Electronic Hessian states

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... 

Liu J, Liang W (2011) Analytical Hessian of electronic excited states in time-dependent density functional theory with Tamm-Dancoff approximation. J Chem Phys 135 014113... 

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 . ... 

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... 

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... 

Table 12.1 The electronic Hessian of an optimized CASSCF/cc-pVDZ ground-state wave function of the H2 molecule at the equilibrium bond distance of 1.40oo (in Eh). Also listed are the Hessian eigenvalues...

To characterize the stationary point and to distinguish local minima from saddle points, we must consider the second-order variation of the energy E k). For ground states, in particular, we must require the electronic Hessian in (10.1.30) to be positive definite (with respect to the nonredundant orbital rotations). At a stationary point, the expression for this Hessian simphfies somewhat. Invoking the Jacobi identity (1.8.17), we obtain... 

In any Newton-based optimization - which, as discussed in Section 10.8, implicitly or explicitly requires the inversion of the Hessian matrix - the inclusion of redundant parameters is not only unnecessary but also undesirable since, at stationary points, these parameters make the electronic Hessian singular. The singularity of the Hessian follows from (10.2.8), which shows that the rows and columns corresponding to redundant rotations vanish at stationary points. Away from the stationary points, however, the Hessian (10.1.30) is nonsingular since the gradient elements that couple the redundant and nonredundant operators in (10.2.5) do not vanish. Still, as the optimization approaches a stationary point, the smallest eigenvalues of the Hessian will tend to zero and may create convergence problems as the stationary point is approached. Therefore, for the optimization of a closed-shell state by a Newton-based method, we should consider only those rotations that mix occupied and virtual orbitals ... 

In Section 10.8.7, we demonstrated how the key step in the orbital-based second-ordra optimization of the Hartree-Fock state - namely, the linear transformation of trial vectors with the electronic Hessian (needed in the iterative solution of the Newton equations) - may be carried out at a cost (n O ) considerably smaller than that of the construction of the electronic Hessian itself (n O ). Unfortunately, the method developed in Section 10.8.7 requires the transframation of the full set of AO integrals to the MO basis, a computationally expensive task. A more efficient scheme is obtained by recasting the equations in terms of AO integrals, as done in the present subsection [18]. 

The essence of such a combined linear response treatment of the electronic and geometric state-variables is that all their mutual interactions are explicitly taken into account in the generalized electronic-nuclear Hessian. The relevant coupling terms... 

The strategies for saddle point optimizations are different for electronic wave functions and for potential energy surfaces. First, in electronic structure calculations we are interested in saddle points of any order (although the first-order saddle points are the most important) whereas in surface studies we are interested in first-order saddle points only since these represent transition states. Second, the number of variables in electronic structure calculations is usually very large so that it is impossible to diagonalize the Hessian explicitly. In contrast, in surface studies the number of variables is usually quite small and we may easily trans-... 

This transition-state-like point is called a bond critical point. All points at which the first derivatives are zero (caveat above) are critical points, so the nuclei are also critical points. Analogously to the energy/geometry Hessian of a potential energy surface, an electron density function critical point (a relative maximum or minimum or saddle point) can be characterized in terms of its second derivatives by diagonalizing the p/q Hessian([Pg.356]

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