Quantum chemistry gradient methods

Pempointner, M., Schwerdtfeger, P. and Hess, B.A. (2000) Accurate electric field gradients for the coinage metal chlorides using the PCNQM method. International Journal of Quantum Chemistry, 76, 371-384. 

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 chemistry 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 chemistry 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. 

All the methods used in this study are response methods. They deserihe the response of an ohservahle sueh as an eleetrie dipole moment /I or quadrupole moment to an external or internal perturhation, e.g., an eleetrie field or field gradient. Response funetions originated in various diseiplines in physies. In statistieal physies, they were used as time-eorrelation functions in the form of Green s functions [44,45]. Linderherg and Ohrn first showed the usefulness of this idea for quantum chemistry [46]. Since then response functions have been derived for many types of electronic wavefunctions. Four of these methods are employed here. 

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],... 

There are some modem methods (such as the memory13 and supermemory14 gradient methods) which are not univariate in nature but which approach the minimum in a sequence of many-dimensional searches. So far, however, such methods have found no use in quantum chemistry and we shall not discuss them further. 

In the one-dimensional search methods there are two principle variations some methods employ only first derivatives of the given function (the gradient methods), whereas others (Newton s method and its variants) require explicit knowledge of the second derivatives. The methods in this last category have so far found very limited use in quantum chemistry, so that we shall refer to them only briefly at the end of this section, and concentrate on the gradient methods. The oldest of these is the method of steepest descent. 

As far as the Fletcher-Reeves method is concerned, it must clearly be the method of choice in linear coefficient optimization as it involves only the storage of gradient and direction vectors between iterations. It has been used by a number of authors (Sleeman,29 Fletcher,5 Kari and Sutcliffe,32 Claxton and Smith,51 and Weinstein and Pauncz52). It [is unfortunately possible, however, to sum up the experience so far gained of the method in quantum chemistry as disappointing, in the sense that in SCF caclulations the authors have found that the calculations proceed significantly more slowly than the conventional iterative procedure, when the conventional procedure converges at all. 

Undoubtedly more effective conjugate-gradient techniques would be of most use in quantum chemistry, perhaps some methods tailored to suit particular functional forms common in the field, but this area of research seems unlikely to be developed further by workers principally interested in optimization and is perhaps a suitable field of endeavour for quantum chemists. 

Sutcliffe attempted to survey those of them, that proved useful or may become useful in quantum chemistry. A common feature of modern effective methods is that they require knowledge of the energy gradient and,profitably, also of the matrix of harmonic force constants or at least a reasonable estimate of the latter. In the forthcoming discussion, however, it should be kept in mind that the cost for the gradient evaluation is at least twice as high as the cost required for a single standard SCF run and that the evaluation of the force constant matrix is even more costly. 

Over the decade 1995-2005, ab initio quantum chemistry has become an important tool in studying imidazole derivatives. Two highly productive approaches are often utilized for the calculations the wave function-based methods (e.g., Hartree-Fock theory and second-order Moller-Plesset perturbation theory (MP2)) and the density functional theory (DFT) based methods (e.g., gradient-corrected (BLYP) and hybrid (B3LYP) methods). 

Since their introduction in the late 1960s, gradient methods, or more properly analytical derivative methods, have become one of the most vigorously developing topics on modern quantum chemistry. They have also acquired considerable significance for the solution of practical chemical problems. The first review on this subject was written in 1974-75, although it was published much later (Pulay, 1977) a short chapter, limited to first derivatives,... 

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