Unitary Transformation Scheme Research Articles

Development of Linear-Scaling Relativistic Quantum Chemistry Covering the Periodic Table

Abstract This Award Account focuses on the author’s studies on the theoretical developments of two-component (2c) relativistic quantum chemistry calculations for large systems with high efficiency and high accuracy, with a review of related studies as the background. The local unitary transformation scheme allows the linear-scaling computation cost to be applied to construct a 2c Hamiltonian, such as an infinite-order two-component version. The divide-and-conquer scheme can lead to linear-scaling computation costs to apply not only a Hartree-Fock (HF) method but also post-HF methods such as the second-order Møller-Plesset perturbation and couple cluster theory with singles and doubles for the 2c Hamiltonian in addition to a non-relativistic version. The frozen core potential scheme can naturally connect pseudopotential calculations with all-electron calculations. The accompanying coordinate expansion with a transfer recurrence relation scheme provides an efficient algorithm for the rapid evaluation of electron repulsion integrals for systems including heavy elements, the orbitals of which have long contractions and high angular momenta, such as f- and g-orbitals. Illustrative applications will help readers realize the advantages and usefulness of these schemes.

The effect of electron–phonon interaction on the thermoelectric properties of defect zigzag nanoribbons

Thermoelectric properties of graphene nanoribbons with periodic edge vacancies and antidot lattice are investigated. The electron-phonon interaction is taken into account in the framework of the Hubbard-Holstein model with the use of the Lang-Firsov unitary transformation scheme. The electron transmission function, the thermopower and the thermoelectric figure of merit are calculated. We have found that the electron-phonon interaction causes a decrease in the peak values of the thermoelectric figure of merit and the shift of the peak positions closer to the center of the bandgap. The effects are more pronounced for the secondary peaks that appear in the structures with periodic antidot.

An analytical solution for the graphene electronic spectrum in the presence of external fields and confinement potential

An algebraic solution of [Formula: see text]-dimensional Dirac equation describing the massless Dirac fermions in the presence of both magnetic field perpendicular to the graphene plane and in-plane homogeneous electric field together with a quadratic confinement potential is investigated. The associated energy eigenvalues are obtained within the framework of unitary transformation scheme. It is shown that the decreasing quadratic confinement potential strength leads to the enhancement of the Landau levels.

Phonon Dispersions of Graphene from Unzipping Carbon Nanotubes

We calculate the phonon spectra of chiral single-walled carbon nanotubes and graphene within a mass and spring model which includes up to third neighbour interactions together with a radial bond-bending interaction. Firstly, the classical Hamiltonian for lattice vibrations of chiral single-walled carbon nanotubes is derived and then it is quantized. The resultant Hamiltonian is diagonalized under a unitary transformation scheme to obtain phonon modes as a function of momentum vector. Using resolvent formalism we analytically obtain phonon dispersions of these structures. We show that our calculated results for chiral SWCNTs reproduce well the results found for achiral SWCNTs. Finally, we introduce an unzipping technique to obtain full analytical phonon dispersion curves of graphene using phonon dispersions of chiral SWCNTs. We compare our analytical results for the phonon dispersions of graphene to the available experimental data and show that they agree well with the experiment. Our analytical results not only provide a basis for understanding phonon dispersions of carbon nanotubes, but also illuminate the bare phonon structure of graphene.

Large‐scale two‐component relativistic quantum‐chemical theory: Combination of the infinite‐order douglas–kroll–hess method with the local unitary transformation scheme and the divide‐and‐conquer method

Large‐scale two‐component (2c) relativistic quantum‐chemical (RQC) theory is reviewed. We briefly discuss the theories, advantages, and extensibilities of an overall linear‐scaling scheme in 2c relativistic theory. The theory is based on the infinite‐order Douglas–Kroll–Hess method, with the local unitary transformation scheme to produce the 2c relativistic Hamiltonian, and the divide‐and‐conquer method to achieve linear‐scaling of Hartree–Fock and electron correlation methods. Furthermore, perspectives for large‐scale RQC are explained to bring the practical usage and treatment of light and heavy elements in 2c relativistic calculations close to those in non‐relativistic methods. © 2014 Wiley Periodicals, Inc.

The role of electron–phonon interaction on the transport properties of graphene based nano-devices

Transport through a nano-device in the presence of electron–phonon interaction has been studied by using unitary-transformation scheme together with the non-equilibrium Green׳s function formalism. The non-interacting Hamiltonian is written within the tight binding model and left/right electrodes are described in wide-band approximation. The interaction between electrons is studied in the mean-field Hubbard model. Our results indicate that both phonon-restricted and phonon-assisted transport through two different configurations of graphene quantum dot occur. Also, the electron–phonon interaction screens the electron–electron interaction, and reduces the spin polarization.

Frozen core potential scheme with a relativistic electronic Hamiltonian: Theoretical connection between the model potential and all-electron treatments

This Letter proposes an accurate scheme using frozen core orbitals, called the frozen core potential (FCP) method, to theoretically connect model potential calculations to all-electron (AE) ones. The present scheme is based on the Huzinaga–Cantu equation combined with spin-free relativistic Douglas–Kroll–Hess Hamiltonians. The local unitary transformation scheme for efficiently constructing the Hamiltonian produces a seamless extension to the FCP method in a relativistic framework. Numerical applications to coinage diatomic molecules illustrate the high accuracy of this FCP method, as compared to AE calculations. Furthermore, the efficiency of the FCP method is also confirmed by these calculations.

Study of the electron–photon interaction on the spin-dependent transport in nano-structures

Transport through a nano device in the presence of light illumination has been studied by using unitary-transformation scheme together with the non-equilibrium Green's function formalism. The non-interacting Hamiltonian is written within the framework of the tight binding model and structureless electrodes are described in wide-band approximation. The interaction between electrons is modeled by the mean-field Hubbard term. Our results indicate that we can tune the "intrinsic magnetization of zigzag edge" and "spin-polarized current" by using light illumination.

Pseudorelativistic operator for a two-electron ion

By means of a unitary transformation scheme, the Coulomb-Dirac operator for two electrons in a central potential is transformed into a pseudorelativistic operator which allows for the decoupling of the electron and positron degrees of freedom to arbitrary order $n$ in the potential strength. In case of $n=2$, relative boundedness properties and positivity of the resulting operator are shown for subcritical potential strength.

Exact decoupling of the Dirac Hamiltonian. II. The generalized Douglas–Kroll–Hess transformation up to arbitrary order

In order to achieve exact decoupling of the Dirac Hamiltonian within a unitary transformation scheme, we have discussed in part I of this series that either a purely numerical iterative technique (the Barysz-Sadlej-Snijders method) or a stepwise analytic approach (the Douglas-Kroll-Hess method) are possible. For the evaluation of Douglas-Kroll-Hess Hamiltonians up to a pre-defined order it was shown that a symbolic scheme has to be employed. In this work, an algorithm for this analytic derivation of Douglas-Kroll-Hess Hamiltonians up to any arbitrary order in the external potential is presented. We discuss how an estimate for the necessary order for exact decoupling (within machine precision) for a given system can be determined from the convergence behavior of the Douglas-Kroll-Hess expansion prior to a quantum chemical calculation. Once this maximum order has been accomplished, the spectrum of the positive-energy part of the decoupled Hamiltonian, e.g., for electronic bound states, cannot be distinguished from the corresponding part of the spectrum of the Dirac operator. An efficient scalar-relativistic implementation of the symbolic operations for the evaluation of the positive-energy part of the block-diagonal Hamiltonian is presented, and its accuracy is tested for ground-state energies of one-electron ions over the whole periodic table. Furthermore, the first many-electron calculations employing sixth up to fourteenth order DKH Hamiltonians are presented.

Exact decoupling of the Dirac Hamiltonian. I. General theory.

Exact decoupling of positive- and negative-energy states in relativistic quantum chemistry is discussed in the framework of unitary transformation techniques. The obscure situation that each scheme of decoupling transformations relies on different, but very special parametrizations of the employed unitary matrices is critically analyzed. By applying the most general power series ansatz for the parametrization of the unitary matrices it is shown that all transformation protocols for decoupling the Dirac Hamiltonian have necessarily to start with an initial free-particle Foldy-Wouthuysen step. The purely numerical iteration scheme applying X-operator techniques to the Barysz-Sadlej-Snijders (BSS) Hamiltonian is compared to the analytical schemes of the Foldy-Wouthuysen (FW) and Douglas-Kroll-Hess (DKH) approaches. Relying on an illegal 1/c expansion of the Dirac Hamiltonian around the nonrelativistic limit, any higher-order FW transformation is in principle ill defined and doomed to fail, irrespective of the specific features of the external potential. It is shown that the DKH method is the only valid analytic unitary transformation scheme for the Dirac Hamiltonian. Its exact infinite-order version can be realized purely numerically by the BSS scheme, which is only able to yield matrix representations of the decoupled Hamiltonian but no analytic expressions for this operator. It is explained why a straightforward numerical iterative extension of the DKH procedure to arbitrary order employing matrix representations is not feasible within standard one-component electronic structure programs. A more sophisticated ansatz based on a symbolical evaluation of the DKH operators via a suitable parser routine is needed instead and introduced in Part II of this work.

Unitary transformation scheme for polaron and bipolaron correlation models

On the basis of the modified Lang-Firsov-transformed Holstein-Hubbard model, effective polaron and bipolaron models are derived within the unified framework of Schrieffer-Wolff-type transformations in the strong correlation limit. The resulting polaronic t-J and bipolaronic local pairing models obtained for large positive and negative on-site interactions, respectively, exhibit different renormalizations of the hopping and interaction terms in the moderately heavy and light polaron and bipolaron cases.

Evaluation of Suitable Mean‐Field (Hartree‐Fock) Representations for Perturbational Studies of Disordered Solids

AbstractA unitary transformation scheme in Hartree‐Fock (HF) space is derived allowing for the definition of a “mixed” representation of the mean‐field operator which forms a suitable self‐consistent‐field (SCF) HF basis for perturbational studies of disordered solids and polymers. The transformed HF operator contains only a small number of off‐diagonal Lagrange multipliers (i.e. coupling terms) between properly defined “defect state(s)” in the filled or empty HF space and the remaining one‐particle functions of the Fermion system which are collected into a prediagonalized subspace. The unitary transformation scheme requires two numerical operations; in the first one the canonical one‐electron functions are transformed into a set of localized valence‐bond‐type orbitals. This one‐particle representation is then transferred into a mixed basis via elementary matrix diagonalizations and multiplications. The relevant theoretical steps are outlined in a few simple examples.