# Exact Solutions

Maxwell s equation are the basis for the calculation of electromagnetic fields. An exact solution of these equations can be given only in special cases, so that numerical approximations are used. If the problem is two-dimensional, a considerable reduction of the computation expenditure can be obtained by the introduction of the magnetic vector potential A =VxB. With the assumption that all field variables are sinusoidal, the time dependence [c.312]

Here, each elementary contribution is refracted at the interface between the coupling medium and the piece. Instead of developing an exact treatment of the refraction at the piece boundary (decomposition of each elementary contribution into its two-dimensional spectrum of plane waves) which leads to a time consuming solution, an approximate solution using the geometrical optics approximation (denoted GO) is proposed. The GO approximation is nothing but an asymptotic solution of the exact solution around the path of stationary phase existing between the source and field points (Fermat s principle). This approximation, initially applied to treat the refraction into a solid isotropic medium has been recently extended to anisotropic materials [9] and is being implemented numerically. [c.736]

B. Exact Solutions to the Capillary Rise Problem [c.12]

A2.3.10 EXACT SOLUTIONS TO THE ISING MODEL [c.543]

That analyticity was the source of the problem should have been obvious from the work of Onsager (1944) [16] who obtained an exact solution for the two-dimensional Ising model in zero field and found that the heat capacity goes to infinity at the transition, a logarithmic singularity tiiat yields a = 0, but not the a = 0 of the analytic theory, which corresponds to a finite discontinuity. (Wliile diverging at the critical point, the heat capacity is synnnetrical without an actual discontinuity, so perhaps should be called third-order.) [c.644]

One may think that the dynamical Jahn-Teller effect is equivalent to take into consideration the GP effect that arises, for example, due to the conical intersection in X3-type systems formed from atoms. We find it pedagogical from our calculations for Lis to distinguish three situations. The first cones-ponds to the calculations in sets (l)-(3) mentioned above, that is, using only one elecbonic adiabatic BO potential energy surface without consideration of the GP effect. The second refers to the generalized BO treatment in which the GP effect is also considered. Finally, one has the exact (or nearly exact) solution, which is obtained by solving the multistate quantum dynamics (vibronic) problem see also Section X.D. If one thinks of the first approach as corresponding to include the dynamic Jahn-Teller effect alone, then Table XIV shows that such an effect has a remarkable importance on the vibrational levels as it may be seen by comparing states with equal sets of quantum numbers (vi,V2,V3) in the point group. In turn, the calculations of sets (4)-(6) would include both the dynamic Jahn-Teller and GP effects. The difference between the above two series of results were then attributable to the GP effect alone see Figures 6-8. This is found [11] to lead to further shifts of [c.595]

Backward Analysis In this type of analysis, the discrete solution is regarded as an exact solution of a perturbed problem. In particular, backward analysis of symplectic discretizations of Hamiltonian systems (such as the popular Verlet scheme) has recently achieved a considerable amount of attention (see [17, 8, 3]). Such discretizations give rise to the following feature the discrete solution of a Hamiltonian system is exponentially close to the exact solution of a perturbed Hamiltonian system, in which, for consistency order p and stepsize r, the perturbed Hamiltonian has the form [11, 3] [c.100]

Theorem 1 ([8]). Let H be analytic. There exists some r > 0, so that for all T < Tt the numerical solution Xk = ) Xo and the exact solution x of the perturbed system H (the sum being truncated after N = 0 1/t) terms) with x(0) = Xq remain exponentially close in the sense that [c.101]

The IE scheme is nonconservative, with the damping both frequency and timestep dependent [42, 43]. However, IE is unconditionally stable or A-stable, i.e., the stability domain of the model problem y t) = qy t), where q is a complex number (exact solution y t) = exp(gt)), is the set of all qAt satisfying Re (qAt) < 0, or the left-half of the complex plane. The discussion of IE here is only for future reference, since the application of the scheme is faulty for biomolecules. [c.238]

The use of a numerical integrator to approximate the exact propagator of a system of ordinary differential equations (ODEs) yields a numerical solution which can be interpreted as the exact solution of a slightly different system of ODEs. If the given system is a Hamiltonian system (aa it is for constant-energy MD), then the slightly different system is Hamiltonian if and only if the integrator is symplectic [21]. In particular, this implies that any given energy surface in phase space is changed only slightly by the use of symplectic [c.319]

Exact Solutions to the Schrodinger Equation [c.49]

For many-electron systems such as atoms and molecules, it is obviously important that approximate wavefiinctions obey the same boundary conditions and symmetry properties as the exact solutions. Therefore, they should be antisynnnetric with respect to interchange of each pair of electrons. Such states can always be constmcted as linear combinations of products such as [c.31]

Caleulations that employ the linear variational prineiple ean be viewed as those that obtain the exaet solution to an approximate problem. The problem is approximate beeause the basis neeessarily ehosen for praetieal ealeulations is not suffieiently flexible to deseribe the exaet states of the quantnm-meehanieal system. Nevertheless, within this finite basis, the problem is indeed solved exaetly the variational prineiple provides a reeipe to obtain the best possible solution in the space spanned by the basis functions. In this seetion, a somewhat different approaeh is taken for obtaining approximate solutions to the Selirodinger equation. [c.46]

Some similarities and differences between perturbation theory and the linear variational principle need to be emphasized. First, neither approach can be used in practice to obtain exact solutions to the Sclnodinger equation for intractable Flamiltonians. In either case, an infinite basis is required neither the sums given by perturbation theory nor the matrix diagonalization of a variational calculation can be carried out. Flence, the strengtiis and weaknesses of the two approaches should be analysed from the point of view that the basis is necessarily truncated. Witliin tliis constraint, diagonalization of H represents the best solution that is possible in the space spaimed by the basis set. In variational calculations, rather severe tmncation of H is usually required, with the effect that its eigenvalues might be poor approximations to the exact values. The problem, of course, is that the basis is not sufficiently flexible to accurately represent the fine quantum-mechanical wavefiinction. In perturbation theory, one can include significantly more fiinctions in the calculation. It films out that the results of a low order perturbation calculation are often superior to a practical variational treatment of the same problem. Unlike variational methods, perturbation theory does not provide an upper bound to the energy (apart from a first-order treatment) and is not even guaranteed to converge. Flowever, in chemistry, it is virtually always energy differences—and not absolute energies—that are of interest, and differences of energies obtained variationally are not themselves upper (or lower) bounds to tlie exact values. For example, suppose a spectroscopic transition energy between the states i. and v . is calculated from the difference X. - Xj obtained by diagonalizing H in a finncated basis. There is no way of knowing whether this value is above or below the exact answer, a situation no different than that associated with taking the difference between two approximate eigenvalues obtained from two separate calculations based on perturbation theory. [c.51]

The equilibrium properties of a fluid are related to the correlation fimctions which can also be detemrined experimentally from x-ray and neutron scattering experiments. Exact solutions or approximations to these correlation fiinctions would complete the theory. Exact solutions, however, are usually confined to simple systems in one dimension. We discuss a few of the approximations currently used for 3D fluids. [c.478]

The interaction between ions of the same sign is assumed to be a pure hard sphere repulsion for r < a. It follows from simple steric considerations that an exact solution will predict dimerization only if i < a/2, but polymerization may occur for o/2 < L = o. However, an approximate solution may not reveal the fiill extent of polymerization that occurs in a more accurate or exact theory. Cummings and Stell [ ] used the model to study chemical association of uncharged atoms. It is closely related to the model for adliesive hard spheres studied by Baxter [70]. [c.500]

This is just the usual classical expression for the probability density. Note that (v) and S(x) are real as long as motion is classically allowed, meaning that E > V(x). If > V(x), then S(x) becomes imaginary and /(v) involves real rather than complex exponentials. At the point of transition between allowed and forbidden regions, i.e., at the so-called turning points of the classical motion, (v) becomes infinite and the solutions above are not valid. However, it is possible to coimect tire solutions on either side of tlie turning point using coimection fonnulas that are detemiined from exact solutions to the Schrodinger equation near the tiimmg point. The reader should consult the standard textbooks [1, 2, 3, 4 and 5, 18] for a detailed discussion of this. [c.1000]

Marquardt R and Quack M 1989 Infrared-multlphoton excitation and wave packet motion of the harmonic and anharmonic oscillators exact solutions and quasiresonant approximation J. Chem. Phys. 90 6320-7 [c.1091]

If we now include the anliannonic temis in equation B 1.5.1. an exact solution is no longer possible. Let us, however, consider a regime in which we do not drive the oscillator too strongly, and the anliannonic temis remain small compared to the hamionic ones. In this case, we may solve die problem perturbatively. For our discussion, let us assume that only the second-order temi in the nonlinearity is significant, i.e. 0 and b = 0 for > 2 in equation B 1.5.1. To develop a perturbational expansion fomially, we replace E(t) by X E t), where X is the expansion parameter characterizing the strength of the field E. Thus, equation B 1.5.1 becomes [c.1268]

If V(R) is known and the mahix elements ffap ate evaluated, then solution of Eq. nO) for a given initial wavepacket is the numerically exact solution to the Schrddinger equation. [c.259]

In the derivation used here, it is clear that two approximations have been made—the configurations are incoherent, and the nuclear functions remain localized. Without these approximations, the wave function fonn Eq. (C.l) could be an exact solution of the Schrddinger equation, as it is in 2D MCTDH form (in fact is in what is termed a natural orbital form as only diagonal configurations are included [20]). [c.318]

Tanford, C., Kirkwood, J. G. Theory of protein titration curves. I. General equations for impenetrable spheres. J. Am. Chem. Soc. 79 (1957) 5333-5339. 6. Garrett, A. J. M., Poladian, L. Refined derivation, exact solutions, and singular limits of the Poisson-Boltzmann equation. Ann. Phys. 188 (1988) 386-435. Sharp, K. A., Honig, B. Electrostatic interactions in macromolecules. Theory and applications. Ann. Rev. Biophys. Chem. 19 (1990) 301-332. [c.194]

We have in mind trajectory calculations in which the time step At is large and therefore the computed trajectory is unlikely to be the exact solution. Let Xnum. t) be the numerical solution as opposed to the true solution Xexact t)- A plausible estimate of the errors in X um t) can be obtained by plugging it back into the differential equation. [c.268]

Unfortunately, this local error Cr cannot be calculated, since we do not know the exact solution to the QCMD equations. The clue to this problem is given by the introduction of an approximation to Let us consider another discrete evolution with an order q > p and define an error estimation via er t + z i) - z t). [c.403]

So called Ilydrogenic atomic orbitals (exact solutions for the hydrogen atom) h ave radial nodes (values of th e distance r where the orbital s value goes to zero) that make them somewhat inconvenient for computation. Results are n ot sensitive to these nodes and most simple calculation s use Slater atom ic orbitals ofthe form [c.269]

At the limit of Knudsen diffusion control it is not reasonable to expect that any of the proposed approximation methods will perform well since, as we know, percentage variations in pressure are quite large. Nevertheless it is interesting to examine their results, which are shown in Figure 11 4 At this limit it is easy to check algebraically that equations (11.54) and (11.55) become the same, while (11.60) differs from the other two. Correspondingly the values of the effectiveness factor calculated using the approximation of Kehoe and Aris coincide with the results of Apecetche et al., and with the exact solution, ile Hite and Jackson s effectiveness factors differ substantially. [c.138]

Iving the Schrodinger equation for atoms with more than one electron is complicated by a. mber of factors. The first complication is that the Schrodinger equation for such systems nnot be solved exactly, even for the helium atom. The helium atom has three particles (two ctrons and one nucleus) and is an example of a three-body problem. No exact solutions can found for systems that involve three (or more) interacting particles. Thus, any solutions 3 might find for polyelectronic atoms or molecules can only be approximations to the real, le solutions of the Schrodinger equation. One consequence of there being no exact solution that the wavefunction may adopt more than one functional form no form is necessarily ore correct than another. In fact, the most general form of the wavefunction will be an finite series of functions. [c.54]

All numerical computations inevitably involve round-off errors. This error increases as the number of calculations in the solution procedure is increased. Therefore, in practice, successive mesh refinements that increase the number of finite element calculations do not necessarily lead to more accurate solutions. However, one may assume a theoretical situation where the rounding error is eliminated. In this case successive reduction in size of elements in the mesh should improve the accuracy of the finite element solution. Therefore, using a P C" element with sufficient orders of interpolation and continuity, at the limit (i.e. when element dimensions tend to zero), an exact solution should be obtaiiied. This has been shown to be true for linear elliptic problems (Strang and Fix, 1973) where an optimal convergence is achieved if the following conditions are satisfied [c.33]

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**Exact Solutions**:

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