Energy of a Cl Wave Function

There are choice other than S for such an orthogonalizing transfonnation, any matrix which has the property X SX = I can be used. 

The variational problem may again be formulated as a secular equation, where the coordinate axes are many-electron functions (Slater determinants), , which are orthogonal (Section 4.2). 

Many problems in computational chemistry can be formulated as an optimization of a multidimensional function/ Optimization is a general term for finding stationary points of a function, i.e. points where tlie first derivative is zero. In the majority of cases the desired stationary point is a minimum, i.e. all the second derivatives should be positive. In some cases the desired point is a first-order saddle point, i.e. the second derivative is negative in one, and positive in all other, directions. Some examples  

The simple-minded approach for minimizing a function is to step one variable at a time until the function has reached a minimum, and then switch to another variable. This requires only the ability to calculate the function value for a given set of variables. However, as tlie variables are not independent, several cycles through tlie whole set are necessary for finding a minimum. This is impractical for more than 5-10 variables, and may not work anyway. Essentially all optimization metliods used in computational chemistry tlius assume that at least the first derivative of the function with respect to all variables, the gradient g, can be calculated analytically (i.e. directly, and not as a numerical differentiation by stepping the variables). Some metliods also assume that tlie second derivative matrix, the Hessian H, can be calculated. 

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Derivative Techniques 240 10.4 Lagrangian Techniques 242 10.5 Coupled Perturbed Hartree-Fock 244 10.6 Electric Field Perturbation 247 10.7 Magnetic Field Perturbation 248 10.7.1 External Magnetic Field 248 13.1 Vibrational Normal Coordinates 312 13.2 Energy of a Slater Determinant 314 13.3 Energy of a Cl Wave Function 315 Reference 315 14 Optimization Techniques 316... 

While singly excited states make relatively small contributions to the correlation energy of a Cl wave function, they are very important when calculating properties (Chapter 10). Molecular properties measure how the wave function changes-when-a,... 

Calculations Based on Perturbation Theory. An alternative to performing a Cl calculation is to use perturbation theory. As in the expansion of a mathematical function in a power series, in perturbation theory the coefficients for and the energy of an MC wave function are expanded in a series. The more... 

This section may be taken as a sort of practical summary of the preceding sections. We will comment on the properties of the Cl wave function and energy, feasibility of Cl calculations, and accuracy achieved in typical chemical applications. However, before doing this we first note an inherent shortcoming of the Cl method and this is its size inconsistency. 

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