Adiabatic potential energy curves

The potential energy curves (Fig. 1), the non-adiabatic coupling, transition dipole moments and other system parameters are same as those used in our previous work (18,19,23,27). The excited states 1 B(0 ) and 2 B( rio) are non-adiabatically coupled and their potential energy curves cross at R = 6.08 a.u. The ground 0 X( Eo) state is optically coupled to both the and the 2 R( nJ) states with the transition dipole moment /ioi = 0.25/xo2-The results to be presented are for the cw field e(t) = A Yll=o cos (w - u pfi)t described earlier. However, for IBr, we have shown (18) that similar selectivity and yield may be obtained using Gaussian pulses too. 

Figure 61. Adiabatic potential energy curves for the atomic hydrogen transmission through the five-membered ring in the case of center approach. Taken from Ref. [47].

The angular-dependent adiabatic potential energy curves of these complexes obtained by averaging over the intermolecular distance coordinate at each orientation and the corresponding probability distributions for the bound intermolecular vibrational levels calculated by McCoy and co-workers provide valuable insights into the geometries of the complexes associated with the observed transitions. The He - - IC1(X, v" = 0) and He + 1C1(B, v = 3) adiabatic potentials are shown in Fig. 3 [39]. The abscissa represents the angle, 9,... 

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.

Figure 4.67 depicts the potential-energy curve for reaction (4.102) along an adiabatic reaction coordinate (R = /Oimc) obtained by stepping along the H—CH3 stretching coordinate with full optimization of geometries at each step. As shown in Fig. 4.67, the reaction exhibits a substantial barrier ( 20.5 kcal mol-1) and overall exothermicity. 

Let us examine the balance between steric and donor-acceptor forces in greater detail for the case of HF- HF. The graph below plots the adiabatic potential-energy curve for H-bond formation (solid line, circles), as well as the steric repulsion energy37 (dotted line)... 

Figure 5.11 The adiabatic potential-energy curve for F HF hydrogen-bond formation (solid line, circles), with the steric repulsion energy (dotted line) and estimated np - oi ip donor-acceptor attraction (dashed line) included for comparison.

Figure 9.5 Adiabatic potential-energy curve according to Eq. (9.35).

The potential energy curves of the species AB, AB+, and AB- are used in figure 4.1 to summarize the definitions of the adiabatic ionization energy and electron affinity of AB. Note that the arrows start and end at vibrational ground states (vibrational quantum number v = 0). 

Figure 4.2 Potential energy curves for the molecules AB and AB+ showing the vertical and the adiabatic ionization energies of AB. r is the A-B bond length, and v represents the vibrational quantum number.

Figure 4. Diabatic (solid lines) and adiabatic (dashed lines) potential-energy curves of Model IVa. The Gaussian wave packet indicates the initial preparation of the system at time t = 0.

Although the phase space of the nonadiabatic photoisomerization system is largely irregular, Fig. 36A demonstrates that the time evolution of a long trajectory can be characterized by a sequence of a few types of quasi-periodic orbits. The term quasi-periodic refers here to orbits that are close to an unstable periodic orbit and are, over a certain timescale, exactly periodic in the slow torsional mode and approximately periodic in the high-frequency vibrational and electronic degrees of freedom. In Fig. 36B, these orbits are schematically drawn as lines in the adiabatic potential-energy curves Wo and Wi. The first class of quasi-periodic orbits we wish to consider are orbits that predominantly... 

Figure 1. Diabatic potential energy curves for Nal with an expanded view of the adiabatic potential curves, e, and Ej, near the diabatic curve crossing.

The wave function of Eqs. (14) and (15) was widely used to obtain BO potential energy curves and adiabatic corrections for the ground state (Kolos et al., 1986 Kotos and Rychlewski 1993, Wolniewicz 1993, 1995a) and electronically excited... 

Fig. 4 The BO (dashed line) and adiabatic (solid line) potential energy curves for the g state of H2. The vertical bar above each of the vibrational energy lines marks the value of = R )...

Figure 3. Computed potential energy curves for the diabatic and adiabatic state in the [HsN-H-NH ] system in the gas phase using 6-31G(d) basis set. The HF and MOVE energy profiles are overlapping.

Fig. 5. The pseudo-Jahn-Teller effect in ammonia (NH3). (a) CCSD(T) ground state potential energy curve breakdown of energy into expectation value of electronic Hamiltonian (He), and nuclear-nuclear repulsion VNN. (b) CASSCF frequency analysis of pseudo-Jahn-Teller effect showing the effect of including CSFs of B2 symmetry is to couple the ground and 1(ncr ) states to give a negative curvature to the adiabatic ground state potential energy surface for the inversion mode.

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. 

Figure 1 Left Enol-keto tautomerism in salicylaldimine (SA) and normal mode displacements for skeleton modes 1 4 and 1/30. Middle H/D diabatic potential energy curves Ua(Qu) for mode i/u (lowest states ground state, bolding and stretching fundamental, first bolding overtone arrows indicate laser excitation). Right two-dimensional (Qj4,Q3o) cuts through the adiabatic PES (obtained upon diagonalizing the field-free part of Eq. (1)) which has dominantly H/D stretching character but includes state and mode couplings (contours from 0 to 7400 cm-1).

Figure 30. Schematic representation of potential energy curves for adiabatic (a, b) and diabatic (c) photoreactions. (Reprinted with permission from Ref. 33).

For most systems, where the velocity of motion of the nuclei is slow relative to the electron velocity, this decoupling of electronic and nuclear motion is valid and is referred to as the adiabatic approximation. Equation (II.3) thus defines an electronic eigenstate (rn,Rv), parametric in the nuclear coordinates, and a corresponding eigenvalue Ek(RN) that is taken to represent the potential-energy curve or surface corresponding to state k. 

By condition 3 we want to ensure that the Born-Oppenheimer approximation can be applied to the description of the simple systems, allowing definition of adiabatic potential-energy curves for the different electronic states of the systems. Since the initial-state potential curve K (f ) (dissociating to A + B) lies in the continuum of the potential curve K+(/ ) (dissociation to A + B + ), spontaneous transitions K ( )->K+(f ) + e" will generally occur. Within the Born-Oppenheimer approximation the corresponding transition rate W(R)—or energy width T( ) = hW(R) of V (R)... 

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