Hessian matrix electronic

Many second-order quantities like this Hessian matrix involve the response of the electronic structure of the system, and their accurate calculation is difficult in standard DFT. Instead, DFPT as outlined below provides a powerful tool enabling access to these derivatives with moderate computational effort but very good precision. 

Another insight into the nature of a covalent bond is provided by analysing the anisotropy of the electron density distribution p (r) at the bond critical point p. For the CC double bond, the electron density extends more into space in the direction of the n orbitals than perpendicular to them. This is reflected by the eigenvalues 2, and k2 of the Hessian matrix, which give the curvatures of p (r) perpendicular to the bond axis. The ratio 2, to /.2 has been used to define the bond ellipticity e according to equation 8S0 ... 

Consider a given molecular system consisting of m atoms. In what follows we adopt the AIM resolution to define the canonical AIM chemical potentials (electron population gradient), p = dE/dN = (fiu fi2,..., fim), and the corresponding AIM hardness matrix (electron population hessian) tj = d2E/dN dN = dp/dN = here all differentiations are carried out for the fixed external potential v. This canonical charge-sensitivity information will be used to generate a variety of system charge sensitivities (CS) that probe the responses of the system to various populational perturbations at constant v. 

Then, analyzing the electron density topology requires the calculation of Vp and of the hessian matrix. After diagonalization one can find the critical points in a covalent bond characterized by a (3, -1) critical point, the positive curvature X3 is associated with the direction joining the two atoms covalently bonded, and the X2, curvatures characterize the ellipticity of the bond by ... 

Sometimes the effect of off-diagonal elements of the Hessian is significant. This occurs, for example, when pairs of floating spherical Gaussians are used to represent p-orbitals [33]. In this case, in-phase and out-of-phase motion of parameters associated with each lobe of the p—orbital have very different frequencies. When the effect of the full Hessian matrix must be incorporated to decrease the width of the electronic parameter frequency spectrum, the parameter kinetic energy can be generalized to include a mass matrix [33]. 

This symmetric solution represents a minimum in the region where all the eigenvalues of the Hessian matrix are positive, Z>ZC = y/2. For values of Z smaller than Zc, the solutions become unsymmetrical with one electron much closer to the nucleus than the other (ri / r2). In order to describe this symmetry breaking, it is convenient to introduce new variables (r, q) of the form... 

The Mode-Tracking idea originated from the fact that the standard quantum chemical calculation of vibrational spectra within the harmonic approximation requires the calculation of the complete Hessian matrix [92]. The Hessian is the matrix of all second derivatives of the electronic energy E i with respect to the nuclear coordinates R. Its calculation gets more computer time demanding the larger the molecule is. However, many if not most vibrations of a supramolecular assembly are of little importance for the function and chemical behavior of this assembly. 

The fact that die number of FICs needed to compute any vibrational hyperpolarizability does not depend upon the size of the molecule leads to important computational advantages. For Instance, the calculation of the longitudinal component of the static y" for l,l-dlamino-6,6-diphosphinohexa-l,3,5-triene requires quartic derivatives of the electronic energy with respect to vibrational displacements (i.e. quartic force constants) [34]. Such fourth derivatives may be computed by double numerical differentiation of the analytical Hessian matrix. With normal coordinates it is necessary to compute the Hessian matrix 3660 times, whereas using FICs only 6 Hessian calculations are required. 

This expression is again written in an abbreviated form using the index permutation operators and Since the symmetric combination of Fock matrix elements is required for the Hessian matrix and the antisymmetric combination for the gradient elements, the matrix F itself may be explicitly constructed and then its contributions included into the other matrix elements as needed. Eq. (166) shows that a larger subset of two-electron integrals are required for the Hessian elements than for the computation of the gradient elements (two unoccupied orbital indices instead of just one), but for typical MCSCF wavefunctions and orbital basis sets this is still only a small fraction of the total. 

Finally the state Hessian matrix M is seen from Eq. (149) to be proportional to the representation of the Hamiltonian operator in the orthogonal complement basis, but with all the eigenvalues shifted by the constant amount (0). The dimension of the matrix M will be one less than the length of the CSF expansion unless it is constructed in the linearly dependent projected basis or the overcomplete CSF expansion set basis. Since the Hamiltonian matrix must usually be constructed in the CSF basis in the MCSCF method anyway, it is most convenient if M and C are also constructed in this basis. The transformation to the projected basis, if explicitly required, involves the projection matrix (1 — cc ). The matrix M only requires the two-electron integral subset that consists of all four orbital indices corresponding to occupied orbitals. 

The discussion above has centred around full second-order optimization methods where no further approximations have been made. Computationally such procedures involve two major steps which consume more than 90% of the computer time the transformation of two-electron integrals and the update of the Cl vector. The latter problem was discussed, to some extent, in the previous section. In order to make the former problem apparent, let us write down the explicit formula for one of the elements of the orbital-orbital parts of the Hessian matrix (31), corresponding to the interaction between two... 

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