Analytical gradients

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 stationary 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, 

Differentiating with respect to some parameter a , we see that it is necessary to evaluate terms such as 

Similar considerations apply to the electric dipole moment the derivatives of the dipole integrals can be easily obtained whilst the derivatives of the density matrix require the use of coupled Hartree-Fock theory (e.g. Gerratt and Mills, 1968). 

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Baker J 1987 An algorithm for geometry optimization without analytical gradients J. Comput. Chem. 8 563... 

The gradient of the PES (force) can in principle be calculated by finite difference methods. This is, however, extremely inefficient, requiring many evaluations of the wave function. Gradient methods in quantum chemistiy are fortunately now very advanced, and analytic gradients are available for a wide variety of ab initio methods [123-127]. Note that if the wave function depends on a set of parameters X], for example, the expansion coefficients of the basis functions used to build the orbitals in molecular orbital (MO) theory. 

Importantly for direct dynamics calculations, analytic gradients for MCSCF methods [124-126] are available in many standard quantum chemistiy packages. This is a big advantage as numerical gradients require many evaluations of the wave function. The evaluation of the non-Hellmann-Feynman forces is the major effort, and requires the solution of what are termed the coupled-perturbed MCSCF (CP-MCSCF) equations. The large memory requirements of these equations can be bypassed if a direct method is used [233]. Modem computer architectures and codes then make the evaluation of first and second derivatives relatively straightforward in this theoretical framework. 

Within some programs, the ROMPn methods do not support analytic gradients. Thus, the fastest way to run the calculation is as a single point energy calculation with a geometry from another method. If a geometry optimization must be done at this level of theory, a non-gradient-based method such as the Fletcher-Powell optimization should be used. 

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. 

MoUer-Plesset perturbation theory energies through fifth-order (accessed via the keywords MP2, MP3, MP4, and MP5), optimizations via analytic gradients for second-order (MP2), third-order (MP3) and fourth-order (without triples MP4SDQ), and analytic frequencies for second-order (MP2). 

Quadratic Cl energies, optionally including triples and quadruples terms (QCISD, QCISDOl, and QCISDfTQI) and optimizations via analytic gradients for QCISD. 

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. 

Finally, there is the question of availablity of analytical derivatives. Minima, maxima and saddle points can be characterized by their first and second derivatives. Over the last 25 years, there has been a rapid development in this area, and analytical gradient formulae are now known for most of the common techniques discussed in this volume. The great advantage is that those methods that use analytical gradients tend to out-perform in speed of execution those methods where gradients have to be estimated numerically. 

To show the principles involved in finding an analytical gradient expression consider an HF-LCAO calculation where the electronic energy comes to... 

Many ab initio packages use the two key equations given above in order to calculate the polarizabilities and hyperpolarizabilities. If analytical gradients are available, as they are for many levels of theory, then the quantities are calculated from the first or second derivative (with respect to the electric field), as appropriate. If analytical formulae do not exist, then numerical methods are used. 

An analytical gradient calculation is invariably faster than a numerical one. To repeat the argument from Chapter 14, with real wavefunction Hamiltonian H (including the field terms) and parameter a (where a is a component of the external electric field)... 

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. 

The dominant practice in Quantum chemistry is optimization. If the geometry optimization, for instance through analytic gradients, leads to symmetry-broken conformations, we publish and do not examine the departure from symmetry, the way it goes. This is a pity since symmetry breaking is a catastrophe (in the sense of Thom s theory) and the critical region deserves attention. There are trivial problems (the planar three-fold symmetry conformation of NH3 is a saddle point between the two pyramidal equilibrium conformations). Other processes appear as bifurcations for instance in the electron transfer... 

Fournier, R., Andzelm, J., Salahub, D. R., 1989, Analytical Gradient of the Linear Combination of Gaussian-Type Orbitals-Local Spin Density Energy , J. Chem. Phys., 90, 6371. 

Shepard R (1995) The analytic gradient method for configuration interaction wave functions. Yarkony DR (ed) In Modern electronic structure theory part I, World Scientific, Singapore, p 345... 

Smooth COSMO solvation model. We have recently extended our smooth COSMO solvation model with analytical gradients [71] to work with semiempirical QM and QM/MM methods within the CHARMM and MNDO programs [72, 73], The method is a considerably more stable implementation of the conventional COSMO method for geometry optimizations, transition state searches and potential energy surfaces [72], The method was applied to study dissociative phosphoryl transfer reactions [40], and native and thio-substituted transphosphorylation reactions [73] and compared with density-functional and hybrid QM/MM calculation results. The smooth COSMO method can be formulated as a linear-scaling Green s function approach [72] and was applied to ascertain the contribution of phosphate-phosphate repulsions in linear and bent-form DNA models based on the crystallographic structure of a full turn of DNA in a nucleosome core particle [74],... 

In addition, by numerically differentiating the analytical gradients, the harmonic vibrational frequencies can be obtained. 

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