Adiabatic curve

In the adiabatic model, the matrix element between the Vy level of the first adiabatic curve Vid(R) and the i 2 level of the second adiabatic curve Vfd(R) (see Fig. 3.5) is reduced to 

TiV is the nuclear kinetic energy operator, which is expressed in the molecular frame as 

Let us ignore the R2 rotational part (R=J-L-S) of this operator, which leads to off-diagonal matrix elements that are proportional to J(J + 1) but still very small compared to the matrix elements of the remaining radial term (Leoni, 1972). The effect of the derivatives with respect to R on the electronic and vibrational wavefunctions, both of which depend on R, is given by 

RdR x RxdR R dR Combining this result with Eq. (3.3.9) yields 

the second term is zero after integration over the electronic coordinates r, since d and 2d are two different solutions of the same equation and must therefore be orthogonal. The off-diagonal matrix elements in Eq. (3.3.10) are often called nonadiabatic corrections to the energies. 

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Figure B3.4.17. When a wavepacket comes to a crossing point, it will split into two parts (schematic Gaussians). One will remain on the same adiabat (difFerent diabat) and the other will hop to the other adiabat (same diabat). The adiabatic curves are shown by fidl lines and denoted by ground and excited die diabatic curves are shown by dashed lines and denoted 1, 2.

Figure 4. Four iriceractirig adiabatic surfaces presented in terms of four adiabatic curves. The points Cj j = 1,2,3, stand for the three conical intersections.

Adiabate, /. adiabatic curve, adiabatisch, a. adiabatic, adiaktlnisch, a. adiactinic. 

Fig. 1. A schematic diagram of the relationship between adiabatic potential curves and reactive resonances, (a) shows the conventional Feshbach resonance trapped in a well of an adiabatic curve, (b) illustrates barrier trapping, which occurs near the energy of the barrier maximum of an adiabatic curve.

Fig. 1. Schematic potential energy curves for a neutral transition metal atom (M) inserting into the H-R bond of a hydrocarbon. Diabatic curves are shown as dashed lines, adiabatic curve shown as a solid line.

The subscript H is used to emphasize that derivatives are along the Hugoniot curve. Now, somewhere along the Hugoniot curve, the adiabatic curve passing through the same point has the same slope as the Hugoniot curve. There, ds2 must be zero and Eq. (5.18) becomes... 

The behavior of a colliding system at the avoided crossing depends on the speed with which the crossing is traversed, as well as the separation and slopes of the adiabatic curves. If the atoms approach slowly enough on the covalent curve, the... 

Figure 17. Approximate one-dimensional ionic and covalent diabatic potentials adapted from Reference 48 for CFjBr. Solid and dashed ionic curves are Rittner-type potentials for parent and fragment ions respectively, and heads and tails are covalent curves. The ionic asymptote is denoted by the arrow. The crossings are avoided dotted curves for the "crossing" near 4.3 A are the adiabatic curves resulting from configuration interaction between the diabatic ionic and covalent curves. (Adiabatic curves for the other crossings are omitted for simplicity.)...

Figure 26.4 Surface temperature and surface fuel mole fraction (a), and NO (6) as functions of inlet composition, along the adiabatic curve, for the stagnation reactor (solid curves) and the PSR (dashed curves). The fuel-lean and fuel-rich regions are indicated. The conditions are pressure of 1 atm, inlet temperature of 25 °C, a strain rate of 1000 s (stagnation reactor), and a residence time of 1 ms (PSR)...

Since most compressors operate along a polytropic path approaching the adiabatic, compressor calculations are generally based on the adiabatic curve. 

The work of Apin et al (Ref 4) on calcn of exponents of a polytropic curve of explosion products of condensed expls was summarized by A.G. Streng in CA 56, 11871-72(1962), as follows The adiabatic curve of explosion products at the front of a detonation wave may be described by the polytropic law p = Av"n. The exponent n depends mainly on the compn of the explosion products the influence of temp... 

K. Troshin, "The Generalized Hugoniot Adiabatic Curve Ibid, p 789 (Overcompressed detonation waves) 56) F.J. Martin D.R. White Formation and Structure of Gaseous Detonatio n Waves, Ibid, 856—65 56a) F.H. 

Quoting Duvall (Ref 6) Figure 6 gives the locus of all states (p, Vv, j, and so on) that can be reached from an initial state (p0, V0, E0) by shock compression. In an analogous way, the ordinary adiabat or adiabatic curve may be defined as the locus of all states that can be reached by adiabatic compression... 

Exponents of a Poly tropic Curve of Explosion Products of Condensed Explosives. Accdg to Apin et al(Ref 1), the adiabatic curve of expln products at the front of a detonation wave may be described by the polytropic law p=Av-n, where p=pressure, A=function of entropy, v=volume of expln products and n=polytropic exponent (See Ref 2, pp D290-R D474-R). The exponent n (also designated as y) depends mainly on the composition. The influence of temperature and pressure may be neglected. Experiments performed with expl compds and mixtures showed that over a wide range of temps and pressures of detonation, the exponent (n) of the polytropic curve of expln products may be obtd from the values of exponents of the individual products ... 

As an example we can take the excited states of NO. It has been shown that there are two excited states of the same symmetry ( 11) whose vibrational levels are best interpreted on the basis of diabatic curves which cross as in Fig. 1 (75-7 7). One of these states (B) arises from the electron excitation to an antibonding valence molecular orbital and the other (C) from excitation to a Rydberg orbital. The Born-Oppenheimer adiabatic curves cannot cross (by virtue of the non-crossing rule which is to be discussed in a later section) and must fullow the dashed curves shown in the figure. 

Historically the first application of symmetry to potential energy surfaces was to prove the so-called non-crossing rule. In its simplest form this may be stated as potential energy curves for states of diatomic molecules of the same symmetry do not cross . We have already seen in section 2 that this should be qualified to apply to adiabatic curves, as in some situations it may be convenient to define diabatic curves wdiich do cross. 

Suppose Va and V/, are diabatic surfaces which cross at some configuration and let cj/2 be the matrix element of the electronic Hamiltonian between the two diabatic states. In other words, the adiabatic curves are the eigenvalues of the potential energy matrix. 

In terms of the adiabatic curves of Figure 24, the electronic wave function of S j BA is represented as follows... 

The potential curves Vm(R) and V+(R) are isolated that is, at the collision energies considered there are no diabatic transitions possible from these curves to other adiabatic curves. 

In principle, excitation transfer at curve crossings may also occur at thermal collision energies, but only under rather restricted conditions, because at low collision energies the system will usually follow the adiabatic curve V (R)- The following two exceptions may arise ... 

Herzberg Continuum (adiabatic curve following two 0(3P2) atoms)... 

Figure 5.20 shows a plot of the experimental temperature decay for run 1, after t = 60 min. An excellent linear regression was obtained, which means that U can be regarded as a constant value. The adiabatic temperature rise curves were calculated using Eq. (5.77) (plots are shown in Fig. 5.19). The adiabatic curves are now ready for a kinetic analysis.

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