Linear potential crossing

The formulas derived in the time-independent framework can be easily transferred into the corresponding time-dependent solutions. The formulas in the time-independent linear potential model, for example, provide the formulas in the time-dependent quadratic potential model in which the two time-dependent diabatic quadratic potentials are coupled by a constant diabatic coupling [1, 13, 147]. The classically forbidden transitions in the time-independent framework correspond to the diabatically avoided crossing case in the time-dependent framework. One more thing to note is that the nonadiabatic tunneling (NT) type of transition does not show up and only the LZ type appears in the time-dependent problems, since time is unidirectional. 

B. Linear Potential Model—Curve-Crossing Case... 

The most fundamental quantum mechanical model of curve crossing is the linear potential model (in coordinate R), in which the diabatic crossing potentials V R) and V2(R) are linear functions of R and the diabatic coupling V(R) is constant (=A). The basic coupled Schrodinger equations are (29)... 

A simplified solution to the problem may be found, as shown by NIKITIN et al./113/, using the linear potential functions (154.11) which represent an approximation of the "diabatic potential curves in the region of the crossing point x = x. Choosing this point as origin of the coordinate system, we set x = 0 and V-j (x ) = 2(Xq) = 0 to obtain the equations... 

Based on the assumption of a linear potential drop (which seems to be more realistic along an ion-transporting channel than along other cross sections of the membrane) Equation 19.17 can be developed into... 

Figure 8.15 HBN radical potential energy surfaces and selected energy levels of 2 and II symmetry. Adiabatic (red, black) and diabatic (violet, magenta) PES. 2D cut at linear geometries, crossing seam shown as a green line ID cuts along BN stretching coordinate for / BH and 6 fixed at 2.1A and 150°, respectively. The component of the A" symmetry of the 11 state shown as a blue line. Assignment based on plots and expansion coefficients of vibrational part of wavefunctions. AU values in cm levels showing resonances are marked.

Superposition of Flows Potential flow solutions are also useful to illustrate the effect of cross-drafts on the efficiency of local exhaust hoods. In this way, an idealized uniform velocity field is superpositioned on the flow field of the exhaust opening. This is possible because Laplace s equation is a linear homogeneous differential equation. If a flow field is known to be the sum of two separate flow fields, one can combine the harmonic functions for each to describe the combined flow field. Therefore, if d)) and are each solutions to Laplace s equation, A2, where A and B are constants, is also a solution. For a two-dimensional or axisymmetric three-dimensional flow, the flow field can also be expressed in terms of the stream function. 

In order to calculate q (Q) all possible quantum states are needed. It is usually assumed that the energy of a molecule can be approximated as a sum of terms involving translational, rotational, vibrational and electronical states. Except for a few cases this is a good approximation. For linear, floppy (soft bending potential), molecules the separation of the rotational and vibrational modes may be problematic. If two energy surfaces come close together (avoided crossing), the separability of the electronic and vibrational modes may be a poor approximation (breakdown of the Bom-Oppenheimer approximation. Section 3.1). 

However, a potential may give rise to more than one type of flux. There are cross-effects A temperature difference can also result in diffusion, called thermal diffusion, and a concentration difference can result in a heat current. The general relation between fluxes 7, and the driving potentials A) is of the form of linear relations... 

The physical meaning of the g (ion) potential depends on the accepted model of an ionic double layer. The proposed models correspond to the Gouy-Chapman diffuse layer, with or without allowance for the Stem modification and/or the penetration of small counter-ions above the plane of the ionic heads of the adsorbed large ions. " The experimental data obtained for the adsorption of dodecyl trimethylammonium bromide and sodium dodecyl sulfate strongly support the Haydon and Taylor mode According to this model, there is a considerable space between the ionic heads and the surface boundary between, for instance, water and heptane. The presence in this space of small inorganic ions forms an additional diffuse layer that partly compensates for the diffuse layer potential between the ionic heads and the bulk solution. Thus, the Eq. (31) may be considered as a linear combination of two linear functions, one of which [A% - g (dip)] crosses the zero point of the coordinates (A% and 1/A are equal to zero), and the other has an intercept on the potential axis. This, of course, implies that the orientation of the apparent dipole moments of the long-chain ions is independent of A. 

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