Approximate solutions of equations

Equation 6 was used to correlate the data of this paper however, a more accurate approximation solution of Equation 3 is given by ... 

In view of the above as well as the fact that all the various approximate solutions of Equation 3 give about the same answer when the reactant concentration is low, it did not seem worthwhile to seek better accuracy in the solution of Equation 3. 

The particle-in-a-box problem, which we considered qualitatively in Chapter 5, turns out to be one of the very few cases in which Schrodinger s equation can be exactly solved. For almost all realistic atomic and molecular potentials, chemists and physicists have to rely on approximate solutions of Equation 6.8 generated by complex computer programs. The known exact solutions are extremely valuable because of the insight... 

Fig. 9.11. Case diagram for the approximate solutions of equation (9.99). Different approximations hold for different values of the Laplace variable, s. and y where y(= XJX.) compares the distance a photogenerated electron diffuses through solution on a particle before participating in a loss reaction with the distance over which the light is absorbed.

An approximate solution of Equation 22 is obtained from Equation 23 by suitably approximating the matrix exponential e K This is accomplished by the Fade approximants of the exponential function. These Fade approximants are rational functions of the form... 

Taking now k=5/4 in Equation (35), one obtains the following value for the physically acceptable and approximated solution of Equation (33), namely,... 

Let us use the letter to represent the 3r coordinates of the r nuclei, relative to axes fixed in space, and the letter x to represent the 3s coordinates of the s electrons, relative to axes determined by the coordinates of the nuclei (for example, as described in Section 48). Let us also use the letter v to represent the quantum numbers associated with the motion of the nuclei, and n to represent those associated with the motion of the electrons. The principal result of Born and Oppenheimer s treatment is that an approximate solution ) of Equation... 

Note that the kernel of coagulation (13.27) does not obey the necessary conditions of existence of a self-similar solution. Therefore we shall consider some results of the numerical solution and obtain an approximate solution of equation (13.33) by the method of moments. 

For a potential flow around the cylinder, the approximate solution of equations (19.26) gives S - = 0.0625. However, at Re 1, a viscous boundary layer of thickness <5 r /V is formed near the cylinder surface, which causes a stronger curving of streamlines near the surface than we would expect for a potential flow. As a result, trajectories are pushed away from the surface. This means that the number of droplets reaching the surface in a unit time decreases. We can ac-... 

In general. Equation 3.69 can be solved numerically for Cj. However, for the purpose of criteria definition from this subdominant (relatively negligible) correction, an approximate solution of Equation 3.69 is enough. Note that the non-isothermal model is completed with Equations 3.27, 3.34 and respective boundary conditions. In the reaction term in the solid phase energy balance, should be replaced by Cq to yield swi/,0 0 This arises from the fact that Equation 3.60 is also... 

H Hamiltonian or energy operator, >. cauci wave tion E exact energy the ground state is considered, i.c., / = 1 0 = is not possible for an n-electron system with > 1 and, therefore, one has to work with approximate solutions of equation (3). Suppose that an approximate form of equation (3), namely... 

Concentration profibs calculated in first approximation from truncated Mac Laurin series, approximate solutions of Equation (4 ), are illustrated in Figure 10. It is interesting to note now that the first of the coupled reactions regulates the second one (see caption of Figure 10) and also that the intermediate substrate-product has its concentration quite effectively stabilised within the membrane with regard to concentration variations of the first substrate outside. Densities of enzyme-sites and diffusion properties of the membrane control the stabilised level. 

For the determination of the approximated solution of this equation the finite difference method and the finite element method (FEM) can be used. FEM has advantages because of lower requirements to the diseretization. If the material properties within one element are estimated to be constant the last term of the equation becomes zero. Figure 2 shows the principle discretization for the field computation. 

To exemplify both aspects of the formalism and for illustration purposes, we divide the present manuscript into two major parts. We start with calculations of trajectories using approximate solution of atomically detailed equations (approach B). We then proceed to derive the equations for the conditional probability from which a rate constant can be extracted. We end with a simple numerical example of trajectory optimization. More complex problems are (and will be) discussed elsewhere [7]. 

Using semi-einpirical methods, which are also based on approximate solutions of the Schrodingcr equation but use parameterized equations, the computation times can be reduced by twu orders of magnitude. HyperChem from Hypercubc,... 

By systematically applying a series of corrections to approximate solutions of the Schroedinger equation the Pople group has anived at a family of computational protocols that include an early method Gl, more recent methods, G2 and G3, and their variants by which one can anive at themiochemical energies and enthalpies of formation, Af and that rival exper imental accuracy. The important thing... 

This equation defines the Galerldn method and a solution that satisfies this equation (for aUj = 1,. . . , °°) is called a weak solution. For an approximate solution, the equation is written once for each member of the trial function, j = 1,. . . , NT — 1, and the boundary condition is apphed. 

In order to find approximate solutions of the equations for Ci t) and gi,..j t) one can use regular approximate methods of statistical physics, such as the mean-field approximation (MFA) and the cluster variation method (CVM), as well as its simplified version, the cluster field method (CFM) . In both MFA and CFM, the equations for c (<) are separated from those for gi..g t) and take the form... 

Morrison, J., and Moss, R., 1980, Approximate solution of the Dirac equation using the Foldy-Wouthuysen Hamiltonian , A/o/. Phys. 41 491. 

The results of the approximate solution of this equation, in terms of the constant C in the energy expression — Ce /dap10, are given in Table III. 

Although the hybrid orbitals discussed in this section satisfactorily account for most of the physical and chemical properties of the molecules involved, it is necessary to point out that the sp orbitals, for example, stem from only one possible approximate solution of the Schrddinger equation. The i and the three p atomic orbitals can also be combined in many other equally valid ways. As we shall see on page 12, the four C—H bonds of methane do not always behave as if they are equivalent. 

However, if u is a solution to some equation like (34), then one can speak, as usual, about the approximation of equation (34) by scheme (21) on a solution of equation (34), about the convergence to a solution of equation (34), etc. 

Homogeneous through execution schemes are quite applicable in the cases where the diffusion coefficient is found as an approximate solution of other equations. For instance, such schemes are aimed at solving the equations of gas dynamics in a heat conducting gas when the diffusion coefficient depends on the density and has discontinuities on the shock waves. 

One is purely formal, it concerns the departure from symmetry of an approximate solution of the Schrodinger equation for the electrons (ie within the Bom-Oppenheimer approximation). The most famous case is the symmetry-breaking of the solutions of the Hartree-Fock equations [1-4]. The other symmetry-breaking concerns the appearance of non symmetrical conformations of minimum potential energy. This phenomenon of deviation of the molecular structure from symmetry is so familiar, confirmed by a huge amount of physical evidences, of which chirality (i.e. the existence of optical isomers) was the oldest one, that it is well accepted. However, there are many problems where the Hartree-Fock symmetry breaking of the wave function for a symmetrical nuclear conformation and the deformation of the nuclear skeleton are internally related, obeying the same laws. And it is one purpose of the present review to stress on that internal link. 

The ultimate goal of most quantum chemical approaches is the - approximate - solution of the time-independent, non-relativistic Schrodinger equation... 

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