Quadrupole-quadrupole interaction?

In summary: So, in summary, the quadrupole-quadrupole interaction between two molecules with zero dipole moment can be calculated using the formula U=(3QQ'/4r^5)[35(k.r)^2(k'.r)^2-20(k'.r)(k.r)(k.k')-5(k.r)^2-5(k'.r)^2+2(k.k')^2+1], which can be derived from combining equations (2.118) and (2.112) in "Classical Electromagnetism" by Franklin (Addison Wesley). This formula has been tested on parallel and crossed model linear quadrupoles, giving the same result for the parallel case but differing for the crossed case. It is also
  • #1
neo143
29
0
Could anyone please tell me how to calculate quadrupole quadrupole interaction between two molecules having dipole moment zero? As quadrupole is a tensor quantity I get a 3*3 matrix for a molecule.
 
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  • #2
I don't know much about quadrupole. Which kind of molecule are you treating, does it suffice to have 3 atoms ?
 
  • #3
suppose the molecule is benzene?
 
  • #4
One sure-fire way is to work with multipole expansions. Expand both field and molecular charge distribution in spherical harmonics -- the quadrapole moment is the coefficient of Y(L=2, M; theta, phi) This might be done in Jackson, and surely is done in books on nuclear physics, and on angular momentum (Edmunds, Angular momentum in QM.)
Regards,
Reilly Atkinson
 
  • #5
The answer is a bit messy, even for symmetric quadrupoles.
I give it here for two symmetric quadrupoles Q and Q' with symmetry axes k and k' a distance r apart. k,k,r are all vectors and Q and Q' are the Qzz component of the Q tensor (with Qxx=Qyy=-Qzz/2).
The energy is
U=(3QQ'/4r^5)[35(k.r)^2(k'.r)^2-20(k'.r)(k.r)(k.k')+2(k.k')^2+(k.k')].
(The k,k',r in the square bracket are all unit vectors.)
I got this by combining Eqs. (2.118) and (2.112) in "Classical Electromagnetism" by Franklin (Addison Wesley).
 
  • #6
Did sdy know if we take ponctual charges, it is clear that 2 are building what is called a dipole...but do 3 point charges suffice to have non-vanishing quadrupole moment ?? Oh..sorry, this is not quantum atomics..
 
  • #7
Even two point charges can have a quad moment. If you want the system to be neutral, at least three charges are needed.
 
  • #8
Thanks Meir for the reply,
I have one more doubt. Suppose I want to calculate quadrupole moment interaction between two symmetrical molecules(with Qxx=Qyy=-Qzz/2).Rest of the components are zero. Should I consider every possible pair of quadrupole moments for two molecules?
like xx for the first molecule and xx for the second
then xx for the first and yy for the second
then xx for the first and zz for the second
....and so on...total terms will be 9.

Thanks once again
Regards
 
  • #9
Meir Achuz said:
The answer is a bit messy, even for symmetric quadrupoles.
I give it here for two symmetric quadrupoles Q and Q' with symmetry axes k and k' a distance r apart. k,k,r are all vectors and Q and Q' are the Qzz component of the Q tensor (with Qxx=Qyy=-Qzz/2).
The energy is
U=(3QQ'/4r^5)[35(k.r)^2(k'.r)^2-20(k'.r)(k.r)(k.k')+2(k.k')^2+(k.k')].
(The k,k',r in the square bracket are all unit vectors.)
I got this by combining Eqs. (2.118) and (2.112) in "Classical Electromagnetism" by Franklin (Addison Wesley).
I believe there is a mistake in that formula. By combining those two mentioned formulas in Franklin's book, I got
U=(3QQ'/4r^5)[35(k.r)^2(k'.r)^2-20(k'.r)(k.r)(k.k')-5(k.r)^2-5(k'.r)^2+2(k.k')^2+1]
which differs a bit. I checked it on two parallel and two crossed model linear quadrupoles. While for parallel case both formulas give same result, for crossed case my gives non-zero interaction and Meir's gives zero interaction. Calculating the energy by simple Coulomb's law one gets non-zero interaction.
 
  • #10
Thanks for the correction, but this post is 5 years old.
 

1. What is a quadrupole-quadrupole interaction?

A quadrupole-quadrupole interaction is a type of intermolecular interaction between two molecules with quadrupole moments. It occurs when the electric field gradients of the two molecules interact with each other, resulting in attractive or repulsive forces.

2. How is a quadrupole moment calculated?

A quadrupole moment is a measure of the distribution of electric charge within a molecule. It can be calculated by taking the second moment of the charge distribution, which involves the integration of the charge density over the entire molecule.

3. What factors influence the strength of a quadrupole-quadrupole interaction?

The strength of a quadrupole-quadrupole interaction is influenced by several factors, including the magnitude and orientation of the quadrupole moments of the molecules, the distance between the molecules, and the dielectric properties of the surrounding medium.

4. How does a quadrupole-quadrupole interaction differ from other intermolecular interactions?

Unlike other intermolecular interactions, such as dipole-dipole or London dispersion forces, a quadrupole-quadrupole interaction is a relatively weak force. It is also highly directional and depends on the specific orientation of the quadrupole moments of the molecules involved.

5. What are some examples of molecules that exhibit quadrupole-quadrupole interactions?

Molecules with permanent quadrupole moments, such as carbon dioxide, water, and sulfur dioxide, are known to exhibit quadrupole-quadrupole interactions. Additionally, some molecules can acquire temporary quadrupole moments due to asymmetric electron distributions, leading to temporary quadrupole-quadrupole interactions.

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