Herbert H. H. Homeier

Organometallic Pt(II) and Ir(III) Triplet Emitters for OLED Applications and the Role of Spin-Orbit Coupling: A Study Based on High-Resolution Optical Spectroscopy

High-resolution optical spectroscopy of organometallic triplet emitters reveals detailed insights into the lowest triplet states and the corresponding electronic and vibronic transitions to the singlet ground state. As case studies, the blue-light emitting materials Pt(4,6-dFppy)(acac) and Ir(4,6-dFppy)2(acac) are investigated and characterized in detail. The compounds’ photophysical properties, being markedly different, are largely controlled by spin–orbit coupling (SOC). Therefore, we study the impact of SOC on the triplet state and elucidate the dominant SOC and state-mixing paths. These depend distinctly on the compounds’ coordination geometry. Relatively simple rules and relations are pointed out. The combined experimental and theoretical results lead us towards structure-efficiency rules and guidelines for the design of new organic light emitting diode (OLED) emitter materials.

A modified Newton method for rootfinding with cubic convergence

We consider a modification of the Newton method for finding a zero of a univariate function. The case of multiple roots is not treated. It is proven that the modification converges cubically. Per iteration it requires one evaluation of the function and two evaluations of its derivative. Thus, the modification is suitable if the calculation of the derivative has a similar or lower cost than that of the function itself. Classes of such functions are sketched and a numerical example is given.

Scalar Levin-type sequence transformations

Abstract Sequence transformations are important tools for the convergence acceleration of slowly convergent scalar sequences or series and for the summation of divergent series. The basic idea is to construct from a given sequence {{sn}} a new sequence {{s n ′}}= T ({{s n }}) where each sn′ depends on a finite number of elements sn1,…,snm. Often, the sn are the partial sums of an infinite series. The aim is to find transformations such that {{sn′}} converges faster than (or sums) {{sn}}. Transformations T ({{s n }},{{ω n }}) that depend not only on the sequence elements or partial sums sn but also on an auxiliary sequence of the so-called remainder estimates ωn are of Levin-type if they are linear in the sn, and nonlinear in the ωn. Such remainder estimates provide an easy-to-use possibility to use asymptotic information on the problem sequence for the construction of highly efficient sequence transformations. As shown first by Levin, it is possible to obtain such asymptotic information easily for large classes of sequences in such a way that the ωn are simple functions of a few sequence elements sn. Then, nonlinear sequence transformations are obtained. Special cases of such Levin-type transformations belong to the most powerful currently known extrapolation methods for scalar sequences and series. Here, we review known Levin-type sequence transformations and put them in a common theoretical framework. It is discussed how such transformations may be constructed by either a model sequence approach or by iteration of simple transformations. As illustration, two new sequence transformations are derived. Common properties and results on convergence acceleration and stability are given. For important special cases, extensions of the general results are presented. Also, guidelines for the application of Levin-type sequence transformations are discussed, and a few numerical examples are given.

Numerical integration of functions with a sharp peak at or near one boundary using Mo¨bius transformations

Abstract A class of coordinate transformations depending on a single parameter is studied as quadrature tool. Working rules for the choice of the parameter are proposed. Numerical tests for the method are presented. They show that these coordinate transformations, when combined with Gauss-Legendre quadrature rules, are well suited for the numerical integration of functions possessing a sharp peak at or near one boundary of the interval of integration. A method to combine the transformations with automatic quadrature routines is also proposed; it seems to be useful for the numerical evaluation of integrals with the same kind of integrand behavior.

Some Properties of the Coupling Coefficients of Real Spherical Harmonics and Their Relation to Gaunt Coefficients

Abstract Spherical harmonics are of considerable importance for computations involving basis functions corresponding to large values of the angular momentum quantum number l. Their use allows efficient coding of programs involving such basis functions because the formulae of the coupling coefficients are simple. The choice of real spherical harmonics allows one to avoid the use of complex quantities in computer programs that increase storage and CPU time requirements. In this paper, certain properties of the coupling coefficients for real spherical harmonics are derived that are necessary for an efficient computation of coupling terms.

Programs for the Evaluation of Overlap Integrals with B Functions

Abstract Programs for the evaluation of overlap integrals and some related integrals with certain exponential-type orbitals (ETOs), the B functions [E. Filter and E.O. Steinborn, Phys. Rev. A 18 (1978) 1] are presented. B functions possess a very simple Fourier transform which results in relatively compact general formulas for molecular integrals derivable using the Fourier transform method. In addition, molecular integrals for the more common Slater-type orbitals (STOs) and other ETOs can be written as finite linear combinations of integrals with B functions because these functions span the space of ETOs. The programs are based on a number of different formulas and representations for the overlap integrals. Simple finite expressions are used in the one-center case, and in the two-center case with equal exponential parameters. In the two-center case, the so-called Jacobi polynomial representation can only be used for largely different exponential parameters. Otherwise, a one-dimensional integral representation [H.P. Trivedi and E.O. Steinborn, Phys. Rev. A 27 (1983) 670] for the two-center overlap integral of B functions with different exponential parameters is used. The numerical properties of this integral representation, and special quadrature methods to deal with them, have been discussed recently [H.H.H. Homeier and E.O. Steinborn, Int. J. Quantum Chem. 42 (1992) 761]. These results show that these Mobius-transformation based quadrature rules, which are well suited for the numerical integration of functions possessing a sharp peak at or near one boundary of integration [H.H.H. Homeier and E.O. Steinborn, J. Comput. Phys. 87 (1990) 61], are superior to several other methods and can be used to calculate the overlap integrals reliably and economically.

Molecular calculations with B functions

A program for molecular calculations with B functions is reported and its performance is analyzed. All the one- and two-center integrals and the three-center nuclear attraction integrals are computed by direct procedures, using previously developed algorithms. The three- and four-center electron repulsion integrals are computed by means of Gaussian expansions of the B functions. A new procedure for obtaining these expansions is also reported. Some results on full molecular calculations are included to show the capabilities of the program and the quality of the B functions to represent the electronic functions in molecules.

On Newton-type methods for multiple roots with cubic convergence

We introduce two families of Newton-type methods for multiple roots with cubic convergence. A further Newton-type method for multiple roots with cubic convergence is presented that is related to quadrature. We also provide numerical tests that show that these new methods are competitive to other known methods for multiple roots.

On Remainder Estimates for Levin-type Sequence Transformations

Remainder estimates for Levin-type sequence transformations are discussed. A new remainder estimate is introduced which is related to the Kummer transformation. Test cases are presented which show that the new remainder estimate can enhance the performance of Levin-type sequence transformations considerably.

Recent Progress on Representations for Coulomb Integrals of Exponential-type Orbitals

Abstract We present some recent work on Coulomb integrals with certain exponential-type orbitals (ETOs), the B functions (E. Filter and E.O. Steinborn, Phys. Rev. A, 18 (1978) 1). The main advantage of B functions is that they possess a very simple Fourier transform (E.J. Weniger and E.O. Steinborn, J. Chem Phys., 78 (1983) 6121), which results in relatively compact general formulas for molecular integrals derivable using the Fourier transform method. Because, in addition, these functions can be taken as a basis of the space of ETOs, molecular integrals for the more common Slater-type orbitals and other ETOs can be written as finite linear combinations of integrals with B functions. Various representations for Coulomb integrals of B functions with equal exponential parameters and three integral representations of Coulomb integrals of B functions with different exponential parameters are given, which could be derived recently. The integral representations are one-dimensional and contain more simple molecular integrals as part of the integrand. Upon insertion of various analytic representations of the latter integrals by finite sums a large number of different integral representations for the Coulomb integrals emerge. The structure of one of the integral representations is completely analogous to a one-dimensional integral representation (H.P. Trivedi and E.O. Steinborn, Phys. Rev. A, 27 (1983) 670) for the two-center overlap integral of B functions with different exponential parameters. Further, a Jacobi polynomial representation for the Coulomb integrals of B functions with different exponential parameters is discussed. Besides finite sums of functions regular at the origin this representation contains irregular solid harmonics and distributional contributions which are displayed explicitly.