Essentials of Computational Chemistry

13.4 EMPIRICAL VALENCE BOND METHODS 479
For the charged VB wave function, we take
−αOH(r
H1 = DOH 1 − e ObH∗ −rOH,eq) 2
+
+
H∈b
+
1
2 kHOH(θ HO b H − θHOH,eq) 2
qiqj
εrij
i∈a j∈b
+
i∈a j∈b
4εij
H=H ∗
σij 12 rij
1
2 kOH(rOH − rOH,eq) 2
6
σij
− + Ɣ2
rij
(13.11)
where, with the exception of the final term, differences between Eqs. (13.10) and (13.11)
simply reflect the differences in O–H ∗ connectivity between Eqs. (13.8) and (13.9). The final
term, Ɣ2, is a parameter that is adjusted to make the relative energies of the two VB wave
functions correct at their respective minima. In the current example Ɣ2 may be determined
as the energy of proton transfer from one water molecule to another in the gas phase (we are
still considering only a two-molecule system), which energy is available from mass spectral
measurements.
If we were to partially optimize the geometries of the systems having the two possible
connectivities, holding fixed only the difference in the two bond lengths from O a and O b
each to H∗ , we would obtain the set of energy curves for the two VB Hamiltonians shown
in Figure 13.7. A crude definition of the energy for the reaction coordinate would then be
simply to take the minimum of H1 or H2 as one proceeds from left to right in the proton
transfer process. Mathematically, that process is equivalent to taking as our energy the lowest
eigenvalue of the 2 × 2matrix
H11 H12
H =
H21 H22
(13.12)
where H11 is taken to be H1, H22 is taken to be H2, andH12 and H21 are taken to be zero.
The matrix then being diagonal, the lowest eigenvalue is simply the lower of H1 or H2.
However, Eq. (13.12) is not simply an odd exercise in matrix algebra. Instead, it suggests a
more chemical approach to obtaining the energy of the reacting system. Along the way from
the VB structure of Eq. (13.8) to that of Eq. (13.9), the system obviously passes through
a region where it is best described as a mixture of these two extreme resonances. The
mathematical way in which this mixing can be accomplished is to allow the off-diagonal
matrix elements in Eq. (13.12) to be non-zero. The most widely used approximation for
these off-diagonal elements in a case like the one discussed thus far is
H12 = H21 = Ae −B(rOO−rOO,‘eq’)
(13.13)
where A, B, andrOO,‘eq’ are parameters to be optimized against experimental data. Thus,
the coupling is designed to decrease exponentially as the two heavy atoms separate from
some maximally coupled distance (for more complicated approaches to model H12 see, for
example, Chang, Minichino, and Miller 1992 and Kim et al. 2000). With a non-zero coupling,
diagonalization of H in Eq. (13.12) will give a curve for the lowest eigenvalue, H0 shown
in Figure 13.7, that smoothly connects the two VB extrema. In addition, the eigenvector