Diabatic potential energy surfaces

Figure 6. Diabatic potential energy surfaces for electron transfer reactions in the system AL/B.

Note that since the profile of the lower adiabatic potential energy surface for the proton depends on the coordinates of the medium molecules, the zeroth-order states and the diabatic potential energy surfaces depend also on the coordinates of the medium molecules. The double adiabatic approximation is essentially used here the electrons adiabatically follow the motion of all nuclei, while the proton zeroth-order states adiabatically follow the change of the positions of the medium molecules. 

Activationless ET processes (1, 4, 6, 9) are described by two diabatic potential energy surfaces crossing at the minimum of the initial surface. This limit is characterized by a rate which decreases with increasing temperature at high T (negative apparent activation energy)... 

It is prerequisite to define localized, diabatic state wave fimctions, representing specific Lewis resonance configurations, in a VB-like method. Although this can in principle be done using an orbital localization technique, the difficulty is that these localization methods not only include orthorgonalization tails, but also include delocalization tails, which make contribution to the electronic delocalization effect and are not appropriate to describe diabatic potential energy surfaces. We have proposed to construct the locahzed diabatic state, or Lewis resonance structure, using a strictly block-localized wave function (BLW) method, which was developed recently for the study of electronic delocalization within a molecule.(28-3 1)... 

Figure 2. To the left, quasi-diabatic potential energy surfaces in the B3LYP/cc-pvtz Dunning s basis set. AA represents a cis state (solid line) BB a trans state (solid line) AB is the excited diradical state spin singlet (dashed line) Triplet is the diradical state S=1 (dotted line). To the right, extrapolated diabatic potential energy surfaces for the same states. The angle used to plot energy entries is a = 2 0. All calculations were done with Gaussian 98 [23].

Not shown to the quasi-diabatic potential energy surfaces in Fig. (3) there is a adiabatic potential energy surface. This is distinguished by the maximum at the crossing point nil. The system has a saddle-point structure. In the regions about the cis and trans attractors there is no difference between them. Between 2%1 i... 

Once the diabatic potential energy surfaces relevant to describing a process, the integration of the sources of external potential (nuclear dynamics) can be done in real space using numerical integration methods. 

Klos J, Chalasinski G, Szczesniak MM, Werner H-J (2001) Ab initio calculations of adiabatic and diabatic potential energy surfaces of cl... HC1 Van der Waals complex. J Chem Phys 115 3085-3098... 

The sudden changes in the adiabatic wavefunctions near avoided crossings make it more convenient to use diabatic potential energy surfaces when simulating photodissociation dynamics. The adiabatic potentials, usually constructed from electronic structure calculation data, should therefore be transformed to diabatic potentials. The adiabatic-diabatic transformation yields diabatic states for which the derivative couplings above approximately vanish. The diabatic potential energy surfaces are obtained from the adiabatic ones by a unitary orthogonal transformation [22,23]... 

H.J. Werner, B. FoUmeg, M.H. Alexander, Adiabatic and diabatic potential-energy surfaces for... 

Alexander, M. H. (1993) Adiabatic and Approximate Diabatic Potential Energy Surfaces for the B...H2 van der Waals Molecule, J. Chem. Phys. 99, 6014-2026. 

The calculation includes three diabatic potential energy surfaces obtained using the diatomic in molecules (DIM) method (36). Using this 2D quantum treatment in hyperspher-ical coordinates the nonadiabatic problem can be solved also for problems involving more than three diabatic surfaces. 

In the present context, the standard BO description corresponds to diagonalizing equation (VIII. 12) in Ref. [25] for all values of the PCB. In contrast, for the GED scheme, these calculations only make sense at the stationary geometries. Our approach makes it apparent that there is no actual physical process associated with the crossings of electronic states occurring within the BO calculations. In contrast, important conical intersections associated with molecular symmetries still find a natural place in the present post-BO approach, as it incorporates the intersections of diabatic potential energy surfaces. 

When the field is absent and the nuclei are viewed as quantum particles, a complete description requires inclusion of electron-phonon and spin-orbit coupling operators. Now, the By( )-coefficients couple electronic states whose diabatic potential energy surfaces jrield degenerate functions for the nuclear dynamics. Finally, these geometry-dependent coefficients couple all diabatic electronic states whenever a fuUy quantum-mechanical molecular system is embedded in an external field. 

In general, in the above considerations the coordinate x is presumed to describe nuclear motion normal to the intersection line L of the diabatic.potential energy surfaces of reactants and products. In particular cases, however, the coordinate x can coincide with a dynamically separable reaction coordinate. Then, the whole manydimensional problem of calculating the transition probability for any energy value is simply reduced to a one-dimensional one. Such is, for instance, the situation in a system of oscillators making harmonic vibrations with the same frequency in both the initial and final state /67/ which we considered in Sec.3.1.1. The diabatic surfaces (50.1) then represent two similar (N+1>dimensional rotational paraboloids which intersect in a N-dimensional plane S, and the intersection... 

B. Research on Nearly Diabatic Potential Energy Surfaces... 

The two methodologies should not be considered as contradictory they may be used in conjunction, as has been mentioned. It may be useful for instance to use the projection approach, defining a valence effective Hamiltonian, which will be later mimicked (as H O by simulation techniques. The diabatization potential energy surfaces might be an important step to define valence states, in regions where non-valence intruder states appear, before simulating them by pseudo-Hamiltonian techniques. 

Figure 1 Hush diagram for intervalence transfer within a class II mixed-valence ion. The dotted lines correspond to diabatic potential energy surfaces. The solid lines are adiabatic potential energy surfaces. Electron transfer can occur either optically (vertical transition with energy, Eop, equaling A) or thermally by moving along the lower adiabatic surface. In the diabatic limit, the barrier height for thermal electron...

Another concept often invoked for qualitative discussions and for calculations when the adiabatic approximation breaks down is that of diabatic potential energy surfaces. There are several nonequivalent ways of defining such surfaces, each of which may be useful under some circumstances. The simplest way is that already illustrated above in conjunction with the NaCl example namely, a diabatic state is the effective potential energy function for nuclear motion when the electronic state is artificially constrained to a state of prespecified pure valency. 

A crossed-beam experiment has been devoted to study the photochemical reaction at threshold Cs(7P)+H2 -> CsH+H. It is shovm that use of lasers (excitation of Cs atoms, detection of CsH products) and calculation of diabatic potential energy surfaces are able to provide a good understanding of this particular reactive process. 

Fig. 8. Diabatic potential energy surfaces for the A (dashed lines) and B (sohd lines) states of H2S as functions of the two HS bond distances for fixed angle a = 92° (left-hand panel) and as functions of one HS bond distance and a with the other HS bond distance fixed at 2.5ao (right-hand panel). The black circles indicate the position of the Pranck-Condon point where the motion on the upper-state potential energy surfaces starts. Redrawn, with permission of the American Institute of Physics, from Ref. 51.

As we assume a vertical transition (Condon approximation), 4 (0) is the ground-state wave packet placed on an excited or ionic diabatic potential energy surface. The autocorrelation function is defined as... 

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