Analytic Gradients and Hessians

The development gradient-corrected XC functionals has definitely been one of the keys to the recent surge in the popularity of DFT. Nevertheless, such functionals could not have had as much impact had it not been for the concurrent development of analytic DFT energy gradients, the first derivatives of the DFT total energy with respect to any of the nuclear coordinates. Without 

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The advantages of MPn perturbation treatments are however clear on both the theoretical and computational points of view. For example, size-consistency is ensured, analytical gradients and Hessians are avalaible, parallelization of the codes is feasable. 

Analytical gradients and Hessians are available for CASSCF, and it is expected that this technology will be extended to the MR-CI and MP2 methods soon. Further, by virtue of the multireference approach, a balanced description of ground and excited states is achieved. Unfortunately, unlike black boxes such as first-order response methods (e.g., time-dependent DFT), CAS-based methods require considerable skill and experience to use effectively. In the last section of this chapter, we will present some case studies that serve to illustrate the main conceptual issues related to computation of excited state potential surfaces. The reader who is contemplating performing computations is urged to study some of the cited papers to appreciate the practical issues. 

Other MP2 based solvent methods consist of the Onsager MP2-SCRF [19], within a relaxed density scheme analogous to the PTDE scheme, and a multipole MP2-SCRF model [28], based on a iterative unrelaxed approach. The analytical gradients and Hessian of the free energy at MP2-PTE level, has been developed within the PCM framework [29],... 

The ground state force field, vibrational normal modes and frequencies have been obtained with MCSCF analytic gradient and hessian calculations [176]. Frequencies computed with the DZ basis set are compared with experimental ones in Table 16. The T - So transition moments were obtained using distorted benzene geometries with atomic displacements along the normal modes, and with the derivatives in Eq. 97 obtained by numerical differentiation. The normal modes active for phosphorescence in benzene are depicted in Fig. 12. The final formula for the radiative lifetime of the k spin sublevel produced by radiation in all (i/f) bands is (ZFS representation x,y,z is used [49]) ... 

To discuss the form and cost of analytic gradient and Hessian evaluations, we consider the simple case of Hartree-Fock (HF) calculations. In nearly all chemical applications of HF theory, the molecular orbitals (MOs) are represented by a linear combination of atomic orbitals (LCAO). In the context of most electronic structure methods, the LCAO approximation employs a more convenient set of basis functions such as contracted Gaussians, rather than using actual atomic orbitals. Taken together, the collection of basis functions used to represent the atomic orbitals comprises a basis set. 

Analytic gradients and hessians for RHF, ROHF, UHF, GVB, and MCSCF wavefunctions that are used to locate stationary points on the potential energy surface and identify their character (local minimum or transition state). 

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],... 

In the following section we discuss the analytical calculation of gradients and Hessians of MP2 energies. 

P. Jorgensen and J. Simons, J. Chem. Phys., 79, 334 (1983). Ab Initio Analytical Molecular Gradients and Hessians. 

Since the late 1970s, significant attention has been devoted to the evaluation of analytical energy derivatives (gradients and hessians) with respect to atomic... 

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. 

A vibrations calculation is the first step of a vibrational analysis. It involves the time consuming step of evaluating the Hessian matrix (the second derivatives of the energy with respect to atomic Cartesian coordinates) and diagonalizing it to determine normal modes and harmonic frequencies. For the SCFmethods the Hessian matrix is evaluated by finite difference of analytic gradients, so the time required quickly grows with system size. 

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