Molecular Hessians

Vibrational frequencies measured in IR experiments can be used as a probe of the metal—ligand bond strength and hence for the variation of the electronic structure due to metal—radical interactions. Theoretical estimations of the frequencies are obtained from the molecular Hessian, which can be straightforwardly calculated after a successful geometry optimization. Pure density functionals usually give accurate vibrational frequencies due to an error cancellation resulting from the neglect of... 

Since H 2) is a rank 2 operator the first term in this expression may be calculated using the technique described for the evaluation of the molecular gradient, using the same effective density matrices. The second and third terms require the solution vectors t(l> of Eq. (171). As the commutators [t, H<0)] in Eq. (171) and [7 <1), H(1>] and [7 (1), TU), H<0)] in Eq. (190) contain operators of third and higher ranks, it does not appear practical to calculate these terms using a density matrix formulation. Since the implementation of CC molecular Hessians has not yet started, we do not discuss the evaluation of CC molecular Hessians in more detail. 

Exercise 10.5 Explain why the second-order contributions to the molecular Hessian matrix consist of the following type of matrix elements F - - P 4>j), ("hj) F + V ... 

The lowest-order molecular properties now correspond to the molecular gradient F and the molecular Hessian G ... 

We now proceed to a consideration of the molecular Hessian - that is, the matrix of second derivatives of the molecular electronic energy with respect to geometrical distortions. Differentiating the molecular gradient in the form of equation (18), we obtain from the chain rule... 

We conclude that for a fully variational wavefunction only the first-order response of the wavefunction dX/dx is required to calculate the energy to second order. In particular, the second-order response of the wavefunction d X/dx is not needed for the evaluation of the molecular Hessian. 

We have established that for a fully variational wavefunction we may calculate the molecular gradient from the zero-order response of the wavefunction (i.e., from the unperturbed wavefunction) and the molecular Hessian from the first-order response of the wavefunction. In general, the 2n + 1 rule is obeyed for fully variational wavefunctions, the derivatives (responses) of the wavefunction to order n determine the derivatives of the energy to order 2n -i- 1. This means, for instance, that we may calculate the energy to third order with a knowledge of the wavefunction to first order, but that the calculation of the energy to fourth order requires a knowledge of the wavefunction response to second order. These relationships are illustrated in Table 1. 

Let us briefly consider the evaluation of the SCF molecular Hessian in second quantization. Proceeding in the manner outlined in Section 4, we arrive at the following expression for the molecular Hessian... 

The time-consuming step in the evaluation of the SCF molecular Hessian is the calculation of two-electron contributions from since each integral now makes 78... 

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