Properties as Energy Derivatives

Let us consider the electronic energy E (x) as a function of some external parameter x. When a molecular electronic system is perturbed in some manner, its total electronic energy changes and may be expressed in terms of a Taylor expansion as 

The coefficients that appear in this expansion describe the response of the molecular system to the external perturbation x and are known as molecular properties. The molecular properties are characteristic of the molecular system and its quantum state. When the perturbation is static, these properties may be calculated by differentiation at x = 0 

Of special interest to us are the properties related to the external application of uniform static electric and magnetic fields, which we will denote by F and B, respectively. To second order, the energy of a closed-shell molecular system may be written as 

Let us consider the analytical evaluation of molecular properties. We shall here write the electronic energy function in the form (x A), where x is a set of external parameters that characterize the physical system, and A a set of wave-function or electronic parameters that determine the electronic 

To keep things simple, we shall first assume that the electronic energy is fully variational with respect to the electronic parameters A. Thus, we shall assume that the electronic energy may be calculated from the expression 

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