Ground State Molecular Structures

To optimise the geometry, the energy must be expressed as a function of atomic displacements. This yields the partial derivatives crucial to automatic minimisation algorithms. The expressions for the total energy derivatives with respect to atomic displacements are quite complex for ab initio and semi-empirical methods but trivial for empirical schemes like Molecular Mechanics (MM). Virtually all modern computer codes provide extensive, efficient facilities for determining ground state molecular geometries. 

Overall, the absolute agreement between theory and experiment currently attainable for TM systems is the reproduction of bond lengths to within about 0.02 A and angles to within a few degrees. However, we must keep in mind that 

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Fig. 11 Relative size and orientation of dipole moment vectors of the ground state black) and the excited states JMLCT (bJA ), 3IL (b3A"), and 3MLCT (a3A") of [Re(Etpy)(CO)3(bpy)]+, projected onto the optimized ground-state molecular structure. Dipole moment vectors originate in the center of charge calculated using Mulliken population analysis. They lie in the molecular symmetry plane. (Calculated by TD-DFT G03/PBE0/vacuum at the optimized ground state geometry.) Reproduced with permission from [76]...

Topsom, R.D. (1987c). Substituent Effects on Ground-State Molecular Structures and Charge Distributions. Prog.Phys.Org.Chem., 16, 85-124. 

It is well known that the ground-state molecular structures of 7r-electron systems are usually well described by molecular orbital theory, even at the one-electron level (93). In the Hiickel Molecular Orbital (HMO) Theory, the relationship between structure and theory arises... 

Figure 22 DBH ground state equilibrium structure 3 and molecular and electronic structure for the computed low-lying real crossing 10,11. In this system the S1(n-ic )/ S0, T2(n-7ill )/T1(rc-7i 1 ) conical intersections and the T1(Tt-7t ,)/S0 and T2(n-7i,l )/S0 triplet/singlet crossings occur at the same molecular structure. The relevant geometrical parameters are in angstrom units.

Figure 12 Ground-state equilibrium structure of methylene. The molecule is chosen to lie in the yz plane with z as symmetry axis. With this convention, the triplet ground state is of 3Bi symmetry. If x and z span the molecular plane, Bi and B2 have to be interchanged.

Fig. 18. Ground state electronic structures for cresyl and o-(methylthio)cresyl phenoxyl radicals. Isosurface representations of molecular orbitals solved by ab initio density functional theory methods for cresyl (ere) and o-(methylthio)cresyl (mtc) phenoxyl radicals. Eigenvalues are listed and for each the SOMO is identihed with an asterisk ( ).

PE spectra can show bands corresponding to ion states that cannot be accessed from the ground-state molecular configuration by the removal of one electron. Such signals can normally be represented by removal of one electron synchronized with excitation of another they are hence referred to as many-electron processes, and the additional satellite structure accompanying the primary band as shake-up structure. A classic example of a two-electron transition is the very weak band found in the spectrum of atomic mercury for the process shown in equation (12) ... 

This expansion of the total electronic wavefunction is very compact, and provides a great deal of physical and chemical visuality. The spin-coupled structure (18) by itself reproduces with very reasonable accuracy all the features of a ground-state molecular potential energy surface. For example the spin-coupled function typically yields 85% of the observed binding energy, and equilibrium internuclear separations are accurate to 0.01 A. This function consequently dominates expansion (17) for all nuclear geometries. The various excited structures provide angular and other types of correlation as an extra quantitative refinement but do not alter the qualitative picture. 

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