The Coulomb Green's function in IR3 and its relation to the harmonic oscillator in IR4

A very simple derivation of the Coulomb Green's function in IR³ is presented, which is based on an application of the Kustaanheimo–Stiefel transformation directly to the equation defining the harmonic oscillator Green's function in IR⁴. The representation that we obtain makes it possible to Fourier transform the Green's function with respect to its space variables by means of a simple Gaussian integration. The method that we use can readily be applied to several other Hamiltonians.     

On the doubly ionized states of ammonia

The Auger electron spectrum of ammonia is theoretically investigated using the ab initio Green's function and configuration interaction methods. Both calculations can quantitatively reproduce the main features of the spectrum. The results allow an unambiguous assignment of the Auger bands, which differs somewhat from that previously proposed on the basis of Hartree-Fock calculations.     

A Green's function approach to the nonrelativistic radial wave equation of hydrogen atom

We report an efficient and consistent method for the solution of Schrödinger wave equation using the Green's function technique and its successful application to the nonrelativistic radial wave equation of hydrogen atom. For the radial wave equation, the Green's function is worked out analytically by means of Laplace transform method and the wave function is proposed under the boundary conditions. Computationally the product of potential term, the proposed wave function and the Green's function are integrated iteratively to get the nonrelativistic radial wave function. The resultant wave after each iteration is normalized and plotted against the standard nonrelativistic radial wave function. The solution converges to the standard wave with the increasing number of iterations. Results are verified for the first 15 states of hydrogen atom. The method adopted here can be extended to many‐body problem and hope that it can enhance our knowledge about complex systems.     

Power moments of hydrogenic Green's functions and Green's functions of the second kind

Power moments for all of the reduced hydrogenic Green's functions have been computed. These moments have the same form as the corresponding moments of the free-particle Green's functions. Green's functions of the second kind are defined, and uses for these objects in model potential theory and the theory of many-body Green's functions are pointed out. In the case of the ground state of the hydrogenic atom, the Green's function of the second kind has been given.     

Autoionization of CO after C 1s→ Zπ* excitation: a comparison with photoemission and auger decay

We have calculated the entire autoionization spectrum of CO following core-to-bound, i.e. C 1s→2π excitation, within a Green's function formalism. Approximate autoionization transition intensities can be related to the Hamiltonian matrix elements. Initialstate screening is important for obtaining realistic autoionization probabilities. For the low-lying ion states there is a one-to-one correspondence between the photoelectron and the autoionization spectrum. The connection between autoionization and Auger decay is discussed.     

Final state effects in the photoionization process

Different methods to take into account many body effects in photoionization processes are discussed. Numerical results are shown for some indirect methods, several approximations within the Green's function formalism, and for potential models. The suitability of the Green's function method for the investigation of satellite structure is considered.     

Dispersion relations and spectral densities

Weyl's eigenvalue theory for ordinary second-order differential equations is discussed for the case of a continuous spectrum. It is demonstrated that the spectral density function obtained from a suitably averaged Green's function, equal to the Weinstein function, can be directly related to the Weyl–Titchmarsh m-function. The explicit connections with scattering theory are derived and it is found that the Weyl and Jost solutions are proportional; the proportionality factor being the reciprocal value of the latter at the origin. The physical interpretation of the complex poles of the spectral density is discussed in relation to Gamow's exponentially growing functions. The advantage of using a formulation that allows for a “perturbation” of boundary conditions is pointed out.     

Many-electron effects in the x-ray emission spectra and the one-particle green's function. Correspondence theorem

Within the framework of reasonable approximation the intensity of spectral bands in the X-ray emission spectrum was shown to be proportional to the imaginary part of the one-particle Green's function. From this ensues a statement about a correspondence between the distribution of the intensity in the X-ray emission and photoelectron spectra — correspondence theorem.     

Spectral variational principle for Green's functions

For a suitable approximation \(\tilde K\) (q, q′, τ) to the Dirac-Feynman Green's function of a quantummechanical system, the parameter \({\mathcal{L}\tilde K}\) is defined, where ?≡i?/?τ??. It is shown thatΔ≧0 andΔ→0 asK→K, the exact Green's function, thus providing a criterion on approximate Green's functions analogous in its role to the variational principle for wavefunctions. A second somewhat weaker criterion is also proposed, based on \(\Sigma \equiv \left[ {tr\tilde K*tr\mathcal{L}^2 \tilde K - |tr\mathcal{L}\tilde K|^2 } \right]_{\tau avg} \geqq 0\) . Recipes are given for projecting out continuum contributions toΔ or∑ and for analyzing for the discrete eigen-value spectrum.     

Green's Function Determination of Force Constants-H2O as Example

Treating isotopically substituted molecule as a perturbed system, Green's function for the perturbation are constructed and related to the force field of vibration. By spectral representation, Green's function is diagonalized in the normal coordinates. Then transforming back to the Cartesian coordinates, the Cartesian force constants are generated without solving the secular equation directly. The relations between the internal force constants and the Cartesian force constants ate given and complete internal force field can be obtained. The results for H₂O are discussed.     

Theory of the excitation spectrum of nondiagonal disordered systems

The spectrum of a two-component solid solution with a nondiagonal disorder is studied in the framework of the average T matrix method. For a one-dimensional system in the nearest-neighbor approximation the criteria for the system parameters are given such that at an in-band resonance, one or two “impurity bands” may be realized, and the corresponding model calculation is performed. In the single-site approximation an expression of the self-energy part of a nondiagonal disordered system Green's function is found taking into account multiple occupancy corrections. The possibility of using it to describe a disordered system excitation spectrum and the calculation of state density moments are discussed.     

Natural energy orbitals and the one-particle Green's function

It is shown how the properties of the one-particle Green's function lead naturally to the definition of the so-called natural energy orbitals. These orbitals allow the fully correlated total energy of a system to be written in Hartree–Fock-like fashion and might therefore provide a bridge between sophisticated correlated wave functions and approximate theories of chemical structure and reactivity based on a Hartree–Fock-like energy expression. Moreover these orbitals form the basis for a self-consistent scheme to calculate the one-particle Green's function. The relation between these natural energy orbitals and the extended Koopmans' theorem is considered. Finally it is shown that the exactness of the lowest extended Koopmans' ionization potential implies the linear independence of the corresponding Dyson orbital from all other Dyson orbitals.     

On the excitonic nature of the first UV absorption peak in polyene

The first absorption peak in the UV spectrum of polyene is interpreted in terms of charge transfer excitons. The exciton spectrum has been calculated from first principles using the Green's function formalism of charge transfer exciton theory. Electron correlation effects on the polyene band structure have been included with the help of second order Møller-Plesset perturbation theory and of the electron polaron model of Toyozawa. The spectrum of bound singlet excitons starts at E_K=0 = 1.86 eV above the top of the valence band. A deeper lying triplet level is observed at 0.72 eV. Further correlation effects on the band gap and dielectric screening of the electron-hole ineraction are discussed.     

Solution of the Pauli master equation via the lanczos algorithm

We present a numerical Green's function approach to solving the Pauli master equation which utilizes the recursive residue generation method (RRGM) developed by Wyatt and co-workers. In most practical applications, only a few Green's function matrix elements are required to express all physical quantities of interest. For such systems, computation time reductions of 10²–10³ over direct diagonalization algorithms are achieved even for systems of moderate size. An application to a simple model for trapping in polymers is given.     

Application of many-body green's functions to the calculation of molecular ionization potentials

We present a method for calculating the one-particle Green's function for molecules. The scheme is essentially that proposed by Schneider and Taylor [1]. The Green's function is obtained through the Dyson equation. Closed expressions result by using the functional derivation technique to truncate an infinite set of coupled equations. A physical interpretation of the approximation is given and a connection with the equations-of-motion method is pointed out. In a numerical application the ionization potentials are obtained for the molecules N₂, H₂O, NH₃ and CH₄.     

Theoretical study of molecular conduction: I. Effective Green's function based on perturbation theory

There have been many experimental and theoretical studies on molecular conduction, as it is a fundamental parameter in the study of molecular‐scale electronics. We have investigated the features of molecular conduction using a Green's function method, which has often been used to solve problems in quantum transport and is also effective in elucidating electron transport in molecules. We have obtained the novel effective Green's functions, including the first‐order energy corrections, by accommodating the self‐energy of the electrodes as perturbation terms. Although these approximate Green's functions only provide information on the first‐order energy corrections, they can involve the elementary properties of molecular conduction. We propose a scheme for the analysis of the relations between molecular orbitals and their roles in molecular conduction and present analytical calculations for normal and cyclic polyenes. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2006     

Unified treatment of the electronic structure of organic conjugated polymers

Theoretical calculations for polaron-type defect structures on organic conjugated polymers are carried out by Green's function and transfer function formalism. Firstly, the renormalization concept is applied to mathematically convert the polymers PPP , PPY , and PTP to an equivalent (trans-PA )-like form. Then, it is shown that Green's functions and polaron wave functions are readily obtained from the equations for trans-PA by the appropriate substitution of the renormalized parameters. It is suggested that the existence of defects such as polarons or bipolarons could be invoked to provide a unified treatment of electric conduction in organic conjugated polymers.     

Partial third-order quasiparticle theory: An application to the photoelectron spectrum of S-tetrazine

An alternative choice of reference state averages in electron propagator theory and retention of all self-energy terms through third-order gives rise to the partial third-order self-energy approximation, P3. Quasiparticle P3 calculations avoid the chief computational bottlenecks of third-order theory and the outer valence Green's function (OVGF). P3 requires no multiplicative factors for scaling self-energy terms and produces better accuracy than OVGF. An application to the photoelectron spectrum of s-tetrazine illustrates the ability of the P3 method to predict correct final state orderings. © 1997 John Wiley & Sons, Inc.     

Integral transform functions and separable potentials

General forms for asymptotic wave functions are derived from properties of the relevant Green's function. The use of separable potentials constructed from the asymptotic functions is described and the relation with integral transform functions is discussed.     

Green's function analysis of the vibrational spectra of polymer chains. II. Fourier method for calculating the green's function of a perfect infinite polymer chain

A new method of calculating the Green's function (GF) for an arbitrary polymer chain is suggested. It makes it possible to obtain analytical expressions for this function in a number of cases of practical value.