Analytic second 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. 

Molecular frequencies depend on the second derivative of the energy with respect to the nuclear positions. Analytic second derivatives are available for the Hartree-Fock (HF keyword). Density Functional Theory (primarily the B3LYP keyword in this book), second-order Moller-Plesset (MP2 keyword) and CASSCF (CASSCF keyword) theoretical procedures. Numeric second derivatives—which are much more time consuming—are available for other methods. 

Numerical optimizations are available for methods lacking analytic gradients (first derivatives of the energy), but they are much, much slower. Similarly, frequencies may be computed numerically for methods without analytic second derivatives. 

Analytical gradient energy expressions have been reported for many of the standard models discussed in this book. Analytical second derivatives are also widely available. The main use of analytical gradient methods is to locate stationaiy points on potential energy surfaces. So, for example, in order to find an expression for the gradient of a closed-shell HF-LCAO wavefunction we might start with the electronic energy expression from Chapter 6,... 

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... 

Johnson, B. G., Frisch, M. J., 1993, Analytic Second Derivatives of the Gradient-Corrected Density Functional Energy. Effect of Quadrature Weight Derivatives , Chem. Phys. Lett., 216, 133. 

Since analytic second derivatives are available for MP2 calculations, numerical difference calculations of CCSD(T) energies are only required for a relatively small basis set. This type of basis set correction approximation is also available in Grow. It is not possible to use some composite methods which, like the G2 and G3 schemes,66 involve adding non-differentiable corrections to the estimated electronic energy. However, there are other recently developed composite methods which might be effectively employed to construct this type of interpolated PES.67... 

Kallay, M., Gauss, J. Analytic second derivatives for general coupled-cluster and configuration-interaction models. J. Chem. Phys. 2004, 120, 6841-8. 

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. 

In practice, then, it is fairly straightforward to convert the potential energy determined from an electronic structure calculation into a wealth of thennodynamic data - all that is required is an optimized structure with its associated vibrational frequencies. Given the many levels of electronic structure theory for which analytic second derivatives are available, it is usually worth the effort required to compute the frequencies and then the thermodynamic variables, especially since experimental data are typically measured in this form. For one such quantity, the absolute entropy 5°, which is computed as the sum of Eqs. (10.13), (10.18), (10.24) (for non-linear molecules), and (10.30), theory and experiment are directly comparable. Hout, Levi, and Hehre (1982) computed absolute entropies at 300 K for a large number of small molecules at the MP2/6-31G(d) level and obtained agreement with experiment within 0.1 e.u. for many cases. Absolute heat capacities at constant volume can also be computed using the thermodynamic definition... 

There are several advantages to this approach compared to energy differences (and assuming that analytical second derivatives are not available or are too costly). First,... 

Values have been corrected by using analytical second derivatives. Scaled values were obtained by scaling the diagonal force constants. d From Reference 132, calculated with a (9s5p2d/4s2p) [4s2p2d/4s2p] basis set. 

The computation is confined to RHF-6-311 + +G(2d,2p) level [16]. At one side, the limitation to the RHF is imposed by the actual availability of analytical second derivatives to this level only, but is also justified from other perspectives. A practical reason for the RHF scheme is clearly seen if translate to the same relative position (with /J3/, reference point as origin) the relaxed potential energy surfaces for the umbrella mode of NH3 computed at different levels (RHF, MP2, CCSD(T), B3LYP/6-311 + + G(2d,2p)). Figure 1 shows that the very different methods give... 

Third and fourth order anharmonic coupling constants are calculated using a combination of analytical second derivatives and finite differences [57]. Specifically, we have used the symmetric expressions [58]... 

Values have been corrected by using analytical second derivatives. Scaled values were obtained by scaling the diagonal force constants. 

Analytical second derivatives for closed-shell (or unrestricted Hartree-Fock (UHF)) SCF wavefunctions are used routinely now. The extension to the MCSCF case is relatively new, however. In contrast to the first derivatives, the coupled perturbed SCF equations have to be solved in order to calculate the second and third energy derivatives. The closed-shell case is relatively straightforward, and will be discussed. The multiconfigurational formalism is... 

As we have seen before, the vibrational spectra of molecules require the calculation of the second derivatives of the energy with respect to the atom coordinates about their equilibrium positions. Many DFT program codes can perform the calculation of the dynamical matrix either using analytical second derivatives or numerical differences. As a result their output includes a set of eigenvectors and vibrational frequencies which are the input to programs that calculate INS spectra ( 5.3). 

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