RELATE ELECTRIC SHIELDING TENSORS TO DERIVATIVES OF ELECTRONIC CHARGE DISTRIBUTION PROPERTIES WITH RESPECT TO NUCLEAR COORDINATE
A molecule is a collection of positive charges (nuclei) and negative charges (electrons) arranged in a fashion as dictated by the laws of quantum mechanics. How the nuclei and electrons are arranged specifically is not important for this general discussion of molecular electrical properties. What is important is that the the molecules is a charge distribution at equilbrium i. e. all of the electric forces in the molecule are balanced. All charge distributions, whether molecular or not, can be characterized by their nonzero multipole moments.
Multipole Moments
Let us quickly discuss multipole moments. A charge distribution that has an overall charge has a nonzero monopole moment. For example, the NO3- ion has an overall charge of -1; therefore has a monopole moment. The La+3 ion also has a monopole moment.
A water molecule has an overall charge of zero; therefore, it has no monopole moment. Even though the number of positive charges equals the number of negative charges, the charge is not evenly distributed throughout the molecule. The region near the oxygen nucleus will have a little more negative charge than the region near the hydrogen nuclei. This type of charge distribution is characterized as a dipole moment. The dipole moment is vector quantity. One can the imagine the vector as pointing from the region with the maximum positive charge distribution to the region with the maximum negative charge distribution.
Now consider charge distribution created by two dipole moments either aligned in a straight line or side by side pointing in anti parallel directions. Neither of these charge distributions has an overall monopole or dipole moment. Both of these distributions is termed a quadrupole moment. Carbon dioxide, O=C=O, being a linear molecule does not a dipole moment. Each of the C=O bonds is polar, but the polarity of each bond faces different directions. Therefore carbon dioxide does not have a dipole moment but does have a quadrupole. Another example of a molecule without a dipole moment but with a quadrupole moment would be trans-dichloroethene. Next order of multipole moment, the octupole, can be constructed from orienting two quadrupoles in different ways. A molecule with a zero quadrupole moment but nonzero octupole moment is methane. A hexadecapole can be constructed from two octupoles and so on.
The moral of the story is that any charge distribution can be characterized by its first nonzero multipole moment and its higher order multipole moments. For example, water has a dipole moment but it also has a quadrupole moment. The quadrupole moment has less electronic importance than the dipole moment. The water molecule also has an octupole, hexadecapole and so on. However, each subsequent multipole has less and less of an impact on the nature of the charge distribution.
Electric Shielding Tensors
To consider what are electric shielding tensors we will assume that the nuclei remain fixed in space so that any redistribution of charge in the molecule is due to a redistribution of electrons. In the above section on multipole moments, we saw that a generic molecule could be characterized by its multipole moments. What was omitted was that for a neutral charge distribution with a nonzero multipole moment, since the charge is not evenly distributed, the distribution will emit an electric field which will be different as we measure the field in different places about the molecule. Let us now consider how an external electric field will affect the molecule. An external electric field will rearrange the charge distribution, since the external field disturbs the equilibrium of electric forces that existed in the molecule before the external field was applied. This new disturbed charge distribution will emit a different field than the undisturbed charge distribution. Electric shielding tensors are the molecular properties that describe how an external electric field incident on a molecule affects the electric field being emitted by the molecule.
The electric shielding tensors are defined as
(A note about notation: the bars over a quantity donates the rank of tensor. One bar indicates a first rank tensor i. e. a vector. A second rank tensor, i. e. 3x3 matrix, is denoted by two bars while a third rank tensor, i. e. a 3x3x3 matrix, has three bars and so on.) We see that the first order shielding tensor is defined by removing the external electric field whereas the quadratic and higher order shielding tensors include the electric fields. A illustration would consider the shielding tensors of the vacuum. (Perhaps a silly example but illustrative) The first order shielding tensor of the vacuum would be a 3x3 matrix with zeroes everywhere, especially on its diagonal. The quadratic shielding tensor would be a 3x3x3 matrix with ones and zeroes in certain positions; specifically, the xxx, yyy, and zzz positions would be one.
We continue the proof of relating electric shielding tensors to derivatives of electronic charge distribution properties with respect to nuclear coordinate.
We first examine the dependence of a molecule's energy on an external electric field by expanding the energy of the molecule as a Taylor series in the electric field.
In the above equation,
, is the dipole moment.
This quantity is the same dipole moment that was
examined from the multipole moment perspective of charge distributions.
The second order quantity,
, is named the polarizability.
The polarizability is very different than
the quadrupole moment so that the parallels seen between the multipole moments
and the molecular energy dependance on the electric field end abruptly.
The polarizability is a measure of how easily a molecule's electronic charge
distribution deforms under the influence of an external electric field.
A wide range of polarizabilities is encompassed when comparing a fluorine
molecule and an iodine molecule. The outer electrons of the fluorine molecule
are 2sp3 hybridized electrons while the outer electrons of the iodine molecule
are 5sp3 hybridized electrons. The 2sp3 electrons are more tightly bound
to the nuclei than the 5sp3 electrons. Thus the 5sp3 electrons are easier
to disturb from an equilibrium position than the 2sp3 electrons. Specifically,
if we apply an electric field to each molecule, the outer electrons of the
iodine molecule will respond easier than the outer electrons of the fluorine
molecule. Thus the iodine molecule has a higher polarizability than the
fluorine molecule. Higher order polarizabilities exist but we shall not
consider them at the moment.
Let us now consider the forces upon a single nucleus, labeled I. One can calculate the force on the charge of the nucleus due to an electric field outside the nucleus.
However one can also find the force on the nucleus by calculating the negative change in the molecule's energy as the nucleus I is moved infinitessimally.
Both of these forces on the nucleus I must be equal if the molecule's charge distribution is at equilbrium. Note that our expression for the electric field in the first force expression comes from our definitions of the electric shielding tensors and the energy derivative. The form of the energy derivative used in the second force expression comes from the Taylor series of the energy expanded about the external electric field.
Equate terms in orders of electric field
Thus the derivative of a molecule's dipole moment with respect to nuclear coordinate is related the electric shielding tensor. And the derivative of a molecule's polarizability with respect to nuclear coordinated is related to the quadratic electric shielding tensor.
RELATE NONLOCAL POLARIZABILITY DENSITIES AND ELECTRIC FIELD SHIELDING TENSORS
Relate molecular polarization to polarizability densities
The field at nucleus I due to a perturbing electric field can be described as
Use integration by parts and the fact 
Therefore,
If fields are uniform, then equate terms in orders of electric field
Schematic of the points referred to inside an arbitrary molecule
RELATE NONLOCAL POLARIZABILITY DENSITIES TO DERIVATIVES OF MOLECULAR ELECTRICAL PROPERTIES WITH RESPECT TO NUCLEAR COORDINATE
The field at a point,
, due to nucleus I is
When the nucleus is shifted, the field at point,, can be expanded as a Taylor series in
. The component of the field can be written as
Now a quick aside is made to emphasize that the nonlocal polarizability density that describes the molecular electronic response to applied external electric fields is the same entity that describes the molecular electronic response to applied internal electric fields. A shift in the nuclear coordinate is an applied internal field relative to the unperturbed internal electric field.
Using the internal electric field from the nucleus,
.
Let
Integrate by parts with respect to recalling that
The polarization operator is a vector, the dipole propagator is a second
rank tensor, and the shift in nuclear coordinate,, is a vector.
Therefore, the above equation can be decomposed as the sum of scalar equations.
Substitute the definition of the dipole propagator,
.
Integrate by parts with respect to 
.
Use the property,
, to obtain the intermediate result,
Setting aside the intermediate result for awhile, the effect of a nuclear shift of a molecule is examined from how the shift affects the Hamiltonian.
We recall that from nondegenerate perturbation time-independent perturbation theory, the first order corrected wavefunction is.
The charge density of the molecule is the expectation value of the charge
density operator. 
Substituting for 
, the first order corrected wavefunction
yields the charge density expectation value.
The first order correction of the charge density is
Since
, we should examine the zeroth order Hamiltonian
=Te + Tn + Ven + Vee + Vnn
Te - kinetic energy of the electrons
Tn - kinetic energy of the nuclei
Ven - potential energy of electron-nucleus attraction
Vee - potential energy of electron-electron repulsion
Vnn - potential energy of nucleu -nucleus repulsion
Only Ven and Vnn are dependent on the nuclear coordinate. Since we are only considering the electronic response, we neglect the change in the nucleus-nucleus repulsive potential energy as it is an additive constant that does not affect the electronic charge distribution. However, a change in the electron-nucleus attractive potential energy with respect to a change in the nuclear position will affect the electronic charge distribution.
Therefore, for a specific nucleus I,
.
Consequently, the perturbation we are considering is
Recall the first order correction to the charge density obtained using perturbation theory,
.
Change the integration variable in the perturbation since the points where we integrate are not dependent on the points of the charge distribution.
Digression on dummy variables
Consider the following indefinite integral
where 
is a constant.
What is important to our result is the bounds of integration, not the label we give to integration variable. Therefore we could write the indefinite integral above as
The integration variable (elephant in the above case) is often called a dummy variable. We may change the label of the dummy variable at our discretion and usually changed to emphasize that the dummy variable is independent for the integrated function (the variable of the integration bounds).
A common way to see the indefinite integral written is with primed dummy variables such as
.
Changing the dummy variable of integration of the perturbation yields
We now substitute this perturbation into our expression for the first order correction to the charge density.
The above expression is the same expression we found using the definition of the polarizability density and a differential change in the electric field at the nucleus.