Intemuclear separation

There can be subtle but important non-adiabatic effects [14, ll], due to the non-exactness of the separability of the nuclei and electrons. These are treated elsewhere in this Encyclopedia.) The potential fiinction V(R) is detennined by repeatedly solving the quantum mechanical electronic problem at different values of R. Physically, the variation of V(R) is due to the fact that the electronic cloud adjusts to different values of the intemuclear separation in a subtle interplay of mutual particle attractions and repulsions electron-electron repulsions, nuclear-nuclear repulsions and electron-nuclear attractions. 

Figure Al.2.1. Potential V(R) of a diatomic molecule as a fiinction of the intemuclear separation i . The equilibrium distance Rq is at the potential minimum.

The close-coupling equations are also applicable to electron-molecule collision but severe computational difficulties arise due to the large number of rotational and vibrational channels that must be retained in the expansion for the system wavefiinction. In the fixed nuclei approximation, the Bom-Oppenlieimer separation of electronic and nuclear motion pennits electronic motion and scattering amplitudes f, (R) to be detemiined at fixed intemuclear separations R. Then in the adiabatic nuclear approximation the scattering amplitude for ... 

With the above in mind, it is sensible to modify the atomic orbital by treating the orbital exponent as a variational parameter. What we could do is vary for each value of the intemuclear separation 7 ab, and for each value of 7 ab calculate the energy with that particular orbital exponent. Just for illustration, I have calculated the energies for a range of orbital exponent and intemuclear distance pairs, and my results are shown as energy contours in Figure 3.3. 

The horizontal axis corresponds to an intemuclear separation running from 1.5 to 2.5 flo and the vertical axis corresponds to an orbital exponent running from 1,0 to 1.4. The potential energy minimum corresponds to an exponent of 1.238, and we note the contraction of the atomic Is orbital on molecule formation. 

If we want to calculate the potential energy curve, then we have to change the intemuclear separation and rework the electronic problem at the new A-B distance, as in the H2 calculation. Once again, should we be so interested, the nuclear problem can be studied by solving the appropriate nuclear Schrodinger equation. This is a full quantum-mechanical equation, not to be confused with the MM treatment. 

Table 16.3 Dipole moment vs intemuclear separation for CO HF/6-311G ...

For a given molecule and a given intemuclear separation a would have a definite value, such as to make the energy level for P+ lie as low as possible. If a happens to be nearly 1 for the equilibrium state of the molecule, it would be convenient to say that the bond is an electron-pair bond if a is nearly zero, it could be called an ionic bond. This definition is somewhat unsatisfactory in that it does not depend on easily observable quantities. For example, a compound which is ionic by the above definition might dissociate adiabatically into neutral atoms, the value of a changing from nearly zero to unity as the nuclei separate, and it would do this in case the electron affinity of X were less than the ionization potential of M. HF is an example of such a compound. There is evidence, given bdow, that the normal molecule approximates an ionic compound yet it would dissociate adiabatically into neutral F and H.13... 

SCF-CI calculations were performed at 20 different intemuclear separations, from 1.2 bohr to 4-00. The lowest separate atom states are, B( P,2p) and H( S) therefore, in order to have a homolytic dissociation and three degenerate 2p orbitals on B we have adopted the closed shell Fock hamiltonian with fractional occupation [23] one electron was placed in the 3(t orbital, correlating with H(ls) at infinite separation, and 1/3 each in the 4it and Itr orbitals correlating with B(2p). 

Figure 1 Relative positions of the potential energy (E) surfaces of the electronic states involved in a hypothetical chemiluminescent reaction as a function of intemuclear separation (r). P and P represent the ground and lowest electronically excited singlet states of the product of the reaction, respectively. R represents the ground electronic state of the reactant. AH is the enthalpy of the dark reaction while AHa is its enthalpy of activation. AH is the enthalpy of activation of the photoreaction, hv denotes the emission of chemiluminescence.

The preferred bond length is rG, because the value on the energy axis is lowest when the intemuclear separation corresponds to this value of bond length. The energy of a bond having exactly this bond length is E0. 

The reason for the insensitivity of sf1 and Kn apparent in Fig.3 is to be found in the weak interaction of the exchange charge distribution p2 — vanishing for large intemuclear separations — with the solvent polarization this is also responsible for the very large value of K22, implying that the solute electronic structure cannot evolve along S2. 

The separate-atom AO basis gives —1.10388 a.u. the single optimised exponent gives — 1.13463 a.u. (exponents 1.0 and 1.333) and the completely optimised basis —1.14518 (both exponents 1.201). We shall return to this point later since, as we have established then, these conclusions are only valid at one value of the intemuclear separation (the experimental value, 1.4 a.u.). 

The conclusion above that optimisation of the non-linear parameters in the AO basis leads to a basis with correct spatial symmetry properties cannot be true for all intemuclear separations. At R = 0 the orbital basis must pass over into the double-zeta basis for helium i.e. two different 1 s orbital exponents. It would be astonishing if this transition were discontinuous at R = 0. While considering the variation of basis with intemuclear distance it is worth remembering that the closed-shell spin-eigenfunction MO method does not describe the molecule at all well for large values of R the spin-eigenfunction constraint of two electrons per spatial orbital is completely unrealistic at large intemuclear separation. With these facts in mind we have therefore computed the optimum orbital exponents as a function of R for three wave functions ... 

Fig. 1. The Hydrogen Molecule total energy as a function of intemuclear separation, RHF, UHF and VB...

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