Uncoupler

It would appear that identical particle pemuitation groups are not of help in providing distinguishing syimnetry labels on molecular energy levels as are the other groups we have considered. However, they do provide very usefiil restrictions on the way we can build up the complete molecular wavefiinction from basis fiinctions. Molecular wavefiinctions are usually built up from basis fiinctions that are products of electronic and nuclear parts. Each of these parts is fiirther built up from products of separate uncoupled coordinate (or orbital) and spin basis fiinctions. Wlien we combine these separate fiinctions, the final overall product states must confonn to the pemuitation syimnetry mles that we stated above. This leads to restrictions in the way that we can combine the uncoupled basis fiinctions. 

Phonons are nomial modes of vibration of a low-temperatnre solid, where the atomic motions around the equilibrium lattice can be approximated by hannonic vibrations. The coupled atomic vibrations can be diagonalized into uncoupled nonnal modes (phonons) if a hannonic approximation is made. In the simplest analysis of the contribution of phonons to the average internal energy and heat capacity one makes two assumptions (i) the frequency of an elastic wave is independent of the strain amplitude and (ii) the velocities of all elastic waves are equal and independent of the frequency, direction of propagation and the direction of polarization. These two assumptions are used below for all the modes and leads to the famous Debye model. 

Kirkwood derived an analogous equation that also relates two- and tlnee-particle correlation fiinctions but an approximation is necessary to uncouple them. The superposition approximation mentioned earlier is one such approximation, but unfortunately it is not very accurate. It is equivalent to the assumption that the potential of average force of tlnee or more particles is pairwise additive, which is not the case even if the total potential is pair decomposable. The YBG equation for n = 1, however, is a convenient starting point for perturbation theories of inliomogeneous fluids in an external field. 

RRKM theory allows some modes to be uncoupled and not exchange energy with the remaining modes [16]. In quantum RRKM theory, these uncoupled modes are not active, but are adiabatic and stay in fixed quantum states n during the reaction. For this situation, equation (A3.12.15) becomes... 

The timescale is just one sub-classification of chemical exchange. It can be further divided into coupled versus uncoupled systems, mutual or non-mutual exchange, inter- or intra-molecular processes and solids versus liquids. However, all of these can be treated in a consistent and clear fashion. 

Figure B2.4.3 shows an example of this in the aldehyde proton spectnim of N-labelled fonnamide. Some lines in the spectnim remain sharp, while others broaden and coalesce. There is no frmdamental difference between the lineshapes in figures B2.4.1 and figures B2.4.3—only a difference in the size of the matrices involved. First, the uncoupled case will be discussed, then the extension to coupled spin systems.

As with the uncoupled case, one solution involves diagonalizing the Liouville matrix, iL+R+K. If U is the matrix with the eigenvectors as cohmms, and A is the diagonal matrix with the eigenvalues down the diagonal, then (B2.4.32) can be written as (B2.4.33). This is similar to other eigenvalue problems in quantum mechanics, such as the transfonnation to nonnal co-ordinates in vibrational spectroscopy. 

These complications require some carefiil analysis of the spin systems, but fiindamentally the coupled spin systems are treated in the same way as uncoupled ones. Measuring the z magnetizations from the spectra is more complicated, but the analysis of how they relax is essentially the same. 

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. 

When the non-adiabatic coupling terms x and x are considered negligibly small and dropped from Eq. (B.15), we get the uncoupled approximate Schrbdinger equation... 

Figure 4. Wavepackec dynamics of photoexcitadon, shown as snapshots of the density (wavepacket amplitude squared) at various times. The model is a 2D model based on a single, uncoupled, state of the butatriene redical cation. The initial structure represents the neutral ground-state vibronic wave function vertically excited onto the A state of the radical cation.

The picture here is of uncoupled Gaussian functions roaming over the PES, driven by classical mechanics. The coefficients then add the quantum mechanics, building up the nuclear wavepacket from the Gaussian basis set. This makes the treatment of non-adiabatic effects simple, as the coefficients are driven by the Hamiltonian matrices, and these elements couple basis functions on different surfaces, allowing hansfer of population between the states. As a variational principle was used to derive these equations, the coefficients describe the time dependence of the wavepacket as accurately as possible using the given... 

P. Deuflhard, W. Huisinga, A. Fischer, Ch. Schiitte. Identification of Almost Invariant Aggregates in Nearly Uncoupled Markov Chains. Preprint, Preprint SC 98-03, Konrad Zuse Zentrum, Berlin (1998)... 

A comparison of Fig. 4 and Fig. 3 shows that this uncoupled QCMD bundle reproduces the disintegration of the full QD solution. However, there are minor quantitative differences of the statistical distribution. Fig. 5 depicts... 

Pig. 5. Comparison of the qi expectation value of the uncoupled QCMD bundle ([g]e o) and full QD ( q)qd) for the test system for e = 1/100 (pictures on top) and e = 1/500 (below). Initial data as in Fig. 3. The shaded domain indicates the funnel between the two curves Qbo and geo (cf. Thm. 5). The light dashed line shows Hagedorn s limit solution qna and the dense lines (q )Qo (left hand pictures) and [ ]e s (right hand pictures). 

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),... 

The force constants k 2 and k2 are the off-diagonal elements of the matrix. If they are zero, the oscillators are uncoupled, but even if they are not zero, the K matrix takes the simple fomi of a symmetrical matrix because ki2 = k2. The matrix is symmetrical even though may not be equal to k22-... 

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. 

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