Complex Basis Function Method Research Articles

The Auger spectrum of benzene.

We present an ab initio computational study of the Auger electron spectrum of benzene. Auger electron spectroscopy exploits the Auger-Meitner effect, and although it is established as an analytic technique, the theoretical modeling of molecular Auger spectra from first principles remains challenging. Here, we use coupled-cluster theory and equation-of-motion coupled-cluster theory combined with two approaches to describe the decaying nature of core-ionized states: (i) Feshbach-Fano resonance theory and (ii) the method of complex basis functions. The spectra computed with these two approaches are in excellent agreement with each other and also agree well with experimental Auger spectra of benzene. The Auger spectrum of benzene features two well-resolved peaks at Auger electron energies above 260eV, which correspond to final states with two electrons removed from the 1e1g and 3e2g highest occupied molecular orbitals. At lower Auger electron energies, the spectrum is less well resolved, and the peaks comprise multiple final states of the benzene dication. In line with theoretical considerations, singlet decay channels contribute more to the total Auger intensity than the corresponding triplet decay channels.

K-Shell photoionization cross sections of one-electron uranium cation by relativistic complex basis function method

The complex basis function (CBF) method is a square-integrable (L2) function method that represents continuum wave functions, and has been applied successfully to various problems, including the calculation of resonance state energies and photoionization cross sections. The previous applications of this method have been limited to non-relativistic problems, and no serious attempt has been made for the relativistic Hamiltonian. In this study, we calculated the photoionization cross sections from the imaginary part of the frequency-dependent polarizability by applying the CBF method to the inhomogeneous Dirac equation of the one-electron uranium cation. The results obtained by employing a pure Coulomb potential to the K-shell photoionization cross sections were in good agreement with previous theoretical values and experimental values for the neutral uranium atom. Phase shifts were obtained by matching the CBF solutions with the numerical ones, which were further propagated by the fourth-order Runge–Kutta method, and thus, the photoionization differential and total cross sections were successfully calculated. The calculated results were analyzed from various viewpoints, including the multipole effect of electromagnetic fields, momentum conservation of photon and photoelectron, interference of multiple perturbation terms, and difference in the orbital angular momentum of the small component wave functions. These analyses demonstrate the effectiveness of the CBF method for the inhomogeneous Dirac equation and thus its applicability to multielectron systems.

Second order Møller-Plesset and coupled cluster singles and doubles methods with complex basis functions for resonances in electron-molecule scattering.

The method of complex basis functions is applied to molecular resonances at correlated levels of theory. Møller-Plesset perturbation theory at second order and equation-of-motion electron attachment coupled-cluster singles and doubles (EOM-EA-CCSD) methods based on a non-Hermitian self-consistent-field reference are used to compute accurate Siegert energies for shape resonances in small molecules including N2-, CO-, CO2-, and CH2O-. Analytic continuation of complex 𝜃-trajectories is used to compute Siegert energies, and the 𝜃-trajectories of energy differences are found to yield more consistent results than those of total energies. The ability of such methods to accurately compute complex potential energy surfaces is investigated, and the possibility of using EOM-EA-CCSD for Feshbach resonances is explored in the context of e-helium scattering.

Calculation of photoionization differential cross sections using complex Gauss-type orbitals.

Accurate theoretical calculation of photoelectron angular distributions for general molecules is becoming an important tool to image various chemical reactions in real time. We show in this article that not only photoionization total cross sections but also photoelectron angular distributions can be accurately calculated using complex Gauss-type orbital (cGTO) basis functions. Our method can be easily combined with existing quantum chemistry techniques including electron correlation effects, and applied to various molecules. The so-called two-potential formula is applied to represent the transition dipole moment from an initial bound state to a final continuum state in the molecular coordinate frame. The two required continuum functions, the zeroth-order final continuum state and the first-order wave function induced by the photon field, have been variationally obtained using the complex basis function method with a mixture of appropriate cGTOs and conventional real Gauss-type orbitals (GTOs) to represent the continuum orbitals as well as the remaining bound orbitals. The complex orbital exponents of the cGTOs are optimized by fitting to the outgoing Coulomb functions. The efficiency of the current method is demonstrated through the calculations of the asymmetry parameters and molecular-frame photoelectron angular distributions of H2+ and H2. In the calculations of H2, the static exchange and random phase approximations are employed, and the dependence of the results on the basis functions is discussed. © 2017 Wiley Periodicals, Inc.

Optimization of complex slater-type functions with analytic derivative methods for describing photoionization differential cross sections.

The complex basis function (CBF) method applied to various atomic and molecular photoionization problems can be interpreted as an L2 method to solve the driven-type (inhomogeneous) Schrödinger equation, whose driven term being dipole operator times the initial state wave function. However, efficient basis functions for representing the solution have not fully been studied. Moreover, the relation between their solution and that of the ordinary Schrödinger equation has been unclear. For these reasons, most previous applications have been limited to total cross sections. To examine the applicability of the CBF method to differential cross sections and asymmetry parameters, we show that the complex valued solution to the driven-type Schrödinger equation can be variationally obtained by optimizing the complex trial functions for the frequency dependent polarizability. In the test calculations made for the hydrogen photoionization problem with five or six complex Slater-type orbitals (cSTOs), their complex valued expansion coefficients and the orbital exponents have been optimized with the analytic derivative method. Both the real and imaginary parts of the solution have been obtained accurately in a wide region covering typical molecular regions. Their phase shifts and asymmetry parameters are successfully obtained by extrapolating the CBF solution from the inner matching region to the asymptotic region using WKB method. The distribution of the optimized orbital exponents in the complex plane is explained based on the close connection between the CBF method and the driven-type equation method. The obtained information is essential to constructing the appropriate basis sets in future molecular applications. © 2017 Wiley Periodicals, Inc.

Complex basis functions revisited: implementation with applications to carbon tetrafluoride and aromatic N-containing heterocycles within the static-exchange approximation.

The method of complex basis functions for computing positions and widths of molecular resonances is revisited. An open-ended and efficient implementation is described. The basis set requirements of the complex basis are investigated within the computationally inexpensive static-exchange approximation, and the results of this investigation lead to a hierarchy of basis sets for complex basis function calculations on small molecules. These basis sets are then applied in static-exchange calculations on some larger molecules with multiple low energy shape resonances: carbon tetrafluoride, benzene, pyridine, pyrimidine, pyrazine, and s-triazine. The results indicate that more sophisticated methods using complex basis functions are worth pursuing in the search for accurate and computationally feasible methods for computing resonance energies in molecular systems.

Construction of complex STO-NG basis sets by the method of least squares and their applications

Electronic resonance state energies and photoionization cross sections of atoms and molecules are calculated with the complex basis function method by using mixture of appropriate complex basis functions representing one-electron continuum orbitals and the usual real basis functions for the remaining bound state orbitals. The choice of complex basis functions has long been a central difficulty in such calculations. To address this challenge, we constructed complex Slater-type orbital represented by N-term Gaussian-type orbitals (cSTO-NG) basis sets using the method of least squares. Three expansion schemes are tested: (1) expansion in complex Gaussian-type orbitals, (2) expansion in real Gaussian-type orbitals, and (3) expansion in even-tempered real Gaussian-type orbitals. By extending the Shavitt–Karplus integral transform expression to cSTO functions, we have established a mathematical foundation for these expansions. To demonstrate the efficacy of this approach, we have applied these basis sets to the calculation of the lowest Feshbach resonance of H2 and the photoionization cross section of the He atom including autoionization features due to doubly excited states. These calculations produce acceptably accurate results compared with past calculations and experimental data in all cases examined here.

Complex configuration interaction calculations of the cross section for the dissociative electron attachment process e− + F2 → F 2− → F + F− using the complex basis function method

The F(2)(-) molecule and the corresponding dynamic processes dealing with electron scattering on the neutral F(2) species have been the subject of many theoretical and experimental investigations in the past. In the context of the Born-Oppenheimer approximation, one of the best theoretical descriptions of the electronic states involves the use of complex basis functions together with configuration interaction (CI) methods. In this work, multireference CI calculations using the complex basis function method have been carried out for the autoionizing ground state of the F(2)(-) molecule. Potential curves and vibrational levels have been obtained for the ground and various excited states of both F(2) and F(2)(-), as well as the variation of the line width of the anionic ground state for the bond distance region in which it is metastable. Cross sections for the dissociative electron attachment process e(-) + F(2) → F(2)(-) → F + F(-) have also been computed within the framework of the boomerang model, and good agreement with available experimental data has been found. In addition, some calculations for the process of vibrational excitation are included which also give good agreement with experiment.

Photoionization cross sections of H2+ and H2with complex Gaussian-type basis functions optimized for the frequency-dependent polarizabilities

Within the framework of the complex basis function method, the photoionization cross sections of H(2)(+) and H(2) were calculated based on the variational principle for the frequency-dependent polarizabilities. In these calculations, complex orbital exponents of Gaussian-type basis functions for the final state continuum wavefunctions were fully optimized for each photon energy with the numerical Newton-Raphson method. In most cases, the use of only one or two complex Gaussian-type basis functions was enough to obtain excellent agreement with previous high precision calculations and available experimental results. However, there were a few cases, in which the use of complex basis functions having various angular momentum quantum numbers was crucial to obtain the accurate results. The behavior of the complex orbital exponents as a function of photon energy was discussed in relation to the scaling relation and the effective charge for photoelectron. The success of this method implies the effectiveness of the optimization of orbital exponents to reduce the number of basis functions and shows the possibility to calculate photoionization cross sections of general molecules using only Gaussian-type basis functions.

Photoionization cross sections with optimized orbital exponents within the complex basis function method

We show a new direction to expand the applicability of the complex basis function method for calculating photoionization cross sections through the imaginary part of the frequency-dependent polarizability. Based on the variational stability of the frequency-dependent polarizability, we made nonlinear optimizations of complex orbital exponents in basis functions representing continuum wave functions, and obtained fairly accurate results for H atom with only one or two complex basis functions particularly with dipole velocity gauge. Results were almost independent of whether Slater-type or Gaussian-type orbitals are used, implying the applicability to general many electron problems. The method was also applied to the (1)S (1s)(2) --> (1)P (1s)(1)(kp)(1) cross section of He atom and the optimized complex orbital exponents were related to those of H atom through the scaling property. The nonlinear optimizations have converged smoothly and the cross sections were in excellent agreement with experiment throughout wide photon energies, which suggest the effectiveness of the approach for many-electron systems.

Calculations of photoionization cross-sections with variationally optimized complex Gaussian-type basis functions

Photoionization cross-sections of various simple systems were calculated with variationally optimizing the complex orbital exponents in Gaussian-type basis functions within the framework of the complex basis function method. The optimizations were performed with the numerical Newton–Raphson method based on the variational principle for the frequency-dependent polarizabilities. Accurate cross-sections were obtained for the Rydberg autoionization of He and Be, and the photodetachment of H − with the optimizations of only one complex basis function. These results show the effectiveness of the optimization of the complex orbital exponents to reduce the number of basis functions significantly and imply the applicability to general problems by adapting many existing techniques developed in bound state theories.

Valence photoionization and autoionizing states of acetylene studied by the complex basis function method in the random phase approximation

The photoionization cross-section of acetylene has a prominent double-hump structure, the peak positions of which are 13.3 and 15.3 eV. The assignment for the lower peak has long been controversial. In this Letter, we settle the problem by calculating the valence photoionization cross-section with the complex basis function method including the channel coupling in the random-phase approximation. At the neutral ground-state geometry, we assigned the lower peak to the 3σ g→3pσ u Rydberg-type autoionizing state, which is perturbed by the 3σ g→3σ u intravalence excited state. We also studied the geometry dependence of the autoionizing state.

Making complex scaling work for long-range potentials

We examine finite basis set implementations of complex scaling procedures for computing scattering amplitudes and cross sections. While ordinary complex scaling, i.e., the technique of multiplying all interparticle distances in the Hamiltonian by a complex phase factor, is known to provide convergent cross-section expressions only for exponentially bounded potentials, we propose a generalization, based on Simon's exterior complex scaling technique, that works for long-range potentials as well. We establish an equivalence between a class of complex scaling transformations carried out on the time-independent Schr\"odinger equation and a procedure commonly referred to as the method of complex basis functions. The procedure is illustrated with a numerical example.

Propagation of wave packets using the complex basis function method

The complex basis function method is applied to the problem of wave packet propagation. Two numerical examples are given: free propagation of a Gaussian wave packet in one dimension and a time-dependent treatment of photoionization of the 2p state of the hydrogen atom. In both cases propagation for longer periods is possible using complex basis functions than with real-valued basis sets of the same size. Possible combinations of the complex basis function method with other propagation techniques are suggested.

Rigorous finite-basis-set approach to the calculation of the angular dependence of photoionization.

A practical solution is provided for the problem of using finite-basis-set methods to calculate the asymmetry parameter, \ensuremath{\beta}(E), which determines the angular dependence of photoejection in the dipole approximation. This problem has been an outstanding limitation of basis-set approaches to photoionization in general, which have heretofore been able only to produce the total cross section. The procedure proposed here is given in two forms: an evaluation of the differential cross section in terms of the outgoing flux, and an alternative prescription involving matching to the outgoing wave function. In both cases the method of complex basis functions is used to compute the necessary matrix elements of the electronic Green's function. Numerical results are reported for \ensuremath{\beta}(E) for ionization of the 2p state of atomic hydrogen.

Complex self-consistent-field and configuration-interaction studies of the lowestP2resonance state ofBe−

The position and width of the lowest $^{2}P$ resonance in electron-beryllium scattering are computed directly in configuration-interaction calculations with the use of the method of complex basis functions. These calculations make use of a reference wave function from a complex self-consistent-field calculation on ${\mathrm{Be}}^{\ensuremath{-}}$ with the use of complex Slater basis functions. The most extensive calculation performed includes 745 single, double, and triple excitations from the reference configuration and yields values for the position and width of 0.323 and 0.296 eV, respectively. Comparison is made with previous theoretical determinations. No experimental values are currently available.

Extension of the Method of Complex Basis Functions to Molecular Resonances

We consider the extension of the complex-coordinate technique to the problem of locating molecular resonances. We suggest a method which uses complex normalizable functions and which becomes equivalent to the usual dilatation transformation asymptotically, but is different for small values of the electronic coordinates. The technique is illustrated by application to the bound states of ${\mathrm{H}}_{2}^{+}$ and to a model nonspherical resonance problem.

Limitations of the method of complex basis functions

The method of complex basis functions proposed by Rescigno and Reinhardt is applied to the calculation of the amplitude in a model problem which can be treated analytically. It is found for an important class of potentials, including some of infinite range and also the square well, that the method does not provide a converging sequence of approximations. However, in some cases, approximations of relatively low order might be close to the correct result. The method is also applied to $S$-wave $e$-H elastic scattering above the ionization threshold, and spurious "convergence" to the wrong result is found. A procedure which might overcome the difficulties of the method is proposed.