Ground state potentials

Many methods have since been developed to asymptotically correct potentials to correct this unfortunate behavior. Any corrections to the ground-state potential are dissatisfying, however, as the resulting potential is not a functional derivative of an energy functional. Even mixing one approximation for fxcCr) and another for /xc has become popular in an attempt to rectify the problem. A more satisfying route to asymptotically correct 

To illustrate the influence of different ground-state potentials consider the N2 molecule. In all the cases discussed below, a SCF step was carried out using the ground-state potential to find the KS levels, which are then used as input to Eq. [47] with the ALDA XC kernel. 

The KS energy levels and KS orbitals for the LDA functional are shown in Table 7. The orbitals are calculated with two different numerical methods, the first is fully numerical basis set free (i.e., solved on a real space grid) while the other uses the Sadlej (52 orbitals) basis set [the OEP results for the EXX (KLI) approximation shown in Table 7 are also calculated basis set free]. Note that the eigenvalues for the higher unoccupied states are positive. This is due to the LDA potential being too shallow and not having the correct asymptotic 

Orbital LDA Basis Set Free S LDA Sadlej OEP Basis Set Free  

Excitation BARE KS AEDA AEDA LB94 OEP Expt  

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Figure Al.6.27. Equipotential contour plots of (a) the excited- and (b), (c) ground-state potential energy surfaces. (Here a hamionic excited state is used because that is the way the first calculations were perfomied.) (a) The classical trajectory that originates from rest on the ground-state surface makes a vertical transition to the excited state, and subsequently undergoes Lissajous motion, which is shown superimposed, (b) Assuming a vertical transition down at time (position and momentum conserved) the trajectory continues to evolve on the ground-state surface and exits from chaimel 1. (c) If the transition down is at time 2 the classical trajectory exits from chaimel 2 (reprinted from [52]).

There are significant differences between tliese two types of reactions as far as how they are treated experimentally and theoretically. Photodissociation typically involves excitation to an excited electronic state, whereas bimolecular reactions often occur on the ground-state potential energy surface for a reaction. In addition, the initial conditions are very different. In bimolecular collisions one has no control over the reactant orbital angular momentum (impact parameter), whereas m photodissociation one can start with cold molecules with total angular momentum 0. Nonetheless, many theoretical constructs and experimental methods can be applied to both types of reactions, and from the point of view of this chapter their similarities are more important than their differences. 

Two colliding atoms approach on tire molecular ground-state potential. During tire molasses cycle witli tire optical fields detuned only about one line widtli to tire red of atomic resonance, tire initial excitation occurs at very long range, around a Condon point at 1800 a. A second Condon point at 1000 takes tire population to a 1 doubly excited potential tliat, at shorter intemuclear distance, joins adiabatically to a 3 potential, drought to be die... 

While it is not essential to the method, frozen Gaussians have been used in all applications to date, that is, the width is kept fixed in the equation for the phase evolution. The widths of the Gaussian functions are then a further parameter to be chosen, although it appears that the method is relatively insensitive to the choice. One possibility is to use the width taken from the harmonic approximation to the ground-state potential surface [221]. 

The two coordinates defined for H4 apply also for the H3 system, and the conical intersection in both is the most symmetric structure possible by the combination of the three equivalent structures An equilateral triangle for H3 and a perfect tetrahedron for H4. These sbnctures lie on the ground-state potential surface, at the point connecting it with the excited state. This result is generalized in the Section. IV. 

Figure 3. Relaxed triangular plot [68] of the U3 ground-state potential energy surface using hyperspherical coordinates. Contours, are given by the expression (eV) — —0.56 -t- 0.045(n — 1) with n = 1,2,..,, where the dashed line indicates the level —0.565 eV. The dissociation limit indicated by the dense contouring implies Li2 X Sg ) -t- Li.

If we require similar information regarding the ground state potential energy surface in a polyatomic molecule the electronic emission specttum may again provide valuable information SVLF spectroscopy is a particularly powerful technique for providing it. 

The photochemical behavior of butadienes has been closely studied. When these compounds are exposed to light, they move from the ground state to an excited state. This excited state eventually returns to one of the ground state conformations via a process that includes a radiationless decay (i.e., without emitting a photon) from the excited state potential energy surface back to the ground state potential energy surface. 

The LUMO-HOMO gap is approximately given by the spectroscopic properties of the SO group. Since DMSO absorbs in the region of 2380 and 2560 A, we obtain96 1.24 x 104/2560 = 4.8eV. This measures the drastic enhancement in redox properties anticipated for this compound. Indeed the 4.8 V value for the ground-state potential covers the whole range of usual redox potentials. The same principle applies to the sulfones because their electrode potential and absorption properties differ little from their sulfoxide analogues. 

Diagonalizing the corresponding 2x2 secular equation and some algebraic manipulation gives the four-electron ground-state potential surface... 

Exercise 2.1. Evaluate the ground-state potential surface for the CH3OCH3—> CH3 + CH30- reaction using the reaction field model, with a cavity radius a = R/2 + 1.5. 

Here (in contrast to the approach taken in Chapter 2) we do not assume that the energy of each valence bond structure is correlated with its solvation-free energy. Instead we use the actual ground-state potential surface to calculate the ground-state free energy. To see how this is actually done let s consider as a test case an SN2 type reaction which can be written as... 

FIGURE 5.8. A downhill trajectory for the proton transfer step in the catalytic reaction of trypsin. The trajectory moves on the actual ground state potential, from the top of the barrier to the relaxed enzyme-substrate complex. 1, 2, and 3 designate different points along the trajectory, whose respective configurations are depicted in the upper part of the figure. The time reversal of this trajectory corresponds to a very rare fluctuation that leads to a proton transfer from Ser 195 to His 57. 

Figure 10.5. Schematic of ground-state potential curve for rearrangement of 71 to 72. Four key structures are shown along the path. [Reproduced with permission from P. S. Zuev, R. S. Sheridan, T. V. Albu, D. G. Truhlar, D. A. Hrovat, and W. T. Borden, Science 2003, 299, 867.]...

Iordanov TD, Davis JL, Masunov AE, Levenson A, Przhonska OV, Kachkovski AD (2009) Symmetry breaking in cationic polymethine dyes, part 1 ground state potential energy surfaces and solvent effects on electronic spectra of streptocyanines. Int J Quantum Chem 109 3592-3601... 

The second mechanism, due to the permutational properties of the electronic wave function is referred to as the permutational mechanism. It was introduced in Section I for the H4 system, and above for pericyclic reactions and is closely related to the aromaticity of the reaction. Following Evans principle, an aromatic transition state is defined in analogy with the hybrid of the two Kekule structures of benzene. A cyclic transition state in pericyclic reactions is defined as aromatic or antiaromatic according to whether it is more stable or less stable than the open chain analogue, respectively. In [32], it was assumed that the in-phase combination in Eq. (14) lies always the on the ground state potential. As discussed above, it can be shown that the ground state of aromatic systems is always represented by the in-phase combination of Eq. (14), and antiaromatic ones—by the out-of-phase combination. 

For a recent list of references to calculations of parts of ground state potential energy hypersurface see Salem, L. Accounts Chem. Res. 4, 322 (1971). 

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