Multipole moments operator

In Cartesian coordinates, the expectation values of multipole moment operators are computed as... 

In Eq. (18), we recognize the first quantization fcth-order electronic multipole moment operator (r - Ro). An analogous expansion of r - Rm 1 would yield the nuclear multipole moment operator (Rm — Rq). The higher-order multipole moments generally depend on the choice of origin however, to simplify the notation, we omit any explicit reference to this dependence. The partial derivatives in Eq. (18) are elements of the so-called interaction tensors defined as... 

In the final square brackets of Eq. (13.17), we can recognize the multipole moment operators for the molecules calculated in their coordinate systems ... 

This takes the form of a single multipole expansion, but this time the multipole moment operators correspond to entire molecules. 

The total multipole moment operator represents a sum of the same operators for the individual particles (of course, they all have to be computed in the same coordinate system) ... 

The multipole moment operators in Eq. (7) are still referred to the global coordinate frame. We now transform them to the local or molecule-fixed frame ... 

In these equations = (1 — R RR ) is a projection operator that removes from its vector operand the component parallel to R. (The symbol is used because Q is the conventional notation for this type of projection operator it should not be confused with the multipole moment operators.)... 

A similar procedure can be applied to the non-local polarizabilities a"".. Here we are dealing (in Eq. (35)) with multipole moment operators Q" and 0". for sites a and a. By representing these, via eq. (17), in terms of moments at a convenient common site a", the non-local polarizability a"" is replaced by local polarizabilities of the form af "though at the cost of introducing higher-rank polarizabilities. 

The perturbation is now readily expressed in terms of the multipole moment operators ... 

J. Cipriani and B. Silvi, Mol. Phys., 45, 259 (1982), Cartesian Expressions of Electric Multipole Moment Operators. 

There is a lot compressed in this expression so it is weU that we spend some time unravelling it First of aU, the subscripts t and u label the angular momenta of the real spherical harmonics and take the Im values 00,10,11c,11s,20,21c,21s,22c,22s,--- (the labels c and s stand for cosine and sine respectively). Qf is the real form of the multipole moment operator of rank t centered on A and expressed in the local-axis system of A, while T/J is the so-called T-tensor that carries the distance and angular dependence. The T-function of ranks h and I2 has a distance dependence where R is the separation between the centers of A and B. 

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