Electronic Hessian

Introducing the following notation for the electronic gradient and the electronic Hessian of the optimized wave function... 

To evaluate the first- and second-order molecular properties, we choose the diagonal representation of the Hamiltonian. In this representation, the electronic energy, the electronic gradient, and the electronic Hessian of the electronic ground state 0) may be written in the following manner... 

Since the electronic Hessian has been evaluated at a stationary point, it no longer contains any gradient-like terms (compare with Eq. 37). Using these expressions in Eq. 18 and Eq. 21, we obtain... 

For the optimization of Hartree-Fock wave functions, it is usually sufficient to apply the SCF scheme described in Sec. 3.1. By contrast, the optimization of MCSCF wave functions requires more advanced methods (e.g., the quasi-Newton method or some globally convergent modification of Newton s method, which involves, directly or indirectly, the calculation of the electronic Hessian as well as the electronic gradient at each iteration) [45]. 

The combination huS — E) is also called the principal propagator M and consists of the electronic Hessian matrix E with the elements... 

Hamiltonian With these operators the two off-diagonal blocks E and E of the electronic Hessian matrix and S and of the overlap matrix become zero [see Exercise 3.12] and the propagator in Eq. (3.159) can be written as... 

Noting, furthermore [see Exercise 3.13] that the state-transfer operators reduce the elements of the diagonal blocks of the electronic Hessian matrix to... 

Exercise 3.12 Show that the off-diagonal blocks and of the electronic Hessian matrix and the off-diagonal blocks and of the overlap matrix vanish if one chooses the state transfer operators, Eq. (3.165), as operators h . ... 

Finding the poles of the propagator corresponds therefore to solving the generalized eigenvalue problem for the electronic Hessian matrix or the principal propagator, which is written out here in more detail... 

Applied to SOPPA this means that the electronic Hessian and overlap matrices are each split in three contributions... 

The solution of the linear equations (equation 66) is usually less time-consuming than the calculation of the two-electron contribution to the Hessian (equation 64), but is treated in somewhat more detail here in order to demonstrate some general principles. The number of orbital rotations is usually so large that the electronic Hessian cannot be constructed or stored explicitly. Instead, iterative techniques are used, where the key step is the evaluation of matrix-vector products such as... 

Restricting ourselves to orthogonal transformations and taking the derivatives of the energy (equation 121) with respect to the elements of the antisymmetric matrix x, we obtain the following expressions for the electronic gradient and the electronic Hessian at = 0 ... 

As expected, the configuration part of the MCSCF gradient is similar to that of the Cl wave function (11.4.23) and its orbital part is similar to that of the Hartiee-Fock wave function (10.1.25). The elements of the MCSCF electronic Hessian are given by... 

The elements of the MCSCF Hessian matrix in (12.2.28)-( 12.2.30) contain contributions that depend on the electronic gradient and vanish at the stationary point. For future reference and manipulation, it is convenient to separate these gradient-containing terms from the remaining contributions to the electronic Hessian. Thus, introducing the notation... 

We are now ready to consider the requirements for an optimized MCSCF state. In the present subsection, we examine the electronic gradient at stationary points in Section 12.2.8, we consider the electronic Hessian. According to (12.2.26) and (12.2.27), the stationary condition for an MCSCF wave function may be written as... 

To analyse the dependence of the level-shifted Newton step (12.3.10) on the numerical parameter p, it is convenient to transform the equations to the diagonal representation of the Hessian. We emphasize that this transformation is carried out only for the purpose of the analysis — in practice, no diagonalization is ever attempted. Diagonalizing the MCSCF electronic Hessian, we obtain... 

In calculations of excited states, the selection of the level-shift parameter is less clear-cut than for minimizations. For example, in an optimization of the first excited state, we would like the final electronic Hessian to have one and only one negative eigenvalue. The search is therefore for a... 

Having outlined the general strategy for optimizing MCSCF wave functions, we now turn our attention to the more practical aspects of the optimization. The number of parameters in an MCSCF wave function is usually so large (up to several million parameters) that the explicit calculation of the electronic Hessian and its subsequent inversion or diagonalization is out of the question. Even the storage of the Hessian matrix is usually not possible. Instead, we solve the level-shifted Newton equations (12.3.9)... 

As noted in the introduction to this section, the electronic Hessian is never explicitly set up in large-scale MCSCF calculations - the number of variational parameters would make this task impossible. Solving instead the MCSCF equations iteratively, the Hessian is needed for only a fairly small number of linear transformations (compared with the number of variational parameters). Such transformations can be carried out quite efficiently using the techniques described in this subsection [ 10]. 

Although the MCSCF electronic Hessian is only very rarely constructed explicitly (except perhaps the orbital-orbital block), we shall briefly comment on its calculation. Again, we concentrate on the nonseparable part of the Hessian (12.5.3). The orbital-orbital block (12.2.43) may be calculated in the same way as the Hartree-Fock Hessian in Section 10.8.6. The configuration-configuration block (12.2.41) is identical to the Cl Hessian and may be obtained using the... 

Table 12.1 contains the electronic Hessian of an optimized valence CASSCF wave function of the H2 molecule at the experimental bond distance of 1.40oo- The wave function is a linear combination of the lo ) and o ) configurations, the orbitals of which have been variationally optimized in the cc-pVDZ basis ... 

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