SCF energy gradients

The energy gradients with respect to a nuclear coordinate, as an example, can be obtained by differentiating the SCF energy expression of Eq. [109]  

This formulation requires one to compute the energy-weighted density matrix using the molecular orbital coefficient matrix C, which must be avoided to 

To obtain equations that are independent of it is necessary to consider the different contributions to the derivative density. As for any matrix representation of operators, it is possible to split the contributions into different subspace projections (compare Eq. [108])  

These projections will be analyzed below. At SCF convergence, the following equations are valid  

In addition, after introducing the perturbation x, the derivative of the idem-potency relation (Eq. [143]) has to be fulfilled  

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Aug. 29, 1986. Implementation of Large FORTRAN Programs on the Loosely Coupled Array of Processors Parallel Computation of Molecular SCF Energy Gradients. 

The second derivatives with respect to nuclear displacements are crucial for characterizing stationary points on a potential hypersurface. They provide as well the normal modes of the system and can be linked within the harmonic approximation to the vibrational frequencies of the system, which can be measured experimentally by IR or Raman spectroscopy. By taking the derivative of the SCF energy gradient expression (Eq. [152]) with respect to another... 

The gradient and second derivative components of the SCF energy can be expressed for both kinds of parametrization (see [28]) as... 

The only remaining question is the nature of the error function. Pulay suggested the difference FDS — SDF (S is the overlap matrix), which is related to the gradient of the SCF energy with respect to the MO coefficients. This has been found to work well in practice. 

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. 

Proton-transfer reactions of iV-tetrachlorosalicylideneaniline (140) and N-tetrachlorosalicylideneaniline-l-pyrenylamine (171) have been investigated by using a semiempirical self-consistent field molecular orbital (SCF MO) method with an energy gradient technique.94 From the calculated potential barriers (81.27 and 87.30kJmol-1) for 140 and 171, respectively, it can be seen that the... 

There is a strong technical aspect to molecular quantum chemistry because of the complexity of SCF and Cl equations and of energy gradient expressions. Cook s book is a good reference to many technical issues to be tackled when the equations are to be implemented in computer programs. 

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