Practical Considerations When Performing ab initio Calculations

4 Practical Considerations When Performing ab initio Calculations 

The initial guess of the density matrix may influence the convergence of the SCF calculation a null matrix is the simplest approach, but better results may be obtained by using a density matrix from a calculation performed at a lower level of theory. For example, the density matrix from a semi-empirical calculation may be used as the starting point for an ab initio calculation. Conversely, such an approach may itself lead to problems if there is a significant difference between the density matrices for the lower and higher levels of theory. 

A more sophisticated method that is often very successful is Pulay s direct inversion of the iterative subspace (DIIS) [Pulay 1980]. Here, the energy is assumed to vary as a quadratic function of the basis set coefficients. In DUS the coefficients for the next iteration are calculated from their values in the previous steps. In essence, one is predicting where the minimum in the energy will lie from a knowledge of the points that have been visited and by assuming that the energy surface adopts a parabolic shape. 

Considerable effort has been spent devising efficient ways of directly calculating the first and second derivatives of the energy with respect to the nuclear coordinates. Derivatives are primarily used during ininimisation procedures for finding equilibrium structures (the first derivative of the energy with respect to its coordinates equals the force on an atom) and are also used by methods which locate transition structures and determine reaction pathways. 

A self-consistent field wavefunction (and thus its energy) can be considered a complicated function of the nuclear coordinates, basis functions and basis function coefficients (and, for a Cl calculation, the coefficients of single determinantal wavefunctions). In order to determine the first, second, etc. derivatives of the energy with respect to the nuclear coordinates [Pulay 1977] it is necessary to consider not only how the energy depends directly on the nuclear coordinates but also whether there is an indirect dependence via other parameters. Indeed, it is only the one-electron part of the Hamiltonian that depends directly upon the nuclear coordinates (H (l), Equation (2.125)), to which is added an intemuclear Coulomb repulsion term. For the other parameters the derivative with respect to the nuclear coordinates is generally determined via the chain rule (for first derivatives). For example, for a generic nuclear coordinate and a generic parameter Xj we can write  

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