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For describing induced dipole I have usually seen the following equation describing the situation, $$P_{i}=\alpha_{i,j}E_{j}+\frac{1}{2}\beta_{i,j,k}E_{j}E_{k}$$ where $P_{i}$ is the $i^{th}$ component of the induced dipole moment, $\alpha$ is the polarizability and $\beta$ is the hyper-polarizability.

However, recently I have been reading a paper where I see quadrupole coupled terms coming in, like dipole-quadrupole polarizability , for example as shown below; $$P_{i}=\alpha_{i,j}E_{j}+\frac{1}{3}A_{i,jk} \frac{d E_{j}}{d r_{k}}+ \frac{1}{2}\beta_{i,j,k}E_{j}E_{k}$$

where in the second term, $A_{i,jk}$ is the dipole-quadrupole polarizability term. This equation is introduced for the case of polar molecule in condensed phase and in the related paper it is for water.

It thus appears that the contribution of dipole-quadrupole polarizability is rather small but maybe higher than hyper-polarizability.

Could someone explain or refer some literature so as to describe:

• The importance of the cross-coupled terms like dipole-quadrupole polarizability,
• And, are there some cases where the cross-coupled terms (second term in second equation) become negligible.

Reference: Batista, E. R.; Xantheas, S. S.; Jónsson, H. Molecular multipole moments of water molecules in ice Ih. The Journal of Chemical Physics 1998, 109 (11), 4546-4551. DOI: 10.1063/1.477058.

For describing an induced dipole, I have usually seen the following equation, $$ P_{i} = \alpha_{ij}E_{j} + \frac{1}{2}\beta_{ijk}E_{j}E_{k} $$ where $P_{i}$ is the $i^{\text{th}}$ component of the induced dipole moment, $\alpha$ is the (dipole) polarizability and $\beta$ is the first (dipole) hyperpolarizability.

However, I recently read a paper with quadrupole-coupled terms coming in, like the dipole-quadrupole polarizability as shown below: $$ P_{i} = \alpha_{ij}E_{j} + \frac{1}{3}A_{ijk} \frac{dE_{j}}{dr_{k}}+ \frac{1}{2}\beta_{ijk}E_{j}E_{k} $$ where in the second term, $A_{ijk}$ is the dipole-quadrupole polarizability term. This equation is introduced for the case of polar molecules in the condensed phase, and in the related paper it is for water.

It appears that the contribution of dipole-quadrupole polarizability is rather small, but maybe larger than the hyperpolarizability.

Could someone explain or refer some literature so as to describe

• the importance of the cross-coupled terms like the dipole-quadrupole polarizability, and
• if there are some cases where the cross-coupled terms (the second term in the second equation) become negligible?

Reference: Batista, E. R.; Xantheas, S. S.; Jónsson, H. Molecular multipole moments of water molecules in ice Ih. J. Chem. Phys. 1998, 109 (11), 4546-4551. DOI: 10.1063/1.477058.

For describing induced dipole I have usually seen the following equation describing the situation, $$P_{i}=\alpha_{i,j}E_{j}+\frac{1}{2}\beta_{i,j,k}E_{j}E_{k}$$ where $P_{i}$ is the $i^{th}$ component of the induced dipole moment, $\alpha$ is the polarizability and $\beta$ is the hyper-polarizability.

However, recently I have been reading a paper where I see quadrupole coupled terms coming in, like dipole-quadrupole polarizability , for example as shown below; $$P_{i}=\alpha_{i,j}E_{j}+\frac{1}{3}A_{i,jk} \frac{d E_{j}}{d r_{k}}+ \frac{1}{2}\beta_{i,j,k}E_{j}E_{k}$$

where in the second term, $A_{i,jk}$ is the dipole-quadrupole polarizability term. This equation is introduced for the case of polar molecule in condensed phase and in the related paper it is for water.

It thus appears that the contribution of dipole-quadrupole polarizability is rather small but maybe higher than hyper-polarizability.

Could someone explain or refer some literature so as to describe:

• The importance of the cross-coupled terms like dipole-quadrupole polarizability,
• And, are there some cases where the cross-coupled terms (second term in second equation) become negligible.

Reference: Molecular multipole moments of water molecules in ice Ih Enrique R. Batista, Sotiris S. Xantheas, Hannes Jónsson; The Journal of Chemical Physics 1998 109:11, 4546-4551 , doi : http://dx.doi.org/10.1063/1.477058

For describing induced dipole I have usually seen the following equation describing the situation, $$P_{i}=\alpha_{i,j}E_{j}+\frac{1}{2}\beta_{i,j,k}E_{j}E_{k}$$ where $P_{i}$ is the $i^{th}$ component of the induced dipole moment, $\alpha$ is the polarizability and $\beta$ is the hyper-polarizability.

However, recently I have been reading a paper where I see quadrupole coupled terms coming in, like dipole-quadrupole polarizability , for example as shown below; $$P_{i}=\alpha_{i,j}E_{j}+\frac{1}{3}A_{i,jk} \frac{d E_{j}}{d r_{k}}+ \frac{1}{2}\beta_{i,j,k}E_{j}E_{k}$$

where in the second term, $A_{i,jk}$ is the dipole-quadrupole polarizability term. This equation is introduced for the case of polar molecule in condensed phase and in the related paper it is for water.

It thus appears that the contribution of dipole-quadrupole polarizability is rather small but maybe higher than hyper-polarizability.

Could someone explain or refer some literature so as to describe:

• The importance of the cross-coupled terms like dipole-quadrupole polarizability,
• And, are there some cases where the cross-coupled terms (second term in second equation) become negligible.

Reference: Batista, E. R.; Xantheas, S. S.; Jónsson, H. Molecular multipole moments of water molecules in ice Ih. The Journal of Chemical Physics 1998, 109 (11), 4546-4551. DOI: 10.1063/1.477058.

For describing induced dipole I have usually seen the following equation describing the situation, $$P_{i}=\alpha_{i,j}E_{j}+\frac{1}{2}\beta_{i,j,k}E_{j}E_{k}$$ where $P_{i}$ is the $i^{th}$ component of the induced dipole moment, $\alpha$ is the polarizability and $\beta$ is the hyper-polarizability.

However, recently I have been reading a paper where I see quadrupole coupled terms coming in, like dipole-quadrupole polarizability , for example as shown below; $$P_{i}=\alpha_{i,j}E_{j}+\frac{1}{3}A_{i,jk}\frac{1}{2}\beta_{i,j,k}E_{j}E_{k}$$

where in the second term, $A_{i,jk}$ is the dipole-quadrupole polarizability term. This equation is introduced for the case of polar molecule in condensed phase and in the related paper it is for water.

It thus appears that the contribution of dipole-quadrupole polarizability is rather small but maybe higher than hyper-polarizability.

Could someone explain or refer some literature so as to describe:

• The importance of the cross-coupled terms like dipole-quadrupole polarizability,
• And, are there some cases where the cross-coupled terms become zero (and vice-versa).

For describing induced dipole I have usually seen the following equation describing the situation, $$P_{i}=\alpha_{i,j}E_{j}+\frac{1}{2}\beta_{i,j,k}E_{j}E_{k}$$ where $P_{i}$ is the $i^{th}$ component of the induced dipole moment, $\alpha$ is the polarizability and $\beta$ is the hyper-polarizability.

However, recently I have been reading a paper where I see quadrupole coupled terms coming in, like dipole-quadrupole polarizability , for example as shown below; $$P_{i}=\alpha_{i,j}E_{j}+\frac{1}{3}A_{i,jk} \frac{d E_{j}}{d r_{k}}+ \frac{1}{2}\beta_{i,j,k}E_{j}E_{k}$$

where in the second term, $A_{i,jk}$ is the dipole-quadrupole polarizability term. This equation is introduced for the case of polar molecule in condensed phase and in the related paper it is for water.

It thus appears that the contribution of dipole-quadrupole polarizability is rather small but maybe higher than hyper-polarizability.

Could someone explain or refer some literature so as to describe:

• The importance of the cross-coupled terms like dipole-quadrupole polarizability,
• And, are there some cases where the cross-coupled terms (second term in second equation) become negligible.

Reference: Molecular multipole moments of water molecules in ice Ih Enrique R. Batista, Sotiris S. Xantheas, Hannes Jónsson; The Journal of Chemical Physics 1998 109:11, 4546-4551 , doi : http://dx.doi.org/10.1063/1.477058