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Method variational

Chemisoq)tion bonding to metal and metal oxide surfaces has been treated extensively by quantum-mechanical methods. Somoijai and Bent [153] give a general discussion of the surface chemical bond, and some specific theoretical treatments are found in Refs. 154-157 see also a review by Hoffman [158]. One approach uses the variation method (see physical chemistry textbooks) ... [Pg.714]

The purpose of this chapter is to provide an introduction to tlie basic framework of quantum mechanics, with an emphasis on aspects that are most relevant for the study of atoms and molecules. After siumnarizing the basic principles of the subject that represent required knowledge for all students of physical chemistry, the independent-particle approximation so important in molecular quantum mechanics is introduced. A significant effort is made to describe this approach in detail and to coimnunicate how it is used as a foundation for qualitative understanding and as a basis for more accurate treatments. Following this, the basic teclmiques used in accurate calculations that go beyond the independent-particle picture (variational method and perturbation theory) are described, with some attention given to how they are actually used in practical calculations. [Pg.4]

B) VARIATIONAL METHODS PROVIDE UPPER BOUNDS TO ENERGIES... [Pg.2186]

MacDonald J K L 1933 Successive approximations by the Rayleigh-Ritz variation method Phys. Rev4Z 830-3... [Pg.2200]

Reddy, J. N., 1986. Applied Functional Analysis and Variational Methods in Engineering, McGraw-Hill, New York. [Pg.110]

Variational methods - theoretically the variational approach offers the most powerful procedure for the generation of a computational grid subject to a multiplicity of constraints such as smoothness, uniformity, adaptivity, etc. which cannot be achieved using the simpler algebraic or differential techniques. However, the development of practical variational mesh generation techniques is complicated and a universally applicable procedure is not yet available. [Pg.195]

The complexity of molecular systems precludes exact solution for the properties of their orbitals, including their energy levels, except in the very simplest cases. We can, however, approximate the energies of molecular orbitals by the variational method that finds their least upper bounds in the ground state as Eq. (6-16)... [Pg.202]

The Seetion entitled The BasiC ToolS Of Quantum Mechanics treats the fundamental postulates of quantum meehanies and several applieations to exaetly soluble model problems. These problems inelude the eonventional partiele-in-a-box (in one and more dimensions), rigid-rotor, harmonie oseillator, and one-eleetron hydrogenie atomie orbitals. The eoneept of the Bom-Oppenheimer separation of eleetronie and vibration-rotation motions is introdueed here. Moreover, the vibrational and rotational energies, states, and wavefunetions of diatomie, linear polyatomie and non-linear polyatomie moleeules are diseussed here at an introduetory level. This seetion also introduees the variational method and perturbation theory as tools that are used to deal with problems that ean not be solved exaetly. [Pg.2]

In applying quantum mechanics to real chemical problems, one is usually faced with a Schrodinger differential equation for which, to date, no one has found an analytical solution. This is equally true for electronic and nuclear-motion problems. It has therefore proven essential to develop and efficiently implement mathematical methods which can provide approximate solutions to such eigenvalue equations. Two methods are widely used in this context- the variational method and perturbation theory. These tools, whose use permeates virtually all areas of theoretical chemistry, are briefly outlined here, and the details of perturbation theory are amplified in Appendix D. [Pg.57]

This upper-bound property forms the basis of the so-ealled variational method in whieh trial wavefunetions are eonstrueted ... [Pg.58]

Variational methods, in particular the linear variational method, are the most widely used approximation techniques in quantum chemistry. To implement such a method one needs to know the Hamiltonian H whose energy levels are sought and one needs to construct a trial wavefunction in which some flexibility exists (e.g., as in the linear variational method where the Cj coefficients can be varied). In Section 6 this tool will be used to develop several of the most commonly used and powerful molecular orbital methods in chemistry. [Pg.59]

The relative strengths and weaknesses of perturbation theory and the variational method, as applied to studies of the electronic structure of atoms and molecules, are discussed in Section 6. [Pg.62]

This Introductory Section was intended to provide the reader with an overview of the structure of quantum mechanics and to illustrate its application to several exactly solvable model problems. The model problems analyzed play especially important roles in chemistry because they form the basis upon which more sophisticated descriptions of the electronic structure and rotational-vibrational motions of molecules are built. The variational method and perturbation theory constitute the tools needed to make use of solutions of... [Pg.73]

The variational method ean be used to optimize the above expeetation value expression for the eleetronie energy (i.e., to make the funetional stationary) as a funetion of the Cl eoeffieients Cj and the ECAO-MO eoeffieients Cv,i that eharaeterize the spin-orbitals. However, in doing so the set of Cv,i ean not be treated as entirely independent variables. The faet that the spin-orbitals ([ti are assumed to be orthonormal imposes a set of eonstraints on the Cv,i ... [Pg.457]

A. Variational Methods Such as MCSCF, SCF, and Cl Produce Energies that are Upper Bounds, but These Energies are not Size-Extensive... [Pg.487]

This characteristic is commonly referred to as the bracketing theorem (E. A. Hylleraas and B. Undheim, Z. Phys. 759 (1930) J. K. E. MacDonald, Phys. Rev. 43, 830 (1933)). These are strong attributes of the variational methods, as is the long and rich history of developments of analytical and computational tools for efficiently implementing such methods (see the discussions of the CI and MCSCF methods in MTC and ACP). [Pg.487]

B. Non-Variational Methods Sueh as MPPT/MBPT and CC do not Produee Upper Bounds, but Yield Size-Extensive Energies... [Pg.489]

In eontrast to variational methods, perturbation theory and eoupled-eluster methods aehieve their energies from a transition formula < H P > rather than from an expeetation value... [Pg.489]

Most of the techniques described in this Chapter are of the ab initio type. This means that they attempt to compute electronic state energies and other physical properties, as functions of the positions of the nuclei, from first principles without the use or knowledge of experimental input. Although perturbation theory or the variational method may be used to generate the working equations of a particular method, and although finite atomic orbital basis sets are nearly always utilized, these approximations do not involve fitting to known experimental data. They represent approximations that can be systematically improved as the level of treatment is enhanced. [Pg.519]

This relationship is often used for computing electrostatic properties. Not all approximation methods obey the Hellmann-Feynman theorem. Only variational methods obey the Hellmann-Feynman theorem. Some of the variational methods that will be discussed in this book are denoted HF, MCSCF, Cl, and CC. [Pg.12]

Elliot C.M., Ockendon J.R. (1982) Weak and variational methods for moving boundary problems. Pitman, Research Notes Math. 59. [Pg.377]

Morel J.-M., Solimini S. (1995) Variational methods in image segmentation. Birkhauser, Boston, Basel, Berlin. [Pg.382]

Mosolov V.P., Myasnikov P.P. (1971) Variational methods in ffow theory of perfect-visco-plastic media. Moscow Univ. (in Russian). [Pg.382]

Vainberg M.M. (1972) A variational method and monotonous operators method. Nauka, Moscow (in Russian). [Pg.385]

Washizu K. (1968) Variational methods in elasticity and plasticity. Perg-amon Press. [Pg.385]

Particle Size. Wet sieve analyses are commonly used in the 20 )J.m (using microsieves) to 150 )J.m size range. Sizes in the 1—10 )J.m range are analyzed by light-transmission Hquid-phase sedimentation, laser beam diffraction, or potentiometric variation methods. Electron microscopy is the only rehable procedure for characterizing submicrometer particles. Scanning electron microscopy is useful for characterizing particle shape, and the relation of particle shape to slurry stabiUty. [Pg.349]

Prenter, P. M. Splines and Variational Methods, Wiley, New York (1975). [Pg.423]

It has been established, that both DN and Ibp form complex compounds with ions Eu(III), Sm(III), Tb(III) and Dy(III), possessing luminescent properties. The most intensive luminescence is observed for complex compounds with ion Tb(III). It has been shown, that complexation has place in low acidic and neutral water solutions at pH 6,4-7,0. From the data of luminescence intensity for the complex the ratio of component Tb Fig was established equal to 1 2 by the continuous variations method. Presence at a solution of organic bases 2,2 -bipyridil, (Bipy) and 1,10-phenanthroline (Phen) causes the analytical signal amplification up to 250 (75) times as a result of the Bipy (Phen) inclusion in inner coordination sphere and formation of different ligands complexes with component ratio Tb Fig Bipy (Phen) = 1 2 1. [Pg.386]


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A More General Variation Method

A non-variational method

Analytic methods variation with temperature

Approximation methods Rayleigh variational principle

Approximations variational methods

Cluster variation method continuous

Cluster-variation method

Comparison of the Variation and Perturbation Methods

Complex Kohn variational method

Component-based variation reduction method

Configuration interaction linear variations method

Constrained space orbital variation method

Continuous variations, method

Contrast variation, method

Design The Method of Variation

Dirac-Slater discrete-variational method

Direct variational methods

Direct variational methods resonance calculations

Discrete variational Xa method

Discrete variational method

Discrete variational method characteristics

Discrete variational method relativistic effects

Discrete variational methods basis functions

Discrete variational methods calculations

Discrete variational methods chemical bonding

Discrete variational methods computational method

Discrete variational methods computations

Discrete variational methods description

Discrete variational methods development

Discrete variational methods efficiency

Discrete variational methods electronic structures

Discrete variational methods first-principles calculations

Discrete variational methods method

Discrete variational methods model clusters

Discrete variational methods procedure

Discrete variational methods results

Discrete variational methods valency

Discrete variational multielectron method

Electronic structure methods variational problem

Electronic structure variational methods

Extension of the Variation Method

Harmonic oscillator and variation method

Harris variational method

Helium atom variation method application

Hylleraas variational method

Jobs Method of Continuous Variations

Kohn variational method

Kohn variational method formation

Kohn variational method scattering

Linear variation method

Linear variation method formulation

Linear variation method hamiltonian

Linear variation method matrix

Mass variation method

Matrix Formulation of the Linear Variation Method

Method of Contrast Variation

Method of continuous variations

Method of variation

Method parameters typical variations

Methods variation-iteration

Molecular potential variational method

NON-VARIATIONAL METHODS WITH SLATER DETERMINANTS

Non-variational method

Particle in a box and variation method

Perturbation theory related to variation method

Rayleigh-Ritz variational method

Reaction path variational method

Reduced variational space method

Resonance energies direct variational methods

Ritz variation method

Scattering theory matrix variational method

Stochastic variational method

Symmetry constraints linear variation method

THE VARIATION METHOD

The Feynman-Hibbs Variational Method

The Method of Continuous Variation

The matrix variational method

The variational method

Time-dependent variational principle method

Time-independent variational methods

Trial variation function method)

Variable time method (variation of P with t)

Variation constants, method

Variation method

Variation method applied to harmonic oscillator

Variation method applied to helium atom

Variation method applied to hydrogen atom in electric field

Variation method applied to particle in a box

Variation method calculations

Variation method excited state energies

Variation method for helium atom

Variation method formulation

Variation method ground state eigenfunctions

Variation method ground state energy

Variation method helium application

Variation method nonlinear

Variation method orbital approximation

Variation method shielding

Variation method spirit

Variation method, in quantum

Variation of constants method

Variation of parameters method

Variation-perturbation method

Variational Method for a Three-dimensional Elasticity Problem

Variational collapse method

Variational method Lagrange-Euler equation

Variational method states

Variational method time-dependent

Variational methods 324 Subject

Variational methods Schrodinger equation

Variational methods for continuum states

Variational methods projection operator

Variational methods, complex atoms

Variational methods. Coupled Hartree-Fock theory

Variational principle,energetic method

Variational quantum Monte Carlo method

Variational theory of linearized methods

Variations of the Conventional Polarographic Method

Variations of the Juge-Stephan Method

Variations of the Method

Variations on the Standard Methods

Virtual Work Equation, Variational Methods and Energy Principles

Wannier variational method

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