Diabatic and adiabatic states

The aim of this work is to obtain the four lowest E curves and wavefunctions of BH at the same level of accuracy and to bring out the interplay of ionic, Rydberg and valence states at energies and internuclear distances which were not previously investigated. We have therefore made use of a method, already put forward by us [16,17] to determine at once quasi-diabatic and adiabatic states, potential energy cnrves and approximate nonadiabatic couplings. We have analogously determined the first three E+ states, of which only the lowest had been theoretically studied... 

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.

Closely related to the above merit of VB methods, the unique definition of diabatic states also allows us to derive the energy profiles for diabatic states. Since for many reactions the whole process can be described with very few resonance structures, the comparison between the diabatic and adiabatic state energy profiles can yield insight into the nature governing the reactions [22-24]. In fact, even for complicated enzymatic reactions, simple VB ideas have shown unparalleled value [25, 26]. However, the further utilization of the VB ideas at the empirical and semi-empirical levels should be carefully verified by benchmark ab initio VB... 

Mo, Y. and Gao, J. (2000). MOVB A Program for Calculating Diabatic and Adiabatic State Energies. Version 1.0. University of Minnesota, Minneapolis, Minnesota. 

The Born-Oppenheimer Approximation Diabatic and Adiabatic States 465... 

It is clear from this formula that diabatic and adiabatic states only differ substantially if the quotient H12/(H22 - Hll) is not too small, i.e. only in the vicinity of the crossing point Rc, which is defined by the relation Hu(Rc) = H22(Rq). 

Fig. 2. Diabatic and adiabatic states around an avoided crossing.

In Fig. 2 the behaviour of diabatic and adiabatic states is depicted. One observes that the energy gap between the two adiabatic states at the crossing point Rc is equal to 2Hl2. From (7) it is clear that the crossing becomes real [i.e. EfRf) = E2 Rq)] if Hl2 is identical to zero. If Htl is electrostatic it can be shown (Landau and Lifshitz, 1967) that Hl2 = 0 if and only if both wave functions (j) j and 2 have different symmetry and multiplicity, as we have seen already. 

Fig. 14.27. Electron transfer in the reaction DA -> D+A , as well as the relation of the Marcus parabolas to the concepts of the conical intersection, diabatic and adiabatic states, entrance and exit channels and the reaction barrier. Panel (a) shows two diabatic surfaces as functions of the and 2 variables that describe the deviation from the comical intersection point (within the...

Coriolis coupling (p. 906 and 912) critical points (p. 888) cross section (p. 901) curvature coupling (p. 906 and 914) cycloaddition reaction (p. 944) democratic coordinates (p. 898) diabatic and adiabatic states (p. 949) donating mode (p. 914) early and late reaction barriers (p. 895) electrophilic attack (p. 938) entrance and exit channels (p. 895) exo- and endothermic reactions (p. 909) femtosecond spectroscopy (p. 889) Franck-Condon factors (p. 962) intrinsic reaction coordinate (IRC) (p. 902) inverse Marcus region (p. 954) mass-weighted coordinates (p. 903)... 

Fig. 14.25. Electron transfer in the reaction DA- -D+A " as well as the relation of the Marcus parabolas to the concepts of the conical intersection, diabatic and adiabatic states, entrance and exit channels and the reaction barrier. Fig. (a) shows two diabatic (and adiabatic) surfaces of the electronic energy as functions of the f and 2 variables that describe the deviation from the conical intersection point (cf. p. 262). Both diabatic surfaces are shown schematically in the form of the two paraboloids one for the reactants (DA), the second for products (D+A ). The region of the conical intersection is also indicated. Fig. (b) also shows the conical intersection, but the surfaces are presented more realistically. The upper and lower parts of Fig. (b) touch at the conical intersection point. On the lower part of the surface we can see two reaction channels each with its reaction barrier (see the text), on the upper part (b) an energy valley is shown that symbolizes a bound state that is separated from the conical intersection by a reaction barrier.

In the case of open-system dynamics, assuming weak coupling of the conical intersection with an environment, the time evolution of the system is determined by the Redfield Eq. (7) for the reduced density matrix. In this case, the time-dependent population probabilities of diabatic and adiabatic states are given by... 

In Chapter IV, Englman and Yahalom summarize studies of the last 15 years related to the Yang-Mills (YM) field that represents the interaction between a set of nuclear states in a molecular system as have been discussed in a series of articles and reviews by theoretical chemists and particle physicists. They then take as their starting point the theorem that when the electronic set is complete so that the Yang-Mills field intensity tensor vanishes and the field is a pure gauge, and extend it to obtain some new results. These studies throw light on the nature of the Yang-Mills fields in the molecular and other contexts, and on the interplay between diabatic and adiabatic representations. 

In what follows we introduce the model Hamiltonian using both diabatic and adiabatic representations. Adopting diabatic electronic basis states /j ), the molecular model Hamiltonian can be written as [162, 163]... 

Figure 25. Diabatic and adiabatic population probabilities of the C (fuU line), B (dotted hne), and X (dashed line) electronic states as obtained for a five-state 16-mode model of the benzene cation.

In addition to such minima the lowest excited states tend to contain numerous minima and funnels at biradicaloid geometries, through which return to the ground state occurs most frequently. Most photochemical reactions then proceed part way in the excited state and the rest of the way in the ground state, and the fraction of each can vary continuously from case to case. (Cf. Figure 6.3, path a.) It is common to label adiabatic only those reactions that produce a spectroscopic excited state of the product (cf. Figure 6.3, path h), so distinction between diabatic and adiabatic reactions would appear to be sharp rather than blurred. But this is only an apparent simplification, since it is hard to unambiguously define a spectroscopic excited state. 

Symmetry properties of the nuclear wavefunction are different in the diabatic and adiabatic representations. The pair of adiabatic electronic states (see (1)) belong to the Al and A2 irreducible representations of the double group of S3. The diabatic states obtained from the adiabatic ones by applying the U matrix form a basis for the two dimensional irreducible representation E of S3. For quartet nuclear spin states, the electronuclear wavefunction, nuclear spin part excluded, must belong to the A2 irreducible representation. This requires the nuclear wavefunction (without nuclear spin) to be of the same E symmetry as the electronic one, because of the identity E x E = Ai + A2 + E. For doublet spin states, the E electronuclear wave-function (nuclear spin excluded) is obtained with an Ai or A2 nuclear wavefunction, combined with the E electronic ones. 

If the diabatic coupling matrix element, He, is -independent, this d/dR matrix element between two adiabatic states must have a Lorentzian H-depen-dence with a full width at half maximum (FWHM) of 46. Evidently, the adiabatic electronic matrix element We(R) is not - independent but is strongly peaked near Rc- Its maximum value occurs at R = Rc and is equal to 1/46 = a/4He. Thus, if the diabatic matrix element He is large, the maximum value of the electronic matrix element between adiabatic curves is small. This is the situation where it is convenient to work with deperturbed adiabatic curves. On the contrary, if He is small, it becomes more convenient to start from diabatic curves. Table 3.5 compares the values of diabatic and adiabatic parameters. The deviation from the relation, We(i )max x FWHM = 1, is due to a slight dependence of He on R and a nonlinear variation of the energy difference between diabatic potentials. When We(R) is a relatively broad curve without a prominent maximum, the adiabatic approach is more convenient. When We (R) is sharply peaked, the diabatic picture is preferable. The first two cases in Table 3.5 would be more convenient to treat from an adiabatic point of view. The description of the last two cases would be simplest in terms of diabatic curves. The third case is intermediate between the two extreme cases and will be examined later (see Table 3.6). 

For 7 1.0, which corresponds to the case for the SiO G and I 1II states, the mixing of the vibrational basis functions is large in both diabatic and adiabatic descriptions. 

Even for this case of strong coupling, the adiabatic picture of two potential curves that avoid crossing is inappropriate. Child has introduced an intermediate coupling picture that takes advantage of both diabatic and adiabatic characteristics. The diabatic curve of the predissociated state is displayed (solid lines) in Fig. 7.36a The corresponding diabatic vibrational levels, Ed, are plotted versus J(J + 1) (solid lines) in Fig. 7.36b. 

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