# Reviwer B(Kanamori)

This article is a good article. I agree with the author that the Goodenough-Kanamori rule should not be changed to the Goodenough-Kanamori-Anderson rule. When the rule was proposed, various mechanisms for the superexchange interaction were not organized in the present form. I discussed that the symmetry relations led us to a definite sign of the interaction nevertheless.

I would be happy if the author could consider the following two points.


1) The paper, K.Ueda, H.Tabata and T.Kawai, Science 280 (1998) 1064, reported that the double perovskite LaFe0.5Cr0.5O3 exhibited bulk ferromagnetism if Cr and Fe atoms occupied alternate (111) planes, while it was antiferromagnetic if they occupied alternate (100) planes; they succeeded in preparing the ordered samples by a careful treatment. I think that this is an example clearly indicating the signs of the interaction between Cr-Fe, Fe-Fe and Cr-Cr pairs.

2) The second point concerns the magnetostriction, exchange striction and cooperative Jahn-Teller distortion discussed near the end of the article. I believe that the papers, J.Kanamori,Prog. Theoret. Phys. 17 (1957) 177-196,197-222 and ;J.Appl.Phys. Suppl.31 (1960)14S were the first papers that discussed the problem, particularly the role of the spin-orbit coupling. I should be happy if the author could mention that in some way. J.Kanamori、

# Reviewer A (Daniel Khomskii)

## REFEREE REPORT on the paper by J.B.Goodenough on "Goodenough-Kanamori rule" for Scholarpedia

This is definitely a good paper which can be put in Scholarpedia. I have only a few questions or suggestions. Here they are, not in order of importance, but as they appear in the text:

1. In the literature Goodenough-Kanamori rule (or rules) are often called also "Goodenough-Kanamori-Anderson rule(s)". I do not think one has to change the term "Goodenough-Kanamori rule" in the paper, it should remain. But may be it is worthwhile to point it once, e.g. after the first appearance of this term in the text. E.g. to formulate it so:

"The Goodenough-Kanamori rule (in the literature also often called Goodenough-Kanamori-Anderson (GKA) rule(s) ), first formulated by Goodenough in 1954, .... " .

2.The first formula in the n.4 on the page 1 is not clear. What are the quantities α1 and β1 there? May be one can simply omit this formula and give directly the expression for effective spin-dependent hopping integrals t (the next formula)?

3. It might be useful to number the formulae.

4. When discussing the virtual electron transfer from a half-filled to an empty orbital, giving rise to ferromagnetic exchange, I would refer to the Hund's rule. For example, to add in the sentence "... favors transfer of a spin that is parallel to the pre-existing spin on the cation (Hund's rule)" (or "...(the first Hund's rule)".

5. I would put comma in the sentence in the third line from the bottom on p.2: "d-electron occupation, not the charge at the cation site".

6. On the p.3, second line in the paragraph starting from "A significant check ...": I do not really understand why "the ΔE of the σ-bond proves to be kTt". Why is that? Why is not it the on-site (Hubbard's) Coulomb repulsion U?

These are all my remarks. After they are clarified and, may be, modifications introduced in the text, the paper can be put on Scholarpedia.

## Additional comment of reviewer A (2008-10-16)

In my first report I never suggested to change Goodenough-Kanamori rule for Goodenough-Kanamori-Anderson rule, neither in the title nor in the body of the paper. What I had in mind is that, rightly or wrongly, the term GKA rule is sometimes used in the literature, and I thought that it might be worthwhile to mention in the paper that it is actually the same, to avoid possible confusion by the readers. But I did not have in mind that one has to change the term in the text. That is why I suggested to add this remark the way I did, in brackets, or may be as a footnote. But it is of course up to the author whether to accept this suggestion or not; I will be perfectly happy with his decision.The author answered all my questions, and the paper can be put in Scholarpedia. It is definitely a very good paper, which will be  quite useful for many scientists working in the field.

# Author

## Response to Referee A

1. The Goodenough-Kanamori rule should not be changed to the Goodenough-Kanamori-Anderson rule even though a few authors have chosen to add Anderson’s name. P.W. Anderson made a seminal contribution to the theory of superexchange in his Ph.D. thesis where he presented a calculation of an antiferromagnetic spin-spin interaction for half-filled orbitals having a spin s = ½. I have made reference to this calculation in the text. However, Anderson had no idea how to obtain a ferromagnetic interaction and therefore could not contribute to a rule pointing out the condition for ferromagnetic versus antiferromagnetic interactions. Formulations of the rule required the recognition, first made by Goodenough, of a cooperative site distortion that led to a cooperative orbital ordering as the site distortions removed a degeneracy of localized-electron orbitals. It was Kanamori, not Anderson, who jumped on this recognition to present a mathematical formulation more palatable to P.W. Anderson. In fact, it was P.W. Anderson who first referred to it as the Goodenough-Kanamori rule.
2. The end of the sentence before Equation (1) has been changed from “between cation spins having an angle $$\theta$$ between their spin axes is” to “from atom 1 to atom 2 having a spin axis oriented at an angle $$\theta$$ with respect to that on atom 1 is”.
3. Numbering of Equations (1) and (2) has been added.
4. (“Hund’s first rule)” has been added as suggested.
5. The comma has been added.
6. To clarify the origin of a $$\Delta E = kT_t$$ in the RMnO3 perovskites the beginning of the second sentence of the paragraph beginning, “A significant check…” has been changed to read, “In LaMnO3, the gap between the Mn4+/Mn3+ and Mn3+/Mn2+ redox energies is opened up by the cooperative orbital ordering below $$T_t$$, so $$\Delta E = kT_t=U$$ of the $$\sigma$$-bond…”

Misprints

Several misprints have been noted and corrected. They are as follows:

1. The phrase, “the Goodenough-Kanamori rule is the same for both superexchange and semicovalent exchange” at the end of the third paragraph should appear as a new sentence at the end of the second paragraph, which begins, “The Goodenough-Kanamori rule states…”
2. Change “semicovalent exchange” to semicovalent antiferromagnetic exchange” in the next-to-last sentence of the third paragraph and join the sentence to the last sentence by changing “... net spin. Where the…” to “…net spin and the…”.
3. After Equation (16) a long paragraph begins, “To illustrate application…” Three illustrations are given in this paragraph. Each illustration should appear as a separate paragraph. Therefore,“As a second example…” and “The ideally ordered La2NiMnO6…” should each begin a new paragraph.
4. The paragraph beginning, “The spinel Ge[Ni2]O4…” has, in the last sentences, the word “strong’s,” which should read “strong.”

## Response to Referee B

1. Reference to Ueda et al has been added as another example in the paragraph beginning, "The ideally ordered..."
2. Reference to Kanamori has been added to opening of the second-to-last paragraph, which begins, "This distinction..."

Two minor typos in the last sentence of the second-to-last paragraph have been corrected.