Zero Eigenvalues of the Hessian

Each teT is a vector in the kernel of the Hessian matrix H(p). Because dim(T) 3, with three coordinates of the displacement tclR, the translational invariance implies the existence of three zero eigenvalues of the Hessian matrix. Equation (67) also means that the rows (or the colimns by symmetry) of H(p) belong to T . 

True zero eigenvalues of the Hessian occur if the potential energy, E,... 

So, of the 9 eartesian displaeements, 3 are of ai symmetry, 3 of b2,2 of bi, and 1 of a2- Of these, there are three translations (ai, b2, and b i) and three rotations (b2, b i, and a2). This leaves two vibrations of ai and one of b2 symmetry. For the H2O example treated here, the three non zero eigenvalues of the mass-weighted Hessian are therefore of ai b2, and ai symmetry. They deseribe the symmetrie and asymmetrie streteh vibrations and the bending mode, respeetively as illustrated below. 

Each maximum, minimum or saddle point occurs at a so-called critical point Tc, where the gradient vanishes. The nature of the critical point is determined by the eigenvalues of the Hessian. All the eigenvalues are real at the critical point, but some of them may be zero. The rank co of the critical point is defined to be the number of non-zero eigenvalues. The signature o is the sum of the signs of the eigenvalues, and critical points are discussed in terms of the pair of numbers (w, o). 

Remark 7 Theorem 2.2.8 provides the conditions for checking the convexity or concavity of a function /( ). These conditions correspond to positive semidefinite (P.S.D.) or negative semidef-inite (N.S.D.) Hessian of /(jc) for all x S, respectively. One test of PSD or NSD Hessian of f x) is based on the sign of eigenvalues of the Hessian. If all eigenvalues are greater than or equal to zero for all jc S, then the Hessian is PSD and hence the function /(jc) is convex. If all eigenvalues are less or equal than zero for all x S then the Hessian is NSD and therefore the function /(jc) is concave. 

Fi. 3a and 3b show the lowest eigenvalues of the Hessian for the RHF wave function as functions of the intemuclear distance. One can see that there is at least one negative eigenvalue at every distance, i.e. the RHF function has a saddle point cheuracter everywhere, CThe existence of the UHF and GHF solutions with energies lower than the RHF would not necessarily impty that conclusion, since the RHF solution could have, in principle, also a true local minimum.) One of the eigenvalues of the Hessian sharply decreases with increasing intemuclear cUstance and at R = 1.793 A it crosses the zero level this is die bi cation point in which the specific UHF/1 solution is appearing. 

The lowest eigenvalues of the Hessian for the GHF/1 solution are depicted in Figs. 5a and 5b. The picture is now somewhat simpler there is a zero eigenvalue at all distances, connected with the rotational arbitrariness mentioned above, but there is no negative eigenvalue until R -1.824 A. which is the point where the GHF/2 solution starts to exist. 

A point K of M where the gradient of E(K) vanishes [where the tangent hyperplane to E(K) is "horizontal"], is a point where the force of deformation is zero, i.e., point K represents an equilibrium configuration. Such a point is called a critical point, and is denoted by K(A,i). Here, the first derivatives being zero, the second partial derivatives of the energy hypersurface are used to characterize the critical points. The first quantity in the parentheses, X, is the critical point index (and not the "order of critical point" as it is sometimes incorrectly called). The index A, of a critical point is defined as the number of negative eigenvalues of the Hessian matrix H(K(A,i)), defined by the elements... 

The energies of most of the defects were minimised using the Newton-Raphson method with BFGS [14] updating of the Hessian. However, it was sometimes necessary to use the more demanding Rational Function Optimiser (RFO) [15] which enforces the required number of imaginary eigenvalues of the Hessian, to be zero at the minimum and one when used to locate a transition state as discussed later. 

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