Energy Derivatives and Properties

Certain intrinsic molecular properties describe how and in what manner a molecule responds to an applied field. These properties offer a good illustration of what is at stake in a direct calculation of a property. For uniform electric fields, the kind developed between two oppositely charged parallel plates, the first-order energy response is the dipole or first moment. The first-order response to a field gradient is the second moment, the quadrupole, and so on. Polarizabilities correspond to the second-order response, and hyperpolarizabilities to third-order and higher order changes. So, each of these is obtained as a derivative. 

The details of electrical properties require some basic information about electric fields. Fields are usually thought of as Cartesian vectors. They arise from or may be defined from a scalar function, the electric potential, V. The x-component of the electric field is dV/dx, which we will d gnate as Vx. If the potential V is independent of x, then there is no x-component of die field. If V has only a linear dependence on x, the field is uniform in die x-direction. Of course, the elearic potential may depend on the spatial coordinates in some 

In this form one can see that any specification of values for the list of elements, V, Vy, Vz, V, Vxy. specifies a particular electric potential. These elements then become parameters, things to choose to pick out a field, or field gradient, or something more complicated. We will designate that list of elements by V, which formally may be called a rank one polytensor. All molecular electrical response properties are derivatives with respea to these parameters, and Table 1 itemizes and categorizes certain of them along with other properties. Each is some type of derivative with respect to a parameter or parameters. 

Third derivatives (First) dipole hyperpolarizability VxVxVx 

means the derivative of the energy with respect to the a direction of the electric field means with respect to the direction SSa is the derivative of the energy with respect to the magnetic field with a direction and SSa is the derivative with respect to the magnetic field arising from the magnetic dipole moment of the nucleus, x is the position coordinate in the x-direction. 

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