides accurate results for the resonant level model for a wide range of temperatures, bias, and gate voltages, correcting the flaws of our recently proposed mapping using action-angle variables. When electron-electron interactions are included, a Gaussian function scheme is required to map the two-electron integrals, leading to quantitative results for the Anderson impurity model. In particular, we show that the current mapping is capable of capturing quantitatively the Coulomb blockade effect and the temperature dependence of the current below and above the blockade.

A Cartesian classical second-quantized many-electron Hamiltonian, for use with the semiclassical initial value representation

A new classical model for the general second-quantized many-electron Hamiltonian in Cartesian coordinates and momenta is presented; this makes semiclassical (SC) calculations using an initial value representation (IVR) more useful than the classical Hamiltonian in action-angle variables given earlier by Miller and White [J. Chem. Phys. 84, 5059-5066 (1986)]. If only 1-electron terms are included in this Hamiltonian, the classical equations of motion for the Cartesian variables are linear, and the SC-IVR gives exact results for the propagator (and thus for transition probabilities, the energy spectrum, etc.), as confirmed by analytic proof and numerical calculations. Though this new Hamiltonian is not exact when 2-electron interactions are included, we observe good results for the SC-IVR transition probabilities for times that are not too long. Test calculations, for example, show that the SC-IVR is accurate for times long enough to obtain good result for the eigenvalue spectrum (i.e., the energy levels of the electronic system).

Quantum Mechanical Origins of the Iczkowski-Margrave Model of Chemical Potential.

Charge flow in materials at the atomistic level is controlled through chemical potential equalization among its constituents. Consequently employing this concept in a simulation requires some model of chemical potential. Current atomistic models of chemical potential, such as the Iczkowski-Margrave (IM) model, are built largely on heuristic arguments and depend linearly on the net charge of each constituent. To gain new insight into the IM model, a many-electron model Hamiltonian is constructed at the atomistic level that is commensurate with the IM model, as opposed to one designed at the one-electron level. For a three-state, two-fragment system, the essential electronegativity and the chemical hardness energies are recovered. However, the model Hamiltonian imparts new charge dependencies not found in the IM model. Decidedly nonlinear, transitional or hopping contributions in those new dependencies are shown to be critical to regulating charge flow. Other modifications to the IM model are illustrated with simple two- and three-fragment systems, involving as many as five states, that act as paradigms for general materials models. Including more than three states in the three-fragment example introduces local bonding refinements to the Mulliken electronegativity and chemical hardness.

Application of a semiclassical model for the second-quantized many-electron Hamiltonian to nonequilibrium quantum transport: The resonant level model

A semiclassical approach is developed for nonequilibrium quantum transport in molecular junctions. Following the early work of Miller and White [J. Chem. Phys. 84, 5059 (1986)], the many-electron Hamiltonian in second quantization is mapped onto a classical model that preserves the fermionic character of electrons. The resulting classical electronic Hamiltonian allows for real-time molecular dynamics simulations of the many-body problem from an uncorrelated initial state to the steady state. Comparisons with exact results generated for the resonant level model reveal that a semiclassical treatment of transport provides a quantitative description of the dynamics at all relevant timescales for a wide range of bias and gate potentials, and for different temperatures. The approach opens a door to treating nontrivial quantum transport problems that remain far from the reach of fully quantum methodologies.

NWChem: A comprehensive and scalable open-source solution for large scale molecular simulations

The latest release of NWChem delivers an open-source computational chemistry package with extensive capabilities for large scale simulations of chemical and biological systems. Utilizing a common computational framework, diverse theoretical descriptions can be used to provide the best solution for a given scientific problem. Scalable parallel implementations and modular software design enable efficient utilization of current computational architectures. This paper provides an overview of NWChem focusing primarily on the core theoretical modules provided by the code and their parallel performance. Program summary Program title: NWChem Catalogue identifier: AEGI_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEGI_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Open Source Educational Community License No. of lines in distributed program, including test data, etc.: 11 709 543 No. of bytes in distributed program, including test data, etc.: 680 696 106 Distribution format: tar.gz Programming language: Fortran 77, C Computer: all Linux based workstations and parallel supercomputers, Windows and Apple machines Operating system: Linux, OS X, Windows Has the code been vectorised or parallelized?: Code is parallelized Classification: 2.1, 2.2, 3, 7.3, 7.7, 16.1, 16.2, 16.3, 16.10, 16.13 Nature of problem: Large-scale atomistic simulations of chemical and biological systems require efficient and reliable methods for ground and excited solutions of many-electron Hamiltonian, analysis of the potential energy surface, and dynamics. Solution method: Ground and excited solutions of many-electron Hamiltonian are obtained utilizing density-functional theory, many-body perturbation approach, and coupled cluster expansion. These solutions or a combination thereof with classical descriptions are then used to analyze potential energy surface and perform dynamical simulations. Additional comments: Full documentation is provided in the distribution file. This includes an INSTALL file giving details of how to build the package. A set of test runs is provided in the examples directory. The distribution file for this program is over 90 Mbytes and therefore is not delivered directly when download or Email is requested. Instead a html file giving details of how the program can be obtained is sent. Running time: Running time depends on the size of the chemical system, complexity of the method, number of cpu's and the computational task. It ranges from several seconds for serial DFT energy calculations on a few atoms to several hours for parallel coupled cluster energy calculations on tens of atoms or ab-initio molecular dynamics simulation on hundreds of atoms.

A numerical method for exact diagonalization of semiconductor quantum dot model

An approach to the exact diagonalization of many-electron Hamiltonian in semiconductor quantum dot (QD) structures is proposed. The QD model is based on 3D finite hard-wall confinement potential and nonparabolic effective-mass approximation (EMA) that render analytical basis functions such as Laguerre polynomials inaccessible for the numerical treatment of this kind of models. In this approach, the many-electron wave function is expanded in a basis of Slater determinants constructed from numerical wave functions of the single-electron Hamiltonian with the nonparabolic EMA which results in a cubic eigenvalue problem from a finite difference discretization. The nonlinear eigenvalue problem is solved by using the Jacobi–Davidson method. The Coulomb matrix elements in the many-electron Hamiltonian are obtained by solving Poisson's problems via GMRES. Numerical results reveal that a good convergence can be achieved by means of a few single-electron basis states.

Electronic Zero-Point Oscillations in the Strong-Interaction Limit of Density Functional Theory

The exchange-correlation energy in Kohn-Sham density functional theory can be expressed exactly in terms of the change in the expectation of the electron-electron repulsion operator when, in the many-electron Hamiltonian, this same operator is multiplied by a real parameter λ varying between 0 (Kohn-Sham system) and 1 (physical system). In this process, usually called adiabatic connection, the one-electron density is kept fixed by a suitable local one-body potential. The strong-interaction limit of density functional theory, defined as the limit λ→∞, turns out to be like the opposite noninteracting Kohn-Sham limit (λ→0) mathematically simpler than the physical (λ = 1) case and can be used to build an approximate interpolation formula between λ→0 and λ→∞ for the exchange-correlation energy. Here we extend the systematic treatment of the λ→∞ limit [Phys. Rev. A 2007, 75, 042511] to the next leading term, describing zero-point oscillations of strictly correlated electrons, with numerical examples for small spherical atoms. We also propose an improved approximate functional for the zero-point term and a revised interpolation formula for the exchange-correlation energy satisfying more exact constraints.