Discrete Green function

Seideman T and Miller W H 1992 Quantum mechanical reaction probabilities via a discrete variable representation-absorbing boundary condition Green function J. Chem. Phys. 97 2499... 

In order to compute the discrete Green-Kubo expression for D we must evaluate correlation function expressions of the form (v L/v ). Consider... 

Reaction Probabilities via a Discrete Variable Representation- Absorbing Boundary Condition Green Function. 

A very effective method to describe scattering and transport is the Green function (GF) method. In the case of non-interacting systems and coherent transport single-particle GFs are used. In this section we consider the matrix Green function method for coherent transport through discrete-level systems. 

The coupling of a finite cluster with bulk metal material is treated through a Green function s method. First, the density of states (DOS) of the bulk contact is calculated as indicated above. Next, the influence of the DOS of the bulk contact on the broadening and shifting of the discrete energy levels of the molecular orbitals (MO) of the cluster is accounted for via our DFT-Green function approach, as will be explained below. This yields the total DOS of the cluster as affected by the continuum. 

In this paper we have taken the Green function for the Fues model potential from [8,9]. The angular part of G (ri, V2) is simply the product of spherical harmonics, while for the radial part we have taken an expansion in Sturm functions, which have only a discrete spectrum ... 

The calculation of the Fourier space contribution is the most time consuming part of the Ewald sum. The essential idea of P M is to replace the simple continuous Fourier transformations in (3) by discrete Fast Fourier Transformations, that are numerical faster to calculate. The charges are interpolated onto a regular mesh. Since this introduces additional errors, the simple Coulomb Green function as used in the second term in (3), is cleverly adjusted in... 

The continuous electronic states of macroscopic contacts affect the discrete electronic states of isolated molecules calculated by solving the Schrodinger equation thus, a combination of Density Functional Theory and Green function theory (DFT-GF) is used to obtain the electrical properties of the metal-molecule-metal junction. 

While for the exact dielectric problem only multipolar expansions are available, the COSMO approximation allows for the analytical calculation of the Green function in the case of a spherical cavity. Nevertheless, all implementations of COSMO are developed for the general case of a molecular shaped cavity, since any restriction to regular cavities such as spheres or ellipsoids is too severe a limitation. In this case the COSMO screening charges have to be calculated numerically. This requires a discretization of the cavity surface into... 

The transition matrix relates the expansion coefficients of the incident and scattered fields. The existence of the transition matrix is postulated by the T-Matrix Ansatz and is a consequence of the series expansions of the incident and scattered fields and the linearity of the Maxwell equations. Historically, the transition matrix has been introduced within the null-field method formalism (see [253,256]), and for this reason, the null-field method has often been referred to as the T-matrix method. However, the null-field method is only one among many methods that can be used to compute the transition matrix. The transition matrix can also be derived in the framework of the method of moments [88], the separation of variables method [208], the discrete dipole approximation [151] and the point matching method [181]. Rother et al. [205] foimd a general relation between the surface Green function and the transition matrix for the exterior Maxwell problem, which in principle, allows to compute the transition matrix with the finite-difference technique. 

In Equation (4.12) the discretization of velocity and pressure is based on different shape functions (i.e. NjJ = l,n and Mil= l,m where, in general, mweight function used in the continuity equation is selected as -Mi to retain the symmetry of the discretized equations. After application of Green s theorem to the second-order velocity derivatives (to reduce inter-element continuity requirement) and the pressure terms (to maintain the consistency of the formulation) and algebraic manipulations the working equations of the U-V-P scheme are obtained as... 

Any data set that consists of discrete classification into outcomes or descriptors is treated with a binomial (two outcomes) or multinomial (tliree or more outcomes) likelihood function. For example, if we have y successes from n experiments, e.g., y heads from n tosses of a coin or y green balls from a barrel filled with red and green balls in unknown proportions, the likelihood function is a binomial distribution ... 

It should be pointed out that, as used here, the Green s function method implicitly assumes the existence of a continuum or quasicontinuum of states to be coupled with some discrete (or set of discrete) state(s). Thus, the generalized perturbation treatment is valid only in the statistical limit for the isolated molecule. 

In this section we explicitly demonstrate how the antiresonant line shapes characteristic of configuration interaction between a discrete BO state and BO continuum can be obtained from the Green s function formalism. We restrict attention to the case of one discrete BO state interacting with one BO continuum. We shall assume that the ground state is connected to both the discrete BO state and the BO continuum by nonvanishing dipole matrix elements. 

The advantage of the IEF-PCM formulation in this respect is that the surface between the two dielectrics is hidden within the Green s function and only the explicit description of the molecular cavity is needed as for bulk IEF-PCM. Although the IEF-PCM implementation was able to overcome several numerical problems as a result of the explicit description of the dielectric interface, the BEM discretization technique for those tesserae in close proximity of the sharp interface could still potentially lead to unphysical divergences. 

The first order Green s function, from the Dyson series and taking into account the exponential decay of electron states, contains two discrete terms, which can be written in terms of the Bardeen matrix elements... 

Duijnen, P.Th. van, Greene S.N. and Richards N.G.J., Time dependent density functional the-ory/discrete reaction field spectra of open-shell systems the visual spectrum of [Felu (PyPepSjz ] in aqueous solution. J.Chem.Phys. (2007) 127 045105. 

In order to implement the approximation to the optical potential we must choose a form for the potential Vopul (7.132) in which the Green s function is calculated (7.135). We choose different types of potential for the discrete and continuum channels of Q space, projected respectively by Q and Q+. For Q space we choose the average potential for the target state i). Its coordinate representation is obtained from (7.48,7.63,7.66)... 

In order to develop the numerical analogs of the electromagnetic field integral representations, we have to discretize, also, the field components and the Green s tensors within the anomalous domain D of integration. We can treat all integral representations, considered in this chapter, as operators acting on the vector functions, x,... 

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