Analytic first derivatives

Quantum mechanical calculations are restricted to systems with relatively small numbers of atoms, and so storing the Hessian matrix is not a problem. As the energy calculation is often the most time-consuming part of the calculation, it is desirable that the minimisation method chosen takes as few steps as possible to reach the minimum. For many levels of quantum mechanics theory analytical first derivatives are available. However, analytical second derivatives are only available for a few levels of theory and can be expensive to compute. The quasi-Newton methods are thus particularly popular for quantum mechanical calculations. 

If classical Coulombic interactions are assumed among point charges for electrostatic interactions between solute and solvent, and the term for the Cl coefficients (C) is omitted, the solvated Eock operator is reduced to Eq. (6). The significance of this definition of the Eock operator from a variational principle is that it enables us to express the analytical first derivative of the free energy with respect to the nuclear coordinate of the solute molecule R ,... 

The treatment described above (which was introduced in Ref. 1) is much simpler than the standard treatment (which uses internal coordinates b, 0, large molecules or small proteins, evaluating the second derivative matrix F numerically, using analytical first derivatives. 

Vibrational spectroscopy is of utmost importance in many areas of chemical research and the application of electronic structure methods for the calculation of harmonic frequencies has been of great value for the interpretation of complex experimental spectra. Numerous unusual molecules have been identified by comparison of computed and observed frequencies. Another standard use of harmonic frequencies in first principles computations is the derivation of thermochemical and kinetic data by statistical thermodynamics for which the frequencies are an important ingredient (see, e. g., Hehre et al. 1986). The theoretical evaluation of harmonic vibrational frequencies is efficiently done in modem programs by evaluation of analytic second derivatives of the total energy with respect to cartesian coordinates (see, e. g., Johnson and Frisch, 1994, for the corresponding DFT implementation and Stratman etal., 1997, for further developments). Alternatively, if the second derivatives are not available analytically, they are obtained by numerical differentiation of analytic first derivatives (i. e., by evaluating gradient differences obtained after finite displacements of atomic coordinates). In the past two decades, most of these calculations have been carried... 

Xie W, Song L, Truhlar DG, Gao J (2008) The variational explicit polarization potential and analytical first derivative of energy towards a next generation force field. J Chem Phys 128(23) 234108... 

Analytical first derivatives in presence of the reaction field are easily derived for a fixed cavity size. The expression for the gradients is the same as the one given in equation (33) of Ref. [18c], with an extra term for the reaction field ... 

Kallay, M., Gauss, J., Szalay, P.G. Analytic first derivatives for general coupled-cluster and configuration interaction models. J. Chem. Phys. 2003, 119, 2991-3004. 

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. 

The utility of Eq. (9.49) depends on the ease with which the Hessian matrix may be constructed. Methods that allow for the analytic calculation of second derivatives are obviously the most efficient, but if analytic first derivatives are available, it may still be worth the time required to determine the second derivatives from finite differences in the first derivatives (where such a calculation requires that the first derivatives be evaluated at a number of perturbed geometries at least equal to the number of independent degrees of freedom for tlie molecule). If analytic first derivatives are not available, it is rarely practical to attempt to construct the Hessian matrix. 

M. Cossi, B. Mennucci, R. Cammi, Analytical first derivatives of molecular surfaces with respect to nuclear coordinates, J. Comput. Chem., 17 (1996) 57-73. 

A typical formula for the analytical first derivative of overlap integrals Sy with respect to coordinates of orbitals is. 

Evaluate the gradient to provide analytic first derivatives dE/dq from which d Eldq dq can be obtained by a displacement in bq. Again, symmetry can be exploited to reduce the number of gradient evaluations. Evaluate the second derivative matrix analytically. 

The title indicates that this paper is about the calculation of vibrational force constants and the geometry optimization of polyatomic molecules however, its primary impact on computational chemistry comes from the methodology for calculating analytic first derivatives with respect to molecular coordinates at the Hartree-Fock (HF) level of theory. Applications of first and higher derivatives of the energies obtained by molecular orbital (MO) calculations have revolutionized computational chemistry, allowing molecular structures and properties to be computed efficiently and reliably [1-5]. Almost all electronic structure codes compute analytic first derivatives of the energy, and Pulay s paper was the first to describe a practical calculational approach. 

Insertion of Eq. (5) and recognition that the terms multiplying the density derivative constitute the Fock matrix, F, yields the following compact expression for the analytic first derivative of the HF energy... 

Pulay demonstrated that analytic first derivatives with respect to geometric parameters can be calculated easily and efficiently for HF energies. Derivatives of correlated methods followed a number of years after SCF derivatives [4, 5]. Extensions of the SCF derivatives to density functional theory methods were straightforward. In the three decades since Pulay s article, hundreds of papers on energy derivatives have been published, and all can trace their roots back to his paper. Energy derivatives have become so useful for calculating molecular structures and properties that, almost universally, first derivatives are formulated and coded soon after a new theoretical method is developed for the energy. 

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