Nuclear configuration

In this section we concentrate on the electronic and vibrational parts of the wavefimctions. It is convenient to treat the nuclear configuration in temis of nomial coordinates describing the displacements from the equilibrium position. We call these nuclear nomial coordinates Q- and use the symbol Q without a subscript to designate the whole set. Similarly, the symbol v. designates the coordinates of the th electron and v the whole set of electronic coordinates. We also use subscripts 1 and ii to designate the lower and upper electronic states of a transition, and subscripts a and b to number the vibrational states in the respective electronic states. The total wavefiinction f can be written... 

There are cases where the variation of the electtonic ttansition moment with nuclear configuration caimot be neglected. Then it is necessary to work with equation (B 1.1.6) keeping the dependence of on Q and integrating it over the vibrational wavefiinctions. In most such cases it is adequate to use only the tenns up to first-order in equation (B 1.1.7). This results in modified Franck-Condon factors for the vibrational intensities [12]. 

When spectroscopists speak of electronic selection niles, they generally mean consideration of the integral over only the electronic coordinates for wavefiinctions calculated at the equilibrium nuclear configuration of the initial state, 2 = 0,... 

Oppenlieimer potential which is defined as the electronic ground-state energy for nuclear configuration T, including the niiclear-niiclear repulsion. 

The discussion in the previous sections assumed that the electron dynamics is adiabatic, i.e. the electronic wavefiinction follows the nuclear dynamics and at every nuclear configuration only the lowest energy (or more generally, for excited states, a single) electronic wavefiinction is relevant. This is the Bom-Oppenlieimer approxunation which allows the separation of nuclear and electronic coordinates in the Schrodinger equation. 

Aspects of the Jahn-Teller symmetry argument will be relevant in later sections. Suppose that the electronic states aie n-fold degenerate, with symmetry at some symmetiical nuclear configuration Qq. The fundamental question concerns the symmetry of the nuclear coordinates that can split the degeneracy linearly in Q — Qo, in other words those that appeal linearly in Taylor series for the matrix elements A H B). Since the bras (/1 and kets B) both transform as and H are totally symmetric, it would appear at first sight that the Jahn-Teller active modes must have symmetry Fg = F x F. There... 

The values of this (approximate) fi (qx) calculated from this equation are smaller than 0.08 kcal/mol over the entire nuclear configuration space involved, and to a very good approximation can be neglected. 

If the symmetries of the two adiabatic functions are different at Rq, then only a nuclear coordinate of appropriate symmeti can couple the PES, according to the point group of the nuclear configuration. Thus if Q are, for example, normal coordinates, xt will only span the space of the totally symmetric nuclear coordinates, while X2 will have nonzero elements only for modes of the correct symmetry. 

The Hamiltonian again has the basic form of Eq. (63). The system is described by the nuclear coordinates, Q, which are relative to a suitable nuclear configuration Q. In conbast to Section in.C, this may be any point in configmation space. As a diabatic representation has been assumed, the kinetic energy operator matrix, T, is diagonal with elements... 

In the transition state region, the spin-pairing change mnst take place. At this nuclear configuration, the electronic wave function may be written as... 

It is important to recall, that the reaction takes place on the gronnd-state surface. Clearly, at the same nuclear configuration, the other combination... 

The most stable nuclear configuration of this system is a pair of H2 molecules. There are three possible spin coupling combinations for H4 corresponding to three distinct stable product H2 pairs H1 H2 with H3 H4, H1 H3 with H2 H4, and H1 H4 with H2 H3. Each H atom contributes one electron, the dot diagrams indicate spin pairing. The three combinations are designated as Hfl), HOT), and H(III), respectively. They may be interconverted via square transition states, Figure 2. 

Let S be any simply connected surface in nuclear configuration space, bounded by a closed-loop L. Then, if 4>(r,R) changes sign when transported adiabatically round L, there must be at least one point on S at which (r, R) is discontinuous, implying that its potential energy surface intersects that of another electronic state. 

Consider the coordinate that transforms the nuclear configuration of H(III) at the minimum energy with the corresponding configuration of TS(I-II). In the foiiner, atoms 1 and 4 are close together, as are atoms 2 and 3. The separation between the two pairs is large. In other words, if Rij is the separation between atoms i and j, we have... 

Yarkoni [108] developed a computational method based on a perturbative approach [109,110], He showed that in the near vicinity of a conical intersection, the Hamiltonian operator may be written as the sum a nonperturbed Hamiltonian Hq and a linear perturbative temr. The expansion is made around a nuclear configuration Q, at which an intersection between two electronic wave functions takes place. The task is to find out under what conditions there can be a crossing at a neighboring nuclear configuration Qy. The diagonal Hamiltonian matrix elements at Qy may be written as... 

Note that the electronic kinetic energy operator does not depend on the nuclear configuration explicitly. Therefore, we can conclude that... 

In the present calculations, the molecule is restricted to Cs symmetry. There are five internal degrees of freedom fthe out-of-plane mode is excluded to preserve C, symmetry). Nuclear configurations will be denoted R = (R(H -O). / (0-H ), / (h2-H ), corresponding to the... 

Molecular quan turn mcchan ics finds the solution to a Sch rddinger equation for an electronic Hamiltonian, H i, that gives a total energy, K(,. (.(R) + (R.R). Repeated solutions at different nuclear configurations, R, lead to some approximate potential energy sur-... 

Atoms with the same number of protons but a different number of neutrons are called isotopes. To identify an isotope we use the symbol E, where E is the element s atomic symbol, Z is the element s atomic number (which is the number of protons), and A is the element s atomic mass number (which is the sum of the number of protons and neutrons). Although isotopes of a given element have the same chemical properties, their nuclear properties are different. The most important difference between isotopes is their stability. The nuclear configuration of a stable isotope remains constant with time. Unstable isotopes, however, spontaneously disintegrate, emitting radioactive particles as they transform into a more stable form. 

Substitution. Substitution products retain the same nuclear configuration as naphthalene. They are formed by the substitution of one or more hydrogen atoms with other functional groups. Substituted naphthalenes of commercial importance have been obtained by sulfonation, sulfonation and alkah fusion, alkylation, nitration and reduction, and chlorination. 

---