Equations of Motion (EOM) Methods for Computing Electron Affinities

Chapter 17 - Equations of motion methods for computing electron affinities and ionization potentials

Response of a Molecule to Adding or Removing an Electron

The ab initio calculation of molecular electron affinities (EA) and ionization potentials (IP) is a difficult task because the energy of interest is a very small fraction of the total electronic energy of the parent species. For example, EAs typically lie in the 0.01-10 eV range, but the total electronic energy of even a small molecule, radical, or ion is usually several orders of magnitude larger. Moreover, the EA or IP is an intensive quantity but the total energy is an extensive quantity, so the difficulty in evaluating EAs and IPs to within a fixed specified (e.g., ±0.1 eV) accuracy becomes more and more difficult as the system's size and number of electrons grows. The situation becomes especially problematic when studying extended systems such as solids, polymers, or surfaces for which the EA or IP is an infinitesimal fraction of the total energy. EOM methods such as the author developed in the 1970s offer a route to calculating the intensive EAs and IPs directly as eigenvalues of a set of working equations. A history of the development of EOM theories as applied to EAs and IPs, their numerous practical implementations, and their relations to Greens function or propagator theories are given in this contribution. EOM methods based upon Møller-Plesset, multiconfiguration self-consistent field, and coupled-cluster reference wave functions are included in the discussion as is the application of EOM methods to metastable states of anions.

In this paper we provide a direct comparison (over the same molecular basis) of the Equations of Motion (EOM) method and Configuration Internation (CI) methods, for the calculation of excitation properties of water and methane. We find that when double particle-hole interactions have been included in the EOM method and double excitations in the CI method, the results from the EOM method are in closer agreement with the experimental values than the results from the CI method. Furthermore, we find the EOM method to be more economical and convenient, from a computational point of view, than the corresponding CI method. The results from the EOM method also tend to be less sensitive to the choice of basis set than the results from the usual CI treatment.

A tight-binding equation of motion (EOM) method for the simulation of electronic transport in complex, and inhomogeneous systems is presented. Conductance is calculated in the linear response regime where chemical potentials can mimic electrochemical potentials. The technique is first elucidated by application to several simple systems to clarify important issues. A calculation of current-perpendicular-to-the-plane giant magnetoresistance (GMR) in a $\mathrm{Co}∕\mathrm{Cu}$ multilayer then follows. A 67% GMR is calculated which originates primarily from spin-dependent interface resistances. The advantages of an EOM method are that complicated geometries can be considered, and interactions such as spin-orbit effects or phonons, for example, may be included easily.

Part I - Dipole Properties of Atoms and Molecules in the Random Phase Approximation: A random phase approximation (RPA) calculation and a direct sum over states is used to calculate second-order optical properties and van der Waals coefficients. A basis set expansion technique is used and no continuum-like functions are included in the basis. However, unlike other methods we do not force the basis functions to satisfy any sum-rule constraints but rather the formalism (RPA) is such that the Thomas Reiche-Kuhn sum rule is satisfied exactly. Central attention is paid to the dynamic polarizability from which most of the other properties are derived. Application is made to helium and molecular hydrogen. In addition to the polarizability and van der Waals coefficients, results are given for the molecular anisotropy of H_2, Rayleigh scattering cross sections and Verdet constants as a function of frequency. Agreement with experiment and other theories is good. Other energy weighted sum-rules are calculated and compare very well with previous estimates. The practicality of our method suggests its applications to larger molecular systems and other properties. Part II - Photoionization Cross Sections for H_2 in the Random Phase Approximation with a Square-Integrable Basis: Total photoionization cross sections for H_2 are calculated in the Random Phase Approximation (RPA) through a numerical analytic continuation procedure applied to the polarizability for complex valuesof the frequency. The representation of the polarizability that is required is obtained from a discrete set of excitation energies and oscillator strengths that satisfies the Thomas-Reich-Kuhn sum rule exactly and other energy-weighted sum rules approximately. The fact that the excitation spectrum is obtained through a solution of the RPA equations with no continuum functions added to the basis makes the method well suited for general molecular photoionization calculations. The results are compared with experiment and good agreement is found. Part III - Oscillator Strengths for the X^1∑^+ - A^1π System in CH^+ from the Equations of Motion Method: The equations of motion method is used to study the X^1∑^+ - A^1π system in CH^+. In a computationally simple scheme, these calculations, which were done in modest sized basis sets, provide transition moments and oscillator strengths that agree well with the best CI calculations to date.

A wavefunction approach to equations of motion—Green's function methods

Studying the spectroscopic properties of nuclei in terms of their nucleonic degrees of freedom is one of the most challenging tasks in nuclear structure physics. In principle, the nuclear Shell Model (SM) allows to solve exactly the nuclear eigenvalue problem. Its actual implementation, however, presents several problems. One has first to turn the eigenvalue problem in the full space into an equivalent one formulated in a restricted model space. In order to achieve this step, it is necessary to find a reliable method for deriving an effective Hamiltonian, acting into the model space, from the bare nucleon nucleon interaction. Another difficulty deals with the dimensions of the SM Hamiltonian matrix. In fact, they grow very rapidly with the number of active nucleons, even in a small model space. It is therefore necessary to search for efficient algorithms which allow to diagonalize large Hamiltonian matrices. The first part of this work has consisted in upgrading an iterative diagonalization algorithm developed few years ago [1], so as to allow large scale nuclear shell model calculations in the uncoupled m-scheme. This new version can generate a large number of extremal eigenstates for each angular momentum and, therefore, is able to provide a complete description of the low energy properties of complex nuclei. The method has been applied to heavy Xenon isotopes. We have investigated first the convergence properties of the iterative procedure, which has allowed to asses the limits of the method [2]. Within these limits, it has been possible to give a complete description of their properties by computing the spectra and E2 and M1 transition strengths [3]. The analysis of the results has allowed to determine the collectivity of the states as well as their proton-neutron symmetry and multiphonon nature. Due to the limited dimensions of the space, shell model calculations are not able to provide a complete picture of collective modes, especially of the high energy resonances. In order to describe correctly these modes, we have reformulated an equation of motion method [4], which generates iteratively a basis of multiphonon states, built of TDA phonons, and uses such a basis to solve the eigenvalue problem. The method, which is free of approximations, has been applied to the closed shell Oxygen 16 and to neutron rich oxygen isotopes, using a realistic effective Hamiltonian in a space which includes up to three-phonon states. We have studied the effect of these complex states on the giant dipole resonance, and, for neutron rich nuclei, on the so-called pygmy resonance. The method and some preliminary results have been presented in recent international conferences. A complete description will be submitted soon for publication on an international journal. [1.] F. Andreozzi, N. Lo Iudice, and A. Porrino, J. Phys. : Nucl. Part. Phys. 29, 2319 (2003). [2.] D. Bianco, F. Andreozzi, N. Lo Iudice, A. Porrino, and F. Knapp, J. Phys. G 38, 025103(2011). [3.] D. Bianco, F. Andreozzi, N. Lo Iudice, A. Porrino, and F. Knapp, Phys. Rev. C 84, 024310 (2011). [4.] F. Andreozzi, F. Knapp, N. Lo Iudice, A. Porrino, and J. Kvasil, Phys. Rev. C 75, 044312 (2007).

Summary form only given. Correlation effects influencing the optical properties of semiconductors that cannot be described at the basis of Hartree-Fock approximation are intensively studied at the present time. Equations of motion methods, the Green's function formalism or the cumulant expansion method with fluctuation-dissipation theorem are successfully used to explain distinctive features of optical spectra of semiconductors. In spite of the fact that all these different theories deal with just the same problem, the physical mechanisms and results obtained by using these theoretical methods are various in detail drastically. Especially it concerns the coherently driven plasmons in high excited semiconductors. Our aim is to derive the equations of motion which contain the results of another theories and allow to include into considerations multi-quantum optical processes involving photon and several mixed phonon-plasmons. We get the semiconductor Bloch equations using the fluctuation-dissipation theorem to calculate four operator expectation values with account of coherent memory effects.

We present implementation of second- and third-order algebraic diagrammatic construction (ADC) theory for efficient and accurate computations of molecular electron affinities (EA), ionization potentials (IP), and densities of states [EA-/IP-ADC(n), n = 2, 3]. Our work utilizes the non-Dyson formulation of ADC for the single-particle propagator and reports working equations and benchmark results for the EA-ADC(2) and EA-ADC(3) approximations. We describe two algorithms for solving EA-/IP-ADC equations: (i) conventional algorithm that uses iterative diagonalization techniques to compute low-energy EA, IP, and density of states and (ii) Green's function algorithm (GF-ADC) that solves a system of linear equations to compute density of states directly for a specified spectral region. To assess the accuracy of EA-ADC(2) and EA-ADC(3), we benchmark their performance for a set of atoms, small molecules, and five DNA/RNA nucleobases. As our next step, we demonstrate the efficiency of our GF-ADC implementation by computing core-level K-, L-, and M-shell ionization energies of a zinc atom without introducing the core-valence separation approximation. Finally, we use EA- and IP-ADC methods to compute the bandgaps of equally spaced hydrogen chains Hn with n up to 150, providing their estimates near thermodynamic limit. Our results demonstrate that EA-/IP-ADC(n) (n = 2, 3) methods are efficient and accurate alternatives to widely used electronic structure methods for simulations of electron attachment and ionization properties.

Electronic Green's functions in a T-shaped multi-quantum dot system

The phase transitions in the Bose-Fermi-Hubbard model are investigated. The boson Green's function is cal- culated in the random phase approximation (RPA) and the formalism of the Hubbard operators is used. The regions of existence of the superuid and Mott insulator phases are established and the (; t) (the boson chemical potential vs. hopping parameter) phase diagrams are built at different values of boson-fermion in- teraction constant (in the regimes of x ed chemical potential or x ed concentration of fermions). The effect of temperature change on this transition is analyzed and the phase diagrams in the (T; ) plane are constructed. The role of thermal activation of the ion hopping is investigated by taking into account the temperature depen- dence of the transfer parameter. The reconstruction of the Mott-insulator lobes due to this effect is analyzed. parameters of the BFHM which describes these objects can be tuned in experiments and results of theoretical calculations can be tested in practice. Dieren t theoretical approaches were used to investigate the BFHM: mean eld theory (2), strong coupling approach (3), density matrix renormalization group (4), quantum Monte Carlo (5) and others. Another example of systems which can be described by the Bose-Fermi-Hubbard-type Hamil- tonian are intercalated crystals (for example, TiO2 intercalated by Li ions). In such systems the impurity ions and electrons play the role of the bosons and fermions, respectively. A lattice gas type models are also used for the description of ionic conductors (7{9). In (10) the pseudospin-electron model of intercalation was formulated and thermodynamics of the model was investigated in the mean eld approximation. It was shown that due to the presence of electrons the eectiv e inter- action between ions is generated and the character of this interaction depends on the lling of the electron band (when the chemical potential of the electrons is near the band edges, the rst order phase transition between uniform phases or phase separation in the regime of the xed electron concentration takes place; such an eect is important when constructing batteries (11)). In this work we consider the BFHM and propose a method for calculation of correlation func- tions and average values of boson and fermion concentrations. The method is based on the intro- duction of Hubbard operators acting in the on-site basis of states and is similar to the composite fermion approach used in (2) (the composite fermions are formed by a fermion and bosons or bosonic holes) but is more general and universal. Introducing the on-site Hubbard operators we can employ a known technique developed for the calculation of a correlation (Green's) function built on these operators, for example, equation of motion method (12) or diagrammatic approach based on the corresponding Wick's theorem (13). In this paper we use the rst of them. As a rst

One and two magnon correlations are examined within the framework of standard perturbation theory using a Hollstein-Primakoff spin wave representation. Within the RPA this method reproduces the results obtained by the equation of motion technique in ibid., vol.6, 323 (1973), provided it is carried out in a certain selfconsistent way. A step by step comparison of the two methods throws new light onto the physical implications involved in the different approximations which seem to be favoured by the two techniques. The present method has in general an advantage over the equation of motion method in casting its results in terms which are easily physically interpretable. This allows the derivation of a simple expression for the neutron scattering cross section in terms of merely three q- omega dependent functions which describe certain free two-magnon correlation functions.

Solution of Potts-3 and Potts-∞ matrix models with the equations of motion method

Highly accurate electron affinities and ionization potentials of chemical systems were described by means of the procedure called GHV-EOM (Valdemoro et al, in Int J Quantum Chem 112:2965, 2012), which combines the G-particle-hole hypervirial (GHV) equation method (Alcoba et al, in Int J Quantum Chem 109:3178, 2009) and that of the equations-of-motion (EOM), by Simons and Smith (Simons and Smith, in J Chem Phys 58:4899, 1973). The present work improves that hybrid method by introducing the point group symmetry within its framework, providing a higher computational efficiency. We report results which show the achievements attained by using the symmetry-adapted methodology. The new formulation turns out to be particularly suitable for characterizing solid models, as cyclic one-dimensional chains.