Analytical Hessians

There are a few points with respect to this procedure that merit discussion. First, there is the Hessian matrix. With elements, where n is the number of coordinates in the molecular geometry vector, it can grow somewhat expensive to construct this matrix at every step even for functions, like those used in most force fields, that have fairly simple analytical expressions for their second derivatives. Moreover, the matrix must be inverted at every step, and matrix inversion formally scales as where n is the dimensionality of the matrix. Thus, for purposes of efficiency (or in cases where analytic second derivatives are simply not available) approximate Hessian matrices are often used in the optimization process - after aU, the truncation of the Taylor expansion renders the Newton-Raphson method intrinsically approximate. As an optimization progresses, second derivatives can be estimated reasonably well from finite differences in the analytic first derivatives over the last few steps. For the first step, however, this is not an option, and one typically either accepts the cost of computing an initial Hessian analytically for the level of theory in use, or one employs a Hessian obtained at a less expensive level of theory, when such levels are available (which is typically not the case for force fields). To speed up slowly convergent optimizations, it is often helpful to compute an analytic Hessian every few steps and replace the approximate one in use up to that point. For really tricky cases (e.g., where the PES is fairly flat in many directions) one is occasionally forced to compute an analytic Hessian for every step. 

There are two types of Hessian calculations semimimerical, using a finite difference of analytic gradients, and fully analytic. The analytic approach employed in our method is usually preferable due to the significantly increased accuracy of the calculated vibrational frequencies as well as its considerable time savings. The relative efficiency and accuracy of analytic Hessians increase with the size of the molecule. All calculations presented were performed with the quantum chemistry program GAMESS [70],... 

Koch H, Jdrgen H, Jensen A, Jorgensen P, Helgaker T, Scuseria GE, Schaefer III HF (1990) Coupled cluster energy derivatives. Analytic Hessian for the closed-shell coupled cluster singles and doubles wave function Theory and applications. J Chem Phys 92 4924-4940... 

Quasi-Newton methods can be viewed as extensions of nonlinear CG methods, in which additional curvature information is used to accelerate convergence. Thus, the required analytic Hessian information, memory, and computational requirements are kept as low as possible, and the main strength of Newton methods—employing curvature information to detect and move away from saddle points efficiently—is retained. 

H. Koch, H. J. Aa. Jensen, P. j0rgensen, T. Helgaker, G. E. Scuseria, and H. F. Schaefer, /. Chem. Phys., 92, 4924 (1990). Coupled-Cluster Energy Derivatives. Analytic Hessian for the Closed-Shell Coupled-Cluster Singles and Doubles Wave Functions Theory and Applications. 

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. 

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... 

J. D. Head and M. C. Zerner, Chem. Phys. Lett., 131,359 (1986). An Approximate Hessian for Molecular Geometry Optimization. (Introduced is an approximate analytical Hessian that decreases the amount of work required by a factor of N in ZDO methods, where N is the size of the basis.)... 

The new development of ROHF-MBPT(2) analytical Hessians offers an important tool for the theoretical description of vibrational spectra but it, too, is not a panacea. Although ROHF does not suffer from spin contamination, ROHF does tend to localize unpaired elearons in unphysical ways, which can cause it to offer a poorer reference than UHF for such problems. A case in point is the C4 molecule, where UHF-MBPT(2) is superior to the ROHF-MBPT(2) description. 

Hessian calculations, on the other hand, are much more expensive and their use in PES exploration methods adds appreciable cost to the calculations. Therefore, as we will show in the later sections of this chapter, estimated and updated Hessians are often used where second derivatives are required by the equations directing movement on the PES. Eor some systems, the assumptions used to estimate the Hessian are not valid. In these cases, or for cases where very accurate force constants are necessary for vibrational energy calculations, computed (either numeric or analytic) Hessians may be necessary. 

Minimization of most small and moderately sized systems is handled very well by QN optimization. For more difficult cases, it is sometimes useful to calculate analytic Hessians at the beginning, every few steps, or even at every step, rather than using updated second derivatives. It may also be useful to compute key elements of the Hessian numerically, particularly those corresponding to coordinates changing rapidly in the optimization [80]. These approaches are discussed in more detail in Section 10.3.5. 

The final factors affecting optimization are the choice for the initial Hessian and the method used to form Hessians at later steps. As discussed in Section 10.3.1, QN methods avoid the costly computation of analytic Hessians by using Hessian updating. In that section, we also showed the mathematical form of some common updating schemes and pointed out that the BEGS update is considered the most appropriate choice for minimizations. What may not have been obvious from Section 10.3.1 is that the initial... 

Table 10.3 Comparison of the number of steps required to minimize geometries (QN with RFO algorithm) using a unit matrix, empirically derived Hessian, and analytic Hessian for the initial Hessian followed by Hessian updating and using all analytic Hessians ...

Molecule Initial unit matrix with updating Initial empirical Hessian with updating Initial analytic Hessian with updating All analytic Hessians... 

Gradients provide structures and their numerical second derivatives characterize critical points while providing vibrational frequencies. However, there are still great advantages if the second derivatives can be determined analytically, too. This unique development was accomplished by Gauss and Stanton [128], who implemented analytical Hessians for CC methods from CCSD to CCSD(T) and the full CCSDT. Open-shell (UHF) second derivatives for CC methods were presented by Szalay et al. [129]. Now this has been done automatically for even higher levels of theory with Kallay and Gauss automated approach [130]. 

The anharmonic correction, taken from Ref. [108], was obtained at the fc-MP2 level of theory with the cc-pVDZ basis set [143]. The cubic force field and those parts of the quartic force field that are required for the determination of the anharmonic effects were obtained by means of the numerical differentiation of the analytical Hessian around the equilibrium structure, as implemented in the Aces II program [83, 144]. The harmonic ZPVEs were obtained at the same level as the equilibrium geometries, i.e. ae-CCSD(T)/cc-pCVTZ, and the anharmonic corrections were computed at the fc-MP2/cc-pVDZ level. This is the only contribution that is obtained at a different equilibrium geometry than the other corrections. 

We have seen that the methods for optimization of molecular geometries can be divided into two broad classes second-order methods which require the exact gradient and Hessian in each iteration, and first-order (quasi-Newton) methods which require the gradient only. Both methods are in widespread use, but the first-order methods are more popular since analytical energy gradients are available for almost all electronic structure methods, whereas analytical Hessians are not. Also, the simpler first-order methods usually perform quite well, converging in a reasonable number of iterations in most cases. 

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