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.     

Generalized formulation of quantum and classical crystallography using Green's functions

Quantum crystallography is a methodology by which structural information about a crystalline material obtained from X‐ray crystallography is combined with quantum mechanical methods. The objective is to enhance the data obtained from the X‐ray diffraction experiment, which are related to the atomic structure of the crystal, and to predict the properties and efficacy of those chemical compounds from which the crystals are derived. One approach in quantum crystallography is to use a projector matrix with a normalized trace. In this approach, quantum mechanical parameters in the projector matrix are fit into crystallographic data. During this fitting, the properties of the projector matrix called idempotency and normalization are used. Throughout this implementation procedure, Clinton's iteration scheme has been used in addition to the least‐squares technique. The purpose of the present study is to generalize Clinton's iterative equations in quantum crystallography by means of single‐particle Green's functions with the aid of the equal atoms model in the theory of direct methods. Convergency characters of the novel iterative equations are discussed by the steepest descent procedure. Furthermore, whether the calculations are valid in nonorthogonal bases was also examined. The iteration schemes widely used in quantum crystallography have been generalized but, in addition, the generalized expressions relating to the phase determination procedure and the probabilities of the sign relations between the structure factors are obtained and discussed comprehensively. The phrase order of crystallography has been put forward as a new concept. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

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     

Polarization Green's function with multiconfiguration self-consistent-field reference states

The polarization Green's-function formalism in the superoperator notation of Goscinski and Lukman is re-derived using a multiconfiguration self-consistent-field (MC -SCF ) reference state to establish the superoperator metric. The potential advantages of employing this more general reference state in Green's-function theories and certain inherent weaknesses associated with the traditional Hartree–Fock or Rayleigh–Schrödinger perturbation theory reference state choices are briefly discussed. The Hermiticity of the superoperators is analyzed within the framework of the MC –SCF reference state. Using a concept of order appropriate for this reference state choice, explicit formulas and computational procedures for the implementation of this Green's-function theory are presented and specialized to include terms consistent through second order.     

Electronic computation of first-order wave functions using green's functions

Green's functions and the symbol manipulative computer language LISP have been used to obtain exact, closed form, first-order functions and second-order energies for the first fourteen states of the hydrogen atom in electric and magnetic fields.     

Spectral analysis of the Green's function

The energy eigenvalue spectrum for a conservative dynamical system is contained implicitly in its Green's function. It becomes explicit in the Fourier transform of either the Green's function or its trace. The trace exists only when the spectrum is entirely discrete. Applications are made to the free particle, the linear harmonic oscillator, and the hydrogen atom. In the latter two cases determination of the Green's function can be considerably simplified by similarity transformations on the Hamiltonian operator.     

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.     

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 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.     

Computational Study of Vertical Ionization Potentials Using Density Functional Theory and Green's Function Methods

Over one hundred vertical ionization potentials (VIPs) were computed using density functional theory (DFT) and Green's function (GF) based methods. The DFT approaches include the unrestricted transition state (uTS) and unrestricted diffuse ionization (uDI) approximations using the Becke88‐Perdew86 exchange‐correlation functional. Green's function methods include the outer‐valence GF (OVGF) approach, the parametrized GF2 (pGF2), and the parametrized GF2 times screened interaction (pGW2) approximations. DFT computations of IPs using the uTS approximation was found to be nearly as accurate as those predicted using the elaborate OVGF method. The much more computationally efficient uDI approximation provides predictions of moderate accuracy and is recommended for computing IPs for larger molecules. We have observed that the average absolute deviations from a uDI calculation using poorer basis set (DZVP) and poorer geometry (AM1 optimization) is only slightly larger.     

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.     

Table of Feynman diagrams of the interacting Fermion Green's function

The Feynman diagrams of the Green's function expansion of fermions interacting with a nonrelativistic 2‐body interaction are displayed in first, second, and third perturbative order of the interaction as 2, 10, and 74 diagrams, respectively. A name convention for the diagrams is proposed and then used to tabulate the diagrams of fourth to seventh order. The Hartree–Fock approximation summons up 2, 8, 40, and 224 of them in first through fourth order. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2007     

Green's function calculations using non-Hartree–Fock orbitals

The single-particle Green's function is used to generate a new zero-order Hamiltonian. The idea to generate a new zero order from the previous zero order by incorporating perturbative corrections up to certain order is attractive since it allows an iterative procedure to repeatedly improve the results by decreasing the perturbation. In particular, in those cases where the Hartree–Fock Hamiltonian is not a good approximation to the full Hamiltonian and where perturbation theory usually does not produce sufficiently accurate results, one might hope that such a repetitive procedure ultimately yields an improved zero order and accurate perturbative corrections from this newly generated zero order. Two such approaches are investigated: first, one in which the ω-independent part of the self-energy is fully incorporated in the zero order and, second, one in which the correlation energy is incorporated in a one-electron potential in an average way. Numerical calculations are reported.     

Green's function analysis of the vibrational spectra of polymer chains. I. Several approaches to the problem

Various methods of solving the dynamical equation for defect polymers are discussed. It is shown that the Green's function (GF) can be conveniently used to reduce the order of the characteristic equation to the number of the degrees of freedom of a repeat unit in the case when the perturbation caused by each of the defects is localized within it. In the general case, the order of the reduced determinant depends on the range of the interaction forces between the defect unit and the regular chain. For a polymer chain containing one or two defect units, the Green's function method allows us to obtain the exact solution of the dynamical equation, while for a high concentration of randomly distributed defect units various approximation methods can be used. As a possible approach to the solution of the latter problem we suggest using either one of the versions of the perturbation theory (the dilute limit, the coherent potential approximation) or the cluster approximation.     

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.     

An investigation into the use of expansion methods in the calculation of chemical shifts and diamagnetic susceptibilities

The chemical shift and the diamagnetic susceptibility of the hydrogenic atom with magnetic dipole and origin of the external magnetic field vector potential noncoincident with the hydrogenic nucleus have been calculated from perturbation theory using a set of expansion functions whose radial parts are single exponent associated Laguerre functions. In contrast to hydrogenic expansion functions these functions give rapid convergence to the exact values of the second-order energy summations when centred at the hydrogenic nucleus. The rate of convergence is fairly insensitive to the choice of expansion function exponent.     

A parallel Green's operator for multidimensional quantum scattering calculations

A parallel algorithm for computing multidimensional scattering wave functions is introduced. The inhomogeneous scattering (Lippmann–Schwinger) equation is solved within the discrete variable representation with absorbing boundary conditions, using iterative (Krylov) methods. A parallel Green's operator enables one to distribute the wave function to orthogonal subspaces in which it is processed in parallel. Application to a model problem of electron scattering in a three-dimensional rectangular quantum wire is given. Speedup is demonstrated with an increasing number of processors and with increasing dimensions and/or sampling density. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 69: 167–173, 1998     

Conformations of polymer chains in voids with an external electric field by Green's function methods

The statistical conformations of a length of polymer chain, such as DNA, trapped in a void within a gel under the influence of an external electric field, have been studied by the method of Green's functions. Based upon a rectangular box approximation for the void shape, the method gives an explicit analytical expression for the end-to-end distance (R_x) as a function of applied field strength, number of chain segments coiled within the void, and size of a chain segment. Results of calculations show that the field compresses the entrained coil into more compact configurations, as would be expected. Such compression is believed to affect the electrophoretic mobility of a long chain molecule like DNA in a dilute gel. © 1995 John Wiley & Sons, Inc.     

Ionization potentials and electron momentum distributions for SiF4 valence-shell orbitals: an (e,2e) spectroscopic investigation and Green's function study

Ionization potentials and electron momentum distributions of SiF₄ valence-shell orbitals have been measured by means of (e,2e) spectroscopy. SCF calculations and a Green's function study have been performed on the same molecule. A comparison between experimental and theoretical results shows that a significant interaction of 2p F with 3d Si orbitals occurs.