Understanding Dipole and Quadrupole Moments for Energy Calculation

[Gavroy]

hi

i have two questions , that i cannot solve by my own:

does anybody know how to calculate the potential energy of two quadrupole moments that are separated by a distance r?( i looked for it in all my physics books, but i did not find anything at all)

and: if you have two dipoles there is an equation for the potential energy that varies with the distance 1/r³ but what exactly does this distance 1/r³ between the two dipoles mean?

for example: if you think of two hydrogen atoms, that possesses a dipole moment, that is caused by an electric field. then what is this distance 1/r³? is it just the distance between the two nuclei or do you have to take account for the fact, that the electron is not in a spherical symmetric orbital anymore if there is an electric field and 1/r³ is some kind of mean value between the electron position and the nucleus of the one atom to this point at the other atom?

 

[SpectraCat]

hi

i have two questions , that i cannot solve by my own:

does anybody know how to calculate the potential energy of two quadrupole moments that are separated by a distance r?( i looked for it in all my physics books, but i did not find anything at all)

and: if you have two dipoles there is an equation for the potential energy that varies with the distance 1/r³ but what exactly does this distance 1/r³ between the two dipoles mean?

for example: if you think of two hydrogen atoms, that possesses a dipole moment, that is caused by an electric field. then what is this distance 1/r³? is it just the distance between the two nuclei or do you have to take account for the fact, that the electron is not in a spherical symmetric orbital anymore if there is an electric field and 1/r³ is some kind of mean value between the electron position and the nucleus of the one atom to this point at the other atom?

Well, the formula for two generic quadrupoles is quite complicated and involves the 3-D angles of both quadrupoles (4 total degrees of freedom) as well as the 6 non-zero elements of both quadrupole tensors. However in some cases there are significant simplifications (e.g. if the quadrupole tensors is diagonal, or if both quadrupoles are identical). I don't have my books with me at home, so I can't look up the precise formulas .. I am pretty sure the scaling with distance goes as 1/r⁴. If you have access to a library, one useful book is "Intermolecular Forces" by Stone.

The r in the formula you asked about refers to the center of mass distance between the two dipoles. So in your example, it would just be the two atomic centers.

 

[Bill_K]

Take a look at this thread: https://www.physicsforums.com/showthread.php?t=491146