Nuclear framework

Atoms have complete spherical synnnetry, and the angidar momentum states can be considered as different synnnetry classes of that spherical symmetry. The nuclear framework of a molecule has a much lower synnnetry. Synnnetry operations for the molecule are transfonnations such as rotations about an axis, reflection in a plane, or inversion tlnough a point at the centre of the molecule, which leave the molecule in an equivalent configuration. Every molecule has one such operation, the identity operation, which just leaves the molecule alone. Many molecules have one or more additional operations. The set of operations for a molecule fonn a mathematical group, and the methods of group theory provide a way to classify electronic and vibrational states according to whatever symmetry does exist. That classification leads to selection rules for transitions between those states. A complete discussion of the methods is beyond the scope of this chapter, but we will consider a few illustrative examples. Additional details will also be found in section A 1.4 on molecular symmetry. 

Some details of END using a multiconfigurational electronic wave function with a complete active space (CASMC) have been introduced in terms of an orthonormal basis and for a fixed nuclear framework [25], and were recently [26] discussed in some detail for a nonoithogonal basis with electron translation factors. 

Another aspect of wave function instability concerns symmetry breaking, i.e. the wave function has a lower symmetry than the nuclear framework. It occurs for example for the allyl radical with an ROHF type wave function. The nuclear geometry has C21, symmetry, but the Cay symmetric wave function corresponds to a (first-order) saddle point. The lowest energy ROHF solution has only Cj symmetry, and corresponds to a localized double bond and a localized electron (radical). Relaxing the double occupancy constraint, and allowing the wave function to become UHF, re-establish the correct Cay symmetry. Such symmetry breaking phenomena usually indicate that the type of wave function used is not flexible enough for even a qualitatively correct description. 

For the sake of simplicity, we will here confine ourselves to consider a system of N electrons moving in a given nuclear framework. The stationary states of such a system are described by the solutions to the Schrodinger equation... 

For practical purposes two different approaches have been used. If the nuclear framework has a center with high degree of symmetry, it may be convenient to expand the Hartree-Fock functions fk(r) in terms of spherical harmonics Ylm(6, q>) around this center ... 

The VB and MO theories are both procedures for constructing approximations to the wavefunctions of electrons, but they construct these approximations in different ways. The language of valence-bond theory, in which the focus is on bonds between pairs of atoms, pervades the whole of organic chemistry, where chemists speak of o- and TT-bonds between particular pairs of atoms, hybridization, and resonance. However, molecular orbital theory, in which the focus is on electrons that spread throughout the nuclear framework and bind the entire collection of atoms together, has been developed far more extensively than valence-bond... 

The system under study is assumed to consist of 2A, electrons, possibly in the presence of a nuclear framework. An orbital picture of the quantum behaviour of the system is then introduced on accepting the validity of an independent-particle model where each electron moves in the field of an effective potential v(r), which afterwards is left substantially unspecified. We emphasize, however, that the choice of v(F) is an essential step of any modeling. Besides semiempirical forms, effective potentials v[ (r)] functionally dependent on the electron numeral density n(r) are intuitively bound to play a prominent role in applications. 

In 2000 Meyer and co-workers reported a novel example of unusual 4-peroxo coordination (complex (146)), as well as of a structurally analogous complex in which the 0-0 linkage is formally cleaved and replaced by two OH units (147), while at the same time the overall tetra-nuclear framework is fully conserved.145... 

If we consider a molecule as having a static but continuous distribution of electronic charge around a rigid nuclear framework, then its electrical or electrostatic potential will have a term similar to Eq. (3.2), with Q. being the positive charges of the nuclei, ZA, and a... 

To achieve non-zero 7ta—7tb conjugation, the pi NBOs of 18 may polarize in opposite directions, leading to a wavefunction of lower symmetry than the nuclear framework. Alternatively, the nuclear framework may distort to diamond-like D2h geometry. However, each such distortion destabilizes what is already a highly unfavorable Lewis-structure wavefunction, so cyclobutadiene is expected to remain highly destabilized relative to other possible polyene topologies. 

The MO theory differs greatly from the VB approach and the basic MO theory is an extension of the atomic structure theory to molecular regime. MOs are delocalized over the nuclear framework and have led to equations, which are computationally tractable. At the heart of the MO approach lies the linear combination of atomic orbitals (LCAO) formahsm... 

The electron density p(r) and the molecular M-nuclear framework ZA, RA together generate an EPS ITr,) in all points of space r, ... 

Equation 15.1 have been derived. The key idea is to replace the continuous density function and the nuclear framework with the following much simpler expression (for a review, see Ref. [9]) ... 

Finally, the remaining (/a, Q) representation describing the equilibrium state of an externally open molecular system with the frozen nuclear framework is examined. The relevant partial Legendre transform of the total electronic energy, which replaces N by /a in the list of independent state-parameters, defines the BO grand-potential ... 

Nuclei move much more slowly than the much-lighter electrons, so when a transition occurs from one electronic state to another, it takes place so rapidly that the nuclei of the vibrating molecule can be assumed to be fixed during the transition. This is called the Franck-Condon principle, and a consequence of it is that an electronic transition is represented by a vertical arrow such as that shown in Figure 2.5 that is, an electronic transition occurs within a stationary nuclear framework. Thus the electronic transition accompanying the absorption of a photon is often referred to as a vertical transition or Franck-Condon transition. 

Under the Born-Oppenheimer approximation, two major methods exist to determine the electronic structure of molecules The valence bond (VB) and the molecular orbital (MO) methods (Atkins, 1986). In the valence bond method, the chemical bond is assumed to be an electron pair at the onset. Thus, bonds are viewed to be distinct atom-atom interactions, and upon dissociation molecules always lead to neutral species. In contrast, in the MO method the individual electrons are assumed to occupy an orbital that spreads the entire nuclear framework, and upon dissociation, neutral and ionic species form with equal probabilities. Consequently, the charge correlation, or the avoidance of one electron by others based on electrostatic repulsion, is overestimated by the VB method and is underestimated by the MO method (Atkins, 1986). The MO method turned out to be easier to apply to complex systems, and with the advent of computers it became a powerful computational tool in chemistry. Consequently, we shall concentrate on the MO method for the remainder of this section. 

Here the R form the set of linear coordinate transformations that leave the nuclear framework invariant, yC are the characters associated with the fi ,-dimensional irreducible representation and g is the order of the point group, G. 

In order to apply group-theoretical descriptions of symmetry, it is necessary to determine what restrictions the symmetry of an atom or molecule imposes on its physical properties. For example, how are the symmetries of normal modes of vibration of a molecule related to, and derivable from, the full molecular symmetry How are the shapes of electronic wave functions of atoms and molecules related to, and derivable from, the symmetry of the nuclear framework ... 

By the symmetry of a normal mode of vibration, we mean tbe symmetry of the nuclear framework under the distortion introduced by the vibration. Pictorially, the symmetry of the normal mode is equal to the symmetry of the pattern of arrows drawn to indicate the directions of the nuclear displacements. The normal modes of vibration of water are the symmetric and antisymmetric stretches, and the angle bend, shown in Figure 6-1. 

If we imagine a continuous deformation of the nuclear framework from the bent to the linear geometry, we expect that the orbitals will change continuously also. 

In summary, the force Fa felt by the nuclear framework due to a displacement of center-a along the x, y, or z axis is given as... 

In this chapter we will familiarize ourselves with basic concepts in molecular symmetry [17]. The presence or absence of symmetry has consequences on the appearance of spectra, the relative reactivity of groups, and many other aspects of chemistry, including the way we will make use of orbitals and their interactions. We will see that the orbitals that make up the primary description of the electronic structure of molecules or groups within a molecule have a definite relationship to the three-dimensional structure of the molecule as defined by the positions of the nuclei. The orientations of the nuclear framework will determine the orientations of the orbitals. The relationships between structural units (groups) of a molecule to each other can often be classified in terms of the symmetry that the molecule as a whole possesses. We will begin by introducing the basic termi-... 

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