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We present a novel generating function (GF) method for the self-condensing vinyl polymerization (SCVP) system with any initial distribution of preexisted polymers. Such a method was proven to be especially useful to investigate the semi-batch SCVP system allowing a sequence of feeding operations during the polymerization. Consequently, the number-, weight-, and z-average molecular weights as well as polydispersity index of hyperbranched polymers can be explicitly given, which are determined by predetermined feeding details and conversions in each polymerization step. These analytical results are further confirmed by the corresponding Monte Carlo simulation. Therefore the present GF method has provided a unified treatment on the semi-batch SCVP system. Accordingly, hyperbranched polymers with desired properties can be prepared by designing feeding details and presetting conversions at each step based on the present GF method.

A test system was developed for true triaxial hydraulic fracturing (HF) and gas fracturing (GF) combined with acoustic emission (AE) monitoring and 3D computerized tomography (CT) scanning reconstruction to carry out a series of HF and GF tests under different true triaxial loading conditions. The critical breakdown pressure characteristics, the AE activities, and the fracture propagation and spatial morphologies were obtained during the fracturing process. The test results showed that the GF breakdown pressure was approximately 29.0–31.3% less than the HF breakdown pressure. Moreover, for the GF specimen, the gas pore pressure induced a large number of new microfractures before critical fracture. However, both HF and GF under true axial compression can only produce simple main longitudinal fractures (LFs, along the borehole) and transverse fractures (TFs, across the borehole) that are strictly dependent on the initial true triaxial stress field. Based on the experimental observations, a hypothesis for the initiation of the HF and GF under true triaxial compression was proposed by using the maximum tangential stress on the borehole wall and the maximum axial stress along the borehole. Accordingly, analytical solutions of the critical breakdown pressure for the HF (impermeable) and GF (permeable) specimens were derived. The theoretical predictions agree well with the observations from the 3D numerical simulations and tests. Lastly, a novel pulsed GF method (dynamic loading fracturing) was proposed to overcome the dependence of the in-situ crustal stress field associated with traditional HF and GF methods (static loading fracturing). Laboratory test results showed that the dynamic pulsed GF at moderate pressure can generate a complex and random fracture network independent of the in-situ stress field in rock.

We have comprehensively investigated the frustrated J1-J2-J3 Heisenberg model on a simple cubic lattice. This model allows three regimes of magnetic order, viz., (π, π, π), (0, π, π) and (0, 0, π,), denoted as AF1, AF2, and AF3, respectively. The effects of the interplay of neighboring couplings on the model are studied in the entire temperature range. The zero temperature magnetic properties of this model are discussed utilizing the linear spin wave (LSW) theory, nonlinear spin wave (NLSW) theory, and Green’s function (GF) method. The zero temperature phase diagrams evaluated by the LSW and NLSW methods are illustrated, and are observed to exhibit different parameter boundaries. In certain regions and along the parameter boundaries, the possible phase transformations driven by the parameters are discussed. The results obtained using the LSW, NLSW, and GF methods are compared with those obtained using the series expansion (SE) method, and are observed to be in good agreement when the value of J2 is not close to the parameter boundaries. The ground state energies obtained using the LSW and NLSW methods are close to that obtained using the SE method. At finite temperatures, only the GF method is employed to evaluate the magnetic properties, and the calculated phase diagram is observed to be identical to the classical phase diagram. The results indicate that at the parameter boundaries, a temperature-driven first-order phase transition between AF1 and AF2 may occur along the boundary line. Along the AF1-AF3 and AF2-AF3 boundary lines, AF3 is less stable than AF1 and AF2. Our calculated critical temperature agrees with that obtained using Monte Carlo simulations and pseudofermion functional renormalization group scheme.

Understanding the impact of atmospheric aerosols on the global radiative balance requires knowing the refractive index (RI) of their components. Currently available methods to estimate this property from molecular structure are mostly empirical and exhibit significant errors (>10%). This work reports a more physically sound model leading to predictions within ±5% from experiment. The root mean square relative error is

Single-particle resonances are crucial for exotic nuclei near and beyond the drip lines. Since the majority of nuclei are deformed, the interplay between deformation and orbital structure near threshold becomes very important and can lead to an improved description of exotic nuclei. In this work, the Green's function (GF) method is applied to solve the coupled-channel Dirac equation with quadrupole-deformed Woods-Saxon potentials for the first time. The detailed formalism for the partial-wave expansion of the Green's function is presented. A new approach getting exact values for energies and widths of resonant states by the GF method is proposed. Numerical checks are carried out by comparing with our previous implementation of the spherical GF method and the results from the deformed complex momentum representation~(CMR), the analytical continuation of the coupling constant (ACCC), and the scattering phase shift (SPS) methods, and it is proved that the GF method is very effective and reliably for describing resonance states, no matter they are narrow or broad, spherical or deformed. Finally, Nilsson levels for bound and resonant orbitals in the halo candidate nucleus $^{37}$Mg are calculated from the deformed GF method over a wide range of deformations and some decisive hints of $p$-wave halo formation are shown in this nucleus, namely, the crossing between the configurations $1/2[321]$ and $5/2[312]$ at deformation parameter $��>0.5$ may enhance the probability to occupy the $1/2[321]$ orbital that is originated from the $2p_{3/2}$ shell., 11pages, 5 figures, 2 tables

Three methods to describe collective motion, the Random Phase Approximation (RPA), Wigner Function Moments (WFM), and Green’s Function (GF) methods, are compared in detail and their physical content analyzed in the example of a simple model, a harmonic oscillator with quadrupole-quadrupole residual interaction. It is shown that they give identical formulae for eigenfrequencies and transition probabilities of all collective excitations of the model. The exact relation between the RPA and WFM variables and the respective dynamical equations is established. The transformation of the RPA spectrum into one of WFM is explained. The very close connection of the WFM method with the GF method is demonstrated. A differential equation describing the current lines of RPA modes is established and the current lines of the scissors mode are analyzed as a superposition of rotational and irrotational components. The orthogonality of the spurious state to all physical states is proved rigorously.

Among the different potential hydrogen storage materials, borohydrides have been largely investigated because of their high gravimetric and volumetric hydrogen content. In the analysis of borohydrides, vibrational spectroscopy plays an important role since it gives information on the local structure of the BH4– ion inside the solid. Here the GF method, developed by Wilson, is used in order to determine the local symmetry of BH4– in solid borohydrides starting from their vibrational spectra. Two different cases of deformations of BH4– are considered. In the first case, the effects of small angular variations on the vibrational spectra of borohydrides will be taken into account; starting from the splitting of the bands corresponding to the deformation modes, the angular deformations will be estimated. In the second one, the BH4– under chemical pressure (in different cubic alkali halides) is considered; in this case, the symmetry of the BH4– remains Td, while the bond lengths change according to the pressure experienced. Different practical examples will be illustrated.

The proposed generalized Falk (GF) method offers an extremely simple and convenient way to solve coupled circuit-field problems in circuit simulators by transforming the discretized governing-field equations into guaranteed stable-and-passive one-dimensional (1D) equivalent-circuit systems, which are then combined with the circuit part of the overall coupled problem. More efficient than the traditional Lanczos-type methods, the GF method transforms a general finite-element system represented by a system of full matrices into an identity capacitance (mass) matrix and a tridiagonal conductance (stiffness) matrix. No positive poles are produced; all transformed matrices remain positive definite. The resulting 1D equivalent-circuit system contains only resistors, capacitors, inductors, and current sources. Several numerical examples are provided.

Heat conduction in fins and slab bodies, a traditional undergraduate topic, is treated with the Green’s function (GF) method. A variety of boundary conditions and heating conditions are included. Computer programs are described which are designed to improve student learning of the GF method. The programs are available from the author.

We develop a methodology to perform finite fault source inversions from strong motion data using Green's functions (GFs) calculated for a three-dimensional (3-D) velocity structure. The 3-D GFs are calculated numerically by inserting body forces at each of the strong motion sites and then recording the resulting strains along the target fault surface. Using reciprocity, these GFs can be recombined to represent the ground motion at each site for any (heterogeneous) slip distribution on the fault. The reciprocal formulation significantly reduces the required number of 3-D finite difference computations to at most 3NS, where NS is the number of strong motion sites used in the inversion. Using controlled numerical resolution tests, we have examined the relative importance of accurate GFs for finite fault source inversions which rely on near-source ground motions. These experiments use both 1-D and 3-D GFs in inversions for hypothetical rupture models in order (1) to analyze the ability of the 3-D methodology to resolve trade-offs between complex source phenomena and 3-D path effects, (2) to address the sensitivity of the inversion results to uncertainties in the 3-D velocity structure, and (3) to test the adequacy of the 1-D GF method when propagation effects are known to be three-dimensional. We find that given “data” from a prescribed 3-D Earth structure, the use of well-calibrated 3-D GFs in the inversion provides very good resolution of the assumed slip distribution, thus adequately separating source and 3-D propagation effects. In contrast, using a set of inexact 3-D GFs or a set of hybrid 1-D GFs allows only partial recovery of the slip distribution. These findings suggest that in regions of complex geology the use of well-calibrated 3-D GFs has the potential for increased resolution of the rupture process relative to 1-D GFs. However, realizing this full potential requires that the 3-D velocity model and associated GFs should be carefully validated against the true 3-D Earth structure before performing the inverse problem with actual data.

We built a Placido’s disc videokeratographic system intended for eye examination of small animals. To derive the corneal topology and other corneal biometric parameters we rely on precise extraction of Placido ring patterns. However, since the images from CCD are noisy, image enhancement is necessary as a pre-processing procedure in order to prevent further noise magnification during the corneal surface topography reconstruction. In this study, two popular image enhancement methods, namely Gabor Filtering (GF) and Multiscale Vessel Enhancement Filtering (MVEF), are quantitatively evaluated in the context of image enhancement before corneal reconstruction. The two methods were tested on 21 images of steel balls with different diameters. Both methods have been demonstrated to be useful for Placido ring extraction; however, the GF method showed better performance on images of steel balls with smaller diameter. Compared to the GF method, the MVEF method performed better on images of steel balls with larger diameter. The result of the present work implies that the two methods should be used cooperatively.

The experimental FTIR and FT-Raman spectra of the antiviral nucleoside analogue 5-iodo-2′-deoxyuridine (IDU) were assigned on the basis of normal coordinate analyses carried out by two different procedures, based on the Wilson GF method: a) using a valency force field transferred from molecular fragments and b) the semiempirical molecular orbital (MO) method, PM3. Assignments are discussed in the light of observed and calculated wavenumbers and isotopic shifts. The results indicate that virtually all normal modes of 5-iodo-2′-deoxyuridine involve some degree of vibrational coupling between the iodouracil base and the deoxyribose moiety. The advantages of both methods for obtaining a reliable assignment of the vibrational spectra of IDU are evaluated as a basis for the study of and drug-target interactions.

A normal coordinate analysis of the acyclovir (9-[(2-hydroxyethoxy)methyl]guanine, acycloguanosine) molecule has been carried out in two ways based on the Wilson GF method: one by constructing a force field transferred from similar molecules; the other based on molecular orbital (MO) calculations by the PM3 semiempirical method. Both analyses have been compared with the experimental IR and Raman spectra of acyclovir and one of its tetradeuterated isotopomers, and with the IR spectra of guanine and 2-methoxyethanol to assign the observed bands.

We have calculated the Debye-Waller factor (DWF) of Cu from a model that was used successfully in earlier calculations of anharmonicity by Cowley and Shukla. The present calculation has been carried out using quasiharmonic theory, the lowest-order (λ2) anharmonic perturbation theory, and a Green's function (GF) method which sums an infinite series of the λ2−type anharmonic terms. The static approximation ω → 0 in the cubic contribution to the self-energy of the GF, introduced in the earlier work on the DWF by Shukla and Hubschle is further justified by showing that in the high-temperature limit the exact results for the λ2 anharmonic contributions (cubic and quartic) to the Helmholtz free energy are given in this approximation. Results for the DWF are also obtained for a modified version of the Morse potential with λ2 perturbation theory (PT) and the GF method. The GF results are in excellent agreement with the experimental Mossbauer and X-ray data in the entire temperature range, 300 K  T  120...

The inspiral of a stellar-mass astrophysical object into a massive black hole may be modeled within perturbation theory of General Relativity via the so-called self-force. In this paper, we present a novel method for the calculation of the self-force which is based on the Green function (GF) of the wave equation satisfied by the field created by the smaller object. We review the results in [M. Casals, S. Dolan, A. C. Ottewill and B. Wardell, Phys. Rev. D 88 (2013) 044022; B. Wardell, C. R. Galley, A. Zenginoğlu, M. Casals, S. R. Dolan and A. C. Ottewill, Phys. Rev. D 89 (2014) 084021] on the GF and the self-force on a scalar charge (as a model for the gravitational case) moving on a Schwarzschild black hole spacetime. This GF method offers an appealing geometrical insight into the origin of the self-force and is a promising candidate for practical self-force calculations.

In the study of neurosciences, and of complex biological systems in general, there is frequently a need to fit mathematical models with large numbers of parameters to highly complex datasets. Here we consider algorithms of two different classes, gradient following (GF) methods and evolutionary algorithms (EA) and examine their performance in fitting a 9-parameter model of a filter-based visual neuron to real data recorded from a sample of 107 neurons in macaque primary visual cortex (V1). Although the GF method converged very rapidly on a solution, it was highly susceptible to the effects of local minima in the error surface and produced relatively poor fits unless the initial estimates of the parameters were already very good. Conversely, although the EA required many more iterations of evaluating the model neuron's response to a series of stimuli, it ultimately found better solutions in nearly all cases and its performance was independent of the starting parameters of the model. Thus, although the fitting process was lengthy in terms of processing time, the relative lack of human intervention in the evolutionary algorithm, and its ability ultimately to generate model fits that could be trusted as being close to optimal, made it far superior in this particular application than the gradient following methods. This is likely to be the case in many further complex systems, as are often found in neuroscience.

Ultraviolet resonance Raman spectra of inosine and its deuterated species (C8-d, Nl-d and C8-, Nl-deuterated derivatives) in aqueous solution have been collected (λexc = 257 and 281 nm) and reported in the spectral region between 400 and 1800 cm−1. The observed vibrational modes have been assigned using the Wilson GF method with an empirical harmonic valence force field and a non-redundant set of internal co-ordinates. This normal mode analysis is based on previous work on the vibrational assignments of the hypoxanthine nucleic base and the furanose ring with various geometries. Here the analysis has been carried out by using three different nucleoside conformations as determined from x-ray studies of molecular crystals. The most striking features of the present work are: (i) the assignments of inosine vibrational modes are very different from those of the hypoxanthine base; (ii) the best fit between measured and calculated vibrational wavenumbers and their isotopic shifts upon selective deuteration has been obtained from inosine structures containing an N-type sugar (C3′-endo) associated with a low-anti base. In addition, we have tested the reliability of the force field in the case of inosine residues involved in a canonical A-RNA helical structure by comparing the calculated wavenumbers of the most characteristic vibrational modes with those observed in off- and on-resonance Raman spectra of poly(rI).

The chapter summarizes important concepts from the theory of molecular vibrations of which normal coordinate analysis is a part. Section 2 describes how molecular vibrations or, more precisely, harmonic oscillators are treated in classical mechanics and how normal coordinates are defined. The quantum mechanics of coupled harmonic oscillators, which is based on normal coordinates, is developed in section 3, together with the selection rules for infrared and Raman spectroscopy. It is followed by excursions into some applications of symmetry concepts and into the domain of real molecules where harmonic oscillators are satisfactory approximations at best (anharmonicity and large-amplitude internal motions). Section 4 introduces the commonly used internal and symmetry coordinates, the transformations between them and the Cartesian and normal coordinates, Wilson's GF method of normal coordinate analysis, and discusses redundant sets of internal coordinates. A short survey of force fields past and present (central, valence, and Urey-Bradley force fields and force fields from semiempirical and ab initio calculations) is given in section 5. The invariance of internal coordinates, the transformation from internal to Cartesian coordinates, and some aspects of redundant coordinates are detailed in appendices. Keywords: anharmonicity; Cartesian coordinates; force constant; force field; harmonic oscillator; internal coordinates; molecular vibrations; normal coordinate analysis; potential energy distribution; quantum mechanics; redundant coordinates; relative intensity; symmetry coordinates; vibrational energy levels; vibrational wave functions; vibrational frequencies; vibrational selection rules

The idea of the thermodynamic Green Functions (GF, GF’s) was originally introduced in statistical physics in order to describe the response of a considered system with respect to an external perturbation. The GF’s allow us to calculate the average values of observables in a direct way, equivalent to the average values obtained by means of the statistical operator procedure without, however, the need to know the explicit form for the statistical operator. The thermodynamic GF technique widely used in solid state physics has been confirmed by successful results in this area. At present, the GF method is applied to electrochemical problems for which it seems to be an equally satisfactory and convenient tool for investigations.

The vibrational spectra of a synthetic purine nucleoside with known antiviral activity, 9-β-D-arabino-furanosyladenine hydrochloride (ara-A.HCl) are reported. The Fourier transform infrared (FT-IR) and Fourier transform Raman (FT-Raman) spectra were recorded in the 4000-30 cm−1 spectral region. The harmonic frequencies and potential energy distributions (PED) of the vibrational modes of ara-A.HCI were calculated by two different methods: a classical molecular mechanics method and a semiempirical molecular orbital (MO) method, PM3. The results of both computational methods, based on the Wilson GF method, are compared with observed spectra, and an assignment of the vibrational modes of ara-A.HCl is proposed on the basis of the potential energy distributions (PED). It is found that the wavenumbers can be calculated with remarkable accuracy (≈1% deviation in most cases), with the classical mechanics method, by transferring a sufficiently large set of available harmonic force constants, thus permitting a reliable assignment. The semiempirical MO method, PM3, is found to be useful for the assignment of experimental frequencies although it is less accurate (≈10% deviation). IR intensities calculated by this method did not coincide with the experimental values. Certain out-of-plane vibrations in the base, not reported in previous studies, have been observed. The performance of both methods was related to the crystallographic and ab initio data available. Previous normal coordinate calculations for the adenine base and the nucleoside 5′-dGMP are compared with our results and discussed, in relation to the crystal structure of Ara-A.HCl.

The FT-inftared and FT-Raman spectra of the 1:1 complex of cadmium (II)-5’-GMP in solid phase have been recorded in order to obtain information on the metal-nncleotide interactions. A previous normal coordinate analysis for this compound has already been done applying a refined force field and the Wilson GF method[1] Now optimized geometry and vibrational calculation for the system have also been performed with the semiempirical PM3 method A comparison of these theoretical results with the observed spectra has been made in order to assign the vibrational bands.

Ultraviolet resonance Raman scattering spectra from aqueous solutions of hypoxanthine and its deuterated species (C8-deuterated, N-deuterated and C8-, N-deuterated derivatives) have been collected and reported in the spectral region between 400 and 1800 cm−1. The laser excitation wavelengths at 281 nm and 257 nm correspond to preresonance and pure resonance conditions, respectively, with the purine strongly allowed π → π* electronic transition: thus the observed experimental Raman features mainly correspond to inplane vibrational modes. The latter were then assigned according to the Wilson GF method by using an empirical harmonic valence force field. Normal mode calculations are based on a non-redundant set of internal coordinates. The calculated vibrational mode wavenumbers and their isotopic shifts upon selective deuterations are in good agreement with the experimental data. The present normal mode analysis rests on the transferability of the guanine and adenine force constants proposed in recent works based on resonance Raman spectroscopy and neutron inelastic scattering data from these major purine bases.

Using only the angular momentum representation a full potential all electron method for the calculation of the electronic structure, solution of Poisson’s equation, and the total energy of ordered and disordered metals and alloys is presented within the multiple scattering theory. The method is applied to niobium and gives excellent agreement with other calculations and experiments.

Neutron inelastic scattering (NIS) and ultraviolet resonance Raman scattering (RRS) experimental data have been obtained from purine nucleic base samples, i.e., adenine, C8-deuterated adenine, and a (N9, C6-amino) - deuterated species. The two types of spectra have made it possible to carry out complete theoretical assignments of the vibrational modes of adenine and its derivatives. Normal mode wavenumbers as well as atomic displacements have been calculated by the Wilson GF method, using a valence force field and a non-redundant set of internal coordinates

Resonance Raman scattering (RRS) spectra of the nucleic acid bases give valuable information on the in-plane modes of vibration, especially those involved in the double-bond stretching motions (1). In this work we report RRS spectra from the purine bases of nucleic acids: adenine, guanine, 2-aminoadenine and hypoxanthine as well as 2’-deoxyinosine, taken with various laser UV excitation wavelengths, i.e. 223-, 257- and 281 nm. These spectra, along with those from the isotopic derivative compounds, obtained by selective deuteration at the C8 site and at the labile hydrogens, gave the opportunity to carry out a normal coordinate analysis of the in-plane modes of these purine bases. The calculations are based on the Wilson GF method and a on a non-redundant set of internal coordinates. Numerical computations were performed by using a home-made interactive program (BORNS).

The mathematical technique for studying the dynamics of perturbed systems, known as the Green’s function (GF) method, was first applied to solid state problems by Lifshitz [3.1] and Koster andSlater [3.2]. They have stimulated the development of the GF method as a powerful tool in the quantum theory of scattering [3.3–5]. Although the general formulation of the GF method in solid state physics has become a textbook subject [3.6’8], only in recent decades has the method been extensively used in the dynamics of real systems, such as solids with defects, disorder, or boundary surfaces [3.9,10].

A brief but clear and complete account is given for two analytical methods of calculating correlation functions in quantum statistical physics from first principles—the widely used method of two-time temperature Green's functions (the GF method) and a new, "direct algebraic" (DA) method. The mathematical and technical clarity and simplicity of the DA method and its resulting practical value are demonstrated for the five most widely used models in quantum statistical physics. Since the DA method is an exactly self-consistent method (in the sense that the expansion coefficients in the equations of motion are chosen from the requirement that the Jacobi operator identity be satisfied exactly), it in principle affords the possibility of an internal check, which is not possible in the GF method. Like the GF method, the DA method permits calculation of the spectra of possible elementary excitations and, hence, of the density of single-particle energy states corresponding to them.

A recently published semi-implicit Runge-Kutta algorithm has been extended to compute parameter sensitivities in dynamical systems. The decomposed direct method (DDM) of sensitivity analysis results in a general and compact FORTRAN program, well-suited even for small computers. The straightforward DDM provides more accurate sensitivities than does the numerically efficient, but more involved Green′s function (GF) approach. Results show that for kinetics problems of moderate complexity the proposed procedure is competitive with the well known GF method in terms of computer-time requirements.

The Raman and IR spectra of the water solution of 30 SiO 4 , 28 SiO 4 , and GeO 4 anions have been recorded. Bands wavenumbers have been recognized as normal vibrations frequencies in isolated anions with T d symmetry. The GVFF force constants have been calculated due to the usage of these frequencies and the application of triangular matrix approximation in GF method. Molecular approximation with C s point symmetry of XO 4 tetrahedra has been used for internal force constants in SiO 4 and GeO 4 anions in the alkaline earth orthosilicates and orthogermanates crystals. The net charge on Si,Ge and O atoms as well as the ionicity of the Si-O and Ge-O bonds were obtained according to the model proposed by K. Ohwada. The nature of the mentioned bonds is discussed due to the comparison of these values for Si-O and Ge-O bonds in solution and in crystals with the alkaline earth cations.

The Green's-function (GF) method has been applied to the investigation of the electronic structure of defects in A/sup N/B/sup 8-N/ semiconductors. Localized states in the forbidden gap and resonance states in the allowed bands induced by group V and group VI impurities, mono- and multivacancies, and complexes of point defects have been considered. It has been shown that the semiempirical variant of the GF method in a basis of localized orbitals makes it possible to describe the behavior of defect levels in the framework of a single approximation for an extensive list of objects on a qualitative or semiquantitative level. Several procedures for the application of the extended defect molecule method, which makes it possible to combine the advantages of the GF and molecular cluster methods, have been discussed.

The relative Raman intensities of the fundamental vibrational modes of the tetrahedral oxoanions SO42−, ClO4−, PO43 and of the plane trigonal oxoanions CO32− and NO3− have been measured from polycrystalline samples. Using the bond polarizability theory, the vibrational eigenvectors of F2(Td) and E' (D3h) species of these anions in different crystalline environments are derived from the above Raman intensity data. The results are consistent with those obtained through Wilson's GF method, although there exists a multiplicity of solutions. Relative values of SO, ClO, PO and NO bond polarizability parameters are reported. Comparison of these results with liquid state absolute values from the literature enables us to suggest a standard for measuring absolute Raman intensities in the solid phase. Copyright © 1980 Heyden & Son Ltd

A hermitian dynamical matrix in terms of Cartesian displacement coordinates is derived starting from a harmonic force constant matrix expressed in terms of the Wilson's internal displacement coordinates. The method accounts both for inter- and intramolecular coupling. The frequency spectrum can be obtained from the dynamical matrix without directly solving the secular equation by the application of Dean's negative eigenvalues theorem.

A general method of calculating accurate and useful many‐electron wavefunctions for atoms and molecules has previously been described. This general method besides achieving lower energies than do the Hartree‐Fock and unrestricted Hartree‐Fock methods still yields an independent particle interpretation of the many‐electron wavefunction, correctly describes dissociation of molecules and solids, and can be used on systems of any total spin. There are several possible approaches for the general method; here we concentrate on one of these, GF method, which is equivalent to optimizing the orbitals in the unrestricted Hartree‐Fock method after spin projection rather than before. The detailed equations (the GF equations), for the GF method are derived here for all values of total spin and number of particles.

A method is proposed for applying the theory of generalized group functions to SCF-GF calculations with large basis sets. A simple procedure for localising the SCF-MO's resulting from a standard SCF calculation is described, with applications to H2O, NH3, CH4 and H2O2. Results compare quite favourably with those obtained by the usual GF method. It is shown that when basis functions are the SCF-MO's and there are only two functions per group, the GF approach is practically equivalent to a configuration interaction treatment where only double excitations within the groups are considered.

The GF method for obtaining accurate many‐electron wavefunctions was described in a previous paper. In this paper, some of the properties of this method are explored, and it is shown that the Hellmann–Feynman, Koopmans', and Brillouin's theorems apply to GF wavefunctions. Calculations are reported on Li_2, CH_4, and CH_3 in order to demonstrate some aspects of the method.

12 pages; International audience; The rotational projection operator previously introduced [12] serves here to build up an explicit formula for the non-relativistic energy associated with a spin-projected determinant. Corresponding orbitals are used throughout, and interchange of configuration space and Euler angles integrations provides rapidly an energy expression previously obtained either by use of Löwdin's projector (via first- and second-order density matrices (8)) or by mixed space and spin permutation operator (GF method (9)).

The magnetic hyperfine splitting constants, aJ, from theoretical calculations on the 32S, 22P, 32P, and 32D excited states of the Li atom are reported. The wave functions were calculated using the author's GF method, which corresponds to optimizing the orbitals of a Slater determinant after spin projection. Thus the wave functions include core polarization, but no appreciable correlation. For the 22P state we calculate a1 / 2=0.2206 a.u. (45.74 Mc/sec for 7Li) which is in good agreement with the experimental value, 0.2227 ± 0.0017 a.u. (46.17 ± 0.35 Mc/sec for 7Li), and the value from configuration interaction calculations, 0.2206. Thus for the Li atom core polarization accounts for most of the error in the Hartree-Fock values of aJ. These calculations yield 〈1 / r^3〉 and spin density, Q(0), in agreement with other accurate theoretical calculations, and in disagreement with the values found using level crossing experiments, indicating that the interpretation of the level crossing experiments in terms of 〈1 / r^3〉 and Q(0) may not be correct. It is found that for both the unrestricted Hartree-Fock and GF methods the use of a different Hamiltonian for each electron leads to virtual orbitals which are good approximations to the actual orbitals of excited states.

The form of the operator O ii , which may be used to determine expectation values of any spin-free operator from Goddard's GI wavefunctions, is derived without the use of group theory. Furthermore in the most frequently used case (the GF method) the form of the operator may be readily determined for an arbitrary number of electrons by the use of a computer.

The vibration frequencies of structures which typify ortho silicate, pyro silicate, chain and sheet have been calculated using an extension of the Wilson GF method, in order to confirm the general interpretation of Raman spectra of binary glasses.

Wavefunctions from spin‐generalized SCF calculations using the GF method, are reported for H2O, OH, and O. Three different basis sets are examined for each of these systems. The shapes and angles between the localized GF orbitals are described in some detail. It is shown that the resulting GF orbitals change in a chemically reasonable manner as we proceed from O to OH to H2O. The dipole and quadrupole moments, electric fields, field gradients, densities, and potentials are reported for the GF wavefunctions.

The adducts of benzoic, p-toluic, mesitoic and cinnamic acids with SbCl5, AlCl3, TiCl4, SnCl4 and ZnCl2 have been prepared, and the IR. absorption spectra of the compounds, in the solid state or in solution in CCl4, studied. The lowering of the carbonyl frequency shows that the acceptor is linked by a dative bond to the carbonyl oxygen atom acting as donor. A calculation, by the Wilson GF method, of force constants and their contributions in potential energy of vibration for monomeric benzoic acid and its adduct with TiCl4 confirmed that the carbonyl stretching force constant, and therefore the CO bond order, is lowered by the formation of the adducts.

The infrared spectra of HNCO and HNCS are quite complicated and the band assignments are still disputed. In this note we try to add some arguments based on the normal coordinate analysis in favour of the recently proposed assignments1,2. The analysis was performed by the standard GF method with the frequency assignment due to Ashby and Werner1 for HNCO, and Durig and Wertz for HNCS2. The frequencies only were used in the iteration procedure and the results are given in the Table.

The standard way of calculating the normal coordinates of a molecule is the well-known GF method of Wilson.(1) In this method one starts from the harmonic force constants of the molecule in order to obtain the different normal coordinates of vibrations and their corresponding energies. These force constants are the second derivatives of the molecular total energies with respect to the different internal coordinates at equilibrium geometry. Therefore, the force constants are obtained by computing the energy hypersurface; in the case of a larger molecule this task is rather time-consuming. Differentiation of the total energy is carried out by calculating the first derivative (gradient), usually analytically, according to the method developed by Pulay,(2) while the second derivative is obtained by numerical methods.(3) It should be noted that, is some cases, both differentiations have been performed analytically after fitting the calculated potential hypersurfaces to quadratic potential-energy functions with the help of least-square methods.(4)

The standard GF method was employed to determine the valence force constants of the dithionitronium cation (NS 2 + ), using the experimental IR and Raman frequencies. The calculation force constants ( f NS = 8.256, f NS,NS = 0.663 and f SNS = 0.236 mdyn A −1 ) reproduce well the 32 SN 34 S + frequencies and suggest a bond order greater than 1, as well as a delocalization of the π electrons in the trinuclear system. CNDO/2 calculations reveal positive charges on sulfur atoms, contrary to simple valence electron count rules. The CNDO/2 preferred geometry is D ∞h rather than a C 2V cyclic structure, in agreement with the X-ray diffraction data. A very important S⋯S interaction has been found.