Uncoupled equations

Hence, in order to contract extended BO approximated equations for an N-state coupled BO system that takes into account the non-adiabatic coupling terms, we have to solve N uncoupled differential equations, all related to the electronic ground state but with different eigenvalues of the non-adiabatic coupling matrix. These uncoupled equations can yield meaningful physical... 

Now, we are in a position to present the relevant extended approximate BO equation. For this purpose, we consider the set of uncoupled equations as presented in Eq. (53) for the = 3 case. The function icq, that appears in these equations are the eigenvalues of the g matrix and these are coi = 2 (02 = —2, and CO3 = 0. In this three-state problem, the first two PESs are u and 2 as given in Eq. (6) and the third surface M3 is chosen to be similar to M2 but with D3 = 10 eV. These PESs describe a two arrangement channel system, the reagent-arrangement defined for R 00 and a product—anangement defined for R —00. 

The procedure we followed in the previous section was to take a pair of coupled equations, Eqs. (5-6) or (5-17) and express their solutions as a sum and difference, that is, as linear combinations. (Don t forget that the sum or difference of solutions of a linear homogeneous differential equation with constant coefficients is also a solution of the equation.) This recasts the original equations in the foiin of uncoupled equations. To show this, take the sum and difference of Eqs. (5-21),... 

We have said that the Schroedinger equation for molecules cannot be solved exactly. This is because the exact equation is usually not separable into uncoupled equations involving only one space variable. One strategy for circumventing the problem is to make assumptions that pemiit us to write approximate forms of the Schroedinger equation for molecules that are separable. There is then a choice as to how to solve the separated equations. The Huckel method is one possibility. The self-consistent field method (Chapter 8) is another. 

Now, we assume that the functions, tcoj, j = 1,. .., N are such that these uncoupled equations are gauge invariant, so that the various % values, if calculated within the same boundary conditions, are all identical. Again, in order to determine the boundary conditions of the x function so as to solve Eq. (53), we need to impose boundary conditions on the T functions. We assume that at the given (initial) asymptote all v / values are zero except for the ground-state function /j and for a low enough energy process, we introduce the approximation that the upper electronic states are closed, hence all final wave functions v / are zero except the ground-state function v /. ... 

These rate laws are coupled through the concentrations. When combined with the material-balance equations in the context of a particular reactor, they lead to uncoupled equations for calculating the product distribution. For a constant-density system in a CSTR operated at steady-state, they lead to algebraic equations, and in a BR or a PFR at steady-state, to simultaneous nonlinear ordinary differential equations. We demonstrate here the results for the CSTR case. 

Use of either one of these for ( rA) t has the effect of uncoupling equation 9.2-28 from -29, and we need only solve the former in terms of A. Thus, equation 9.2-28 becomes ... 

The last term was obtained using the identity (6.34) again, this time with f as the vector and then recognizing that the term V x V(V tyf) vanishes by identity (6.37). At last, (6.39) separates into two uncoupled equations... 

Another case which may be treated analytically is the case of exact resonance, WA = WB, if in addition, I aa - bb — 0 and VAB — VgAt i.e. the coupling matrix element is real. In this case Eqs. (14.16) are readily decoupled, leading to two identical uncoupled equations. The equation for CA(f) is... 

Under the assumption that the coupling elements daa> are very small, Equation (2.16) may be solved by first-order perturbation theory the coefficients aa/ (t) on the right-hand side are replaced by their initial values at t = 0. The evolution of each final state (/ i) is then governed by the (uncoupled) equation... 

Equation (3.125) is the required transformed Hamiltonian, and we see that in the representation in which (3 is diagonal, the Dirac equation decomposes into uncoupled equations for the upper and lower components of the wave function, i.e. for electron and positron wave functions. Setting (3 equal to +1 gives the positive energy (electron) states, whilst (3 equals -1 gives the negative energy (positron) states. 

The flow term, which involves the transport of all ionic species, can be solved on the basis of the Navier-Stokes equation tind can be expressed in terms of Let us mzike one more restriction, namely that the surface potential is not very low, so that the countercharge is essentially determined by counterions the terms with (C (eq)-C ( >)l for co-ions are then relatively small. In that case, (4.8.17) can be written as a set of uncoupled equations for each individual ionic species. For the co-ions the result is simple ... 

The utility of the steady-state CSTR methods lies in the fact that the problem of the steady-state PSD may be couched in terms of uncoupled equations that are considerably easier to solve than the full problem. Some caution must be exercised in omitting the stochastic term. The results of the comprehensive calculations of Thompson and Stevens (1977) show that in fxrrtain instances, the steady-state PSD is sensitive to stochastic broadening. 

Rouse (1953) transformed this matrix equation into a set of uncoupled equations—that is, into Ns independent equations for the normal modes --------------------------... 

If we compare Eqs. 5.1.14 with the conservation equation (Eq. 5.1.2) for a binary system and the pseudo-Fick s law Eq. 5.1.15, with Eq. 3.1.1 then we can see that from the mathematical point of view these pseudomole fractions and pseudofluxes behave as though they were the corresponding variables of a real binary mixture with diffusion coefficient D-. The fact that the are real, positive, and invariant under changes of reference velocity strengthens the analogy. If the initial and boundary conditions can also be transformed to pseudocompositions and fluxes by the same similarity transformation, the uncoupled equations represent a set of independent binary-type problems, n - 1 in number. Solutions to binary diffusion problems are common in the literature (see, e.g.. Bird et al., 1960 Slattery, 1981 Crank, 1975). Thus, the solution to the corresponding multicomponent problem can be written down immediately in terms of the pseudomole fractions and fluxes. Specifically, if... 

We have developed the solution to the linearized equations as a special case of an exact solution in Section 8.3.5 in order to emphasize the close relationship that exists between the two methods. It should be noted, however, that this is not the way in which Toor or Stewart and Prober obtained their results. Indeed, these equations are not to be found in this form in the papers that first presented the linearized theory. Both Toor and Stewart and Prober obtained their results using the procedure described in Chapter 5 that is, by diagonalizing the matrix [D] and solving sets of uncoupled equations. The final result is... 

Note, however, that uncoupled equations (29) and (30) hold only for k> p. This means that they describe the evolution of all the Bogoliubov coefficients only if p = 1. Then all the functions r j (f) are identically equal to zero because of the initial conditions (32) consequently, no photon can be created from vacuum. Moreover, in the next section we show that the total number of photons (but not the total energy) is an integral of motion in this specific case. 

In these equations, V) (R) is the potential energy curve of the predissociated state and V2 (R) is the potential curve of the predissociating state. Note that these equations determine Xv and xe independently from each other. These equations are uncoupled equations. For a coupled equation approach, see Section 7.12. 

The set of variables in Eqs. (11.4.2) and (11.4.3) must include all of the slowly relaxing variables. When the Hamiltonian has certain symmetry properties, the set of Eqs. (11. 3.26) and (11.4.2) can be separated into groups of uncoupled equations. Since, in general, we do not know how to compute the time-correlation functions (F(r), F+(0)), the elements of f should be regarded as quantities to be determined from a comparison between theory and experiment. However, symmetry can be used to relate the off-diagonal elements of V to each other and thereby to reduce the number of independent quantities. 

The eigenvectors VJ are called natural thermal modes, and the methodology employed for uncoupling equations of thermal balance is called modal analysis. 

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