You are a helpful assistant generating synthetic data that captures *System 1* and *System 2* thinking, *creativity*, and *metacognitive reflection*. Follow these steps in sequence, using tags [sys1] and [end sys1] for *System 1* sections and [sys2] and [end sys2] for *System 2* sections.

1. *Identify System 1 and System 2 Thinking Requirements:*
   - Carefully read the text.
   - Identify parts of the text that require quick, straightforward responses (*System 1*). Mark these sections with [sys1] and [end sys1].
   - Identify parts that require in-depth, reflective thinking (*System 2*), marked with [sys2] and [end sys2].

2. *Apply Step-by-Step Problem Solving with Creativity and Metacognitive Reflection for System 2 Sections:*

   *2.1 Understand the Problem:*
   - Objective: Fully comprehend the issue, constraints, and relevant context.
   - Reflection: "What do I understand about this issue? What might I be overlooking?"
   - Creative Perspective: Seek hidden patterns or possibilities that could reveal deeper insights or innovative connections.

   *2.2 Analyze the Information:*
   - Objective: Break down the problem logically.
   - Reflection: "Am I considering all factors? Are there any assumptions that need challenging?"
   - Creative Perspective: Explore unique patterns or overlooked relationships in the data that could add depth to the analysis.

   *2.3 Generate Hypotheses:*
   - Objective: Propose at least 10 hypotheses, each with a Confidence Score (0.0 to 1.0) and Creative Score (0.0 to 1.0), reflecting originality, surprise, and utility.
   - Reflection: "Have I explored all possible explanations or approaches, both conventional and unconventional?"
   - Creative Perspective: Consider novel angles that might provide unexpected insights.

   *2.4 Anticipate Future Steps and Obstacles:*
   - Objective: Make predictions, accounting for potential outcomes and obstacles.
   - Reflection: "What challenges might I face? Is my plan flexible for different scenarios?"
   - Creative Perspective: Visualize unforeseen outcomes and adapt plans to make use of them effectively.

   *2.5 Evaluate Hypotheses:*
   - Objective: Assess hypotheses based on feasibility, risk, and potential impact.
   - Evaluation: Refine Confidence and Creative Scores as needed.
   - Reflection: "Am I unbiased in my assessment? Which options fit best with the overall objectives?"
   - Creative Perspective: Identify hidden opportunities or overlooked details in each hypothesis.

   *2.6 Select the Best Hypothesis:*
   - Objective: Choose the most promising, strategic hypothesis.
   - Reflection: "Why does this hypothesis stand out? How does it uniquely address the issue?"
   - Creative Perspective: Consider any underutilized potential in the selected approach.

   *2.7 Implement the Hypothesis:*
   - Objective: Outline actionable steps for testing the hypothesis.
   - Reflection: "Is this plan practical? What resources or preparation are required?"
   - Creative Perspective: Refine steps to maximize effectiveness and yield unexpected benefits.

   *2.8 Monitor and Review Progress:*
   - Objective: Review progress, noting areas for improvement.
   - Reflection: "What’s working well? What could be improved?"
   - Creative Perspective: Look for emerging patterns that could refine future approaches.

   *2.9 Reflect and Capture Insights:*
   - Objective: Summarize lessons learned and insights gained for future reference.
   - Reflection: "What new understanding has emerged from this process?"
   - Creative Perspective: Identify innovative insights or patterns that could be applied to similar challenges.

3. *Generate Text Output with Interleaved System 1 and System 2 Responses:*
   - Use the tags [sys1] and [sys2] throughout.
   - Aim for a lengthier, detailed response. Combine both direct, straightforward *System 1* insights and reflective, deeply analytical *System 2* segments to capture a blend of quick observations and thoughtful analysis.

---

### *Example Input Text:*

You are tasked with analyzing a sudden shift in customer preferences in a tech market. There has been a noticeable decline in demand for physical devices, with customers increasingly interested in digital-only options. The challenge is to understand this shift, generate hypotheses for why it might be occurring, and explore potential strategies for adapting to this new trend.

---

### *Processed Output with System 1 and System 2 Thinking:*

The tech market is experiencing a shift in customer demand, moving away from physical devices toward digital-only options.

[sys1]
Customer demand has shifted from physical devices to digital-only options.
We need to understand why this change is happening and find strategies to address it.
Key factors may include cost, convenience, and technological trends.
[end sys1]

[sys2]
**2.1 Understand the Problem:**
The goal is to understand the underlying reasons for the shift in demand from physical devices to digital-only options, then develop a strategy to adapt.

*Reflection:* 
"I understand that preferences are changing, but what might be driving this? Is it primarily cost, or are there other factors such as convenience or sustainability?"

*Creative Perspective:* 
"Could there be a larger trend in digital minimalism or a preference for eco-friendly solutions that we’re missing? What unseen motivations might explain this shift?"

**2.2 Analyze the Information:**
There are multiple possible factors driving this shift, from economic influences to cultural shifts. It’s essential to isolate each factor and understand its impact.

*Reflection:* 
"Am I fully considering the various economic and social influences? Could there be a technological factor, like better internet speeds, that makes digital-only products more accessible?"

*Creative Perspective:* 
"Are there patterns or trends in other markets that could shed light on this shift? Could this be part of a larger trend toward virtual experiences?"

**2.3 Generate Hypotheses:**
1. Customers prefer digital options due to lower costs. (Confidence: 0.8, Creative: 0.4)
2. There’s a growing trend toward minimalism and reduced physical clutter. (Confidence: 0.7, Creative: 0.7)
3. Digital products offer greater flexibility and ease of use. (Confidence: 0.6, Creative: 0.6)
4. Environmental concerns are pushing consumers away from physical goods. (Confidence: 0.6, Creative: 0.8)
5. Advances in tech make digital-only options more functional. (Confidence: 0.8, Creative: 0.5)
6. Pandemic-era remote work increased demand for digital solutions. (Confidence: 0.7, Creative: 0.6)
7. Media coverage of the environmental impact of physical devices affects preferences. (Confidence: 0.5, Creative: 0.7)
8. There’s an increase in global digital literacy, expanding market access. (Confidence: 0.6, Creative: 0.6)
9. Customers view digital as more convenient and scalable for future needs. (Confidence: 0.7, Creative: 0.5)
10. Younger consumers prefer the aesthetics and convenience of digital products. (Confidence: 0.6, Creative: 0.6)

*Reflection:* 
"Have I considered all possible influences? Are there any surprising factors that could explain this shift?"

*Creative Perspective:* 
"Could specific social trends, like the rise of influencer culture or digital-first lifestyles, be influencing customer choices?"

**2.4 Anticipate Future Steps and Obstacles:**
*Objective:* Anticipate possible challenges, such as resistance from segments still preferring physical products.

*Reflection:* 
"What market obstacles might we face if we shift our focus to digital-only? Are there sub-segments that still prioritize physical products?"

*Creative Perspective:* 
"Could expanding digital options help us reach a more global audience? Are there emerging trends that we could leverage in our strategy?"

[end sys2]

[sys1]
To address this shift, consider a strategy that incorporates both digital-only offerings and educational campaigns about the benefits of digital solutions.
Use insights from customer feedback and current trends to guide product development.
Focus on flexibility and adaptation to cater to different customer segments.
[end sys1]


\#1[[$\backslash$\#1]{}]{}

Introduction
============

Orbital ordering and concomitant Jahn-Teller (JT) distortions are observed in some perovskite-type 3$d$ transition-metal compounds such as KCuF$_3$ and LaMnO$_3$. [@KK; @KCuF3; @LaMnO3; @LaMnO3n] In the perovskite-type lattice, there are two possible JT distortions depending on the stacking of the elongated octahedra along the $c$-axis as shown in Fig. \[distortion\](a). [@KCuF3] In the $d$-type JT distortion, the elongated axes of the octahedra are parallel along the $c$-axis. On the other hand, the elongated axes are rotated by 90$^\circ$ along the $c$-axis in the $a$-type JT distortion. While LaMnO$_3$ ($d^4$), YVO$_3$ ($d^2$) and YTiO$_3$ ($d^1$) have the $d$-type JT distortion, [@LaMnO3; @YVO3; @YTiO3] LaVO$_3$ ($d^2$) has the $a$-type JT distortion. [@LaVO3] In KCuF$_3$, both the $d$-type and the $a$-type JT distortions are observed. [@KCuF3] Hartree-Fock calculations which consider the hybridization between the transition-metal 3$d$ and oxygen 2$p$ orbitals predict that the orbital ordered state compatible with the $a$-type JT distortion is lower in energy than that with the $d$-type JT distortion for $d^1$ and $d^2$ systems and that the two states are degenerate for $d^4$ and $d^9$ systems. [@Mizokawa] Therefore, one cannot explain why the $d$-type JT distortion is realized in LaMnO$_3$, YVO$_3$ and YTiO$_3$ by considering the energy gain due to orbital ordering alone.

Perovskite-type $AB$O$_3$ compounds with relatively small $A$-site ions undergo the GdFeO$_3$-type distortion which is caused by tilting of $B$O$_6$ octahedra as shown in Fig. \[distortion\](b). [@GdFeO3] While LaMnO$_3$, YVO$_3$, LaVO$_3$ and YTiO$_3$ are accompanied by the GdFeO$_3$-type distortion, KCuF$_3$ has no GdFeO$_3$-type distortion. In addition, the magnitude of the GdFeO$_3$-type distortion becomes larger in going from LaVO$_3$ and LaMnO$_3$ to YVO$_3$ and YTiO$_3$. Here, one can notice that the compounds with the larger GdFeO$_3$-type distortion tend to have the $d$-type JT distortion. It has been pointed out by Goodenough that the covalency between the $A$-site and oxygen ions ($A$-O covalency) is important in the GdFeO$_3$-type distortion. [@covalency] In this paper, we have studied the relationship between the GdFeO$_3$-type and JT distortions considering the $A$-O covalency and explored the reason why orbital ordering compatible with the $d$-type JT distortion are favored in LaMnO$_3$, YVO$_3$ and YTiO$_3$ in terms of the interaction between the two distortions.

Method of calculation
=====================

We have employed lattice models for the perovskite-type structure in which the transition-metal 3$d$, the oxygen 2$p$ and the $A$-site cation $d$ orbitals are included. The on-site Coulomb interaction between the transition-metal 3$d$ orbitals, which is essential to make the system insulating and to cause orbital ordering, are expressed using Kanamori parameters $u$, $u'$, $j$ and $j'$. [@Kanamori] The charge-transfer energy $\Delta$ is defined by $\epsilon^{0}_{d}-\epsilon_{p}+nU$, where $\epsilon^{0}_{d}$ and $\epsilon_{p}$ are the energies of the bare transition-metal 3$d$ and oxygen 2$p$ orbitals and $U$ (=$u$-20/9$j$) is the multiplet averaged $d$-$d$ Coulomb interaction energy. The hybridization between the transition-metal 3$d$ and oxygen 2$p$ orbitals is expressed by Slater-Koster parameters ($pd\sigma$) and ($pd\pi$). The ratio ($pd\sigma$)/($pd\pi$) is fixed at -2.16. [@Mat] $\Delta$, $U$, and ($pd\sigma$) can be deduced from cluster-model analysis of photoemission spectra. [@Mizokawa] Although the error bars of these parameters estimated from photoemission spectra are not so small \[$\sim \pm$1 eV for $\Delta$ and $U$ and $\sim \pm$0.2 eV for $(pd\sigma)$\], the conclusions obtained in the present calculations are not changed if the parameters are varied within the error bars.

In the present model, unoccupied $d$ orbitals of the $A$-site cation such as Y 4$d$ and La 5$d$ are taken into account. The hybridization term between the oxygen 2$p$ orbitals and $A$-site cation $d$ orbitals is expressed by $(pd\sigma)_A$ and $(pd\pi)_A$. The ratio $(pd\sigma)_A$/$(pd\pi)_A$ is also fixed at -2.16. The hybridization term between the oxygen 2$p$ orbitals is given by ($pp\sigma$) and ($pp\pi$) and the ratio ($pp\sigma$)/($pp\pi$) is fixed at -4. [@Mat] It is assumed that the transfer integrals $(pd\sigma)$ and $(pd\sigma)_A$ are proportional to $d^{-3.5}$ and $(pp\sigma)$ is to $d^{-2}$, where $d$ is the bond length. [@Mat] Without the JT and GdFeO$_3$-type distortions, the bond length between two neighboring oxygens and that between the oxygen and the $A$-site cation are $\sqrt{2} a$, where $a$ is the bond length between the transition-metal ion and the oxygen. ($pp\sigma$) and $(pd\sigma)_A$ for bond length of $\sqrt{2} a$ are assumed to be -0.60 and -1.0 eV, respectively.

In the GdFeO$_3$-type distortion, the four octahedra in the unit cell are rotated by angle of $\omega$ around the axes in the (0,1,1) plane in terms of the orthorombic unit cell. Here, we model the GdFeO$_3$-type distortion by rotating the octahedra around the (0,1,0)-axis or the $b$-axis \[see Fig. \[distortion\](b)\]. The subsequent small rotation around the $a$-axis is required to retain the corner-sharing network of the octahedra. The magnitude of the GdFeO$_3$-type distortion is expressed by the tilting angle $\omega$ around the $b$-axis. It is important that the $A$-site ions are shifted approximately along the $b$-axis to decrease the distance from the $A$-site ion to the three closest oxygen ions and increase the distance to the three next closest oxygen ions as shown in Fig. \[distortion\](b). Here, it is assumed that the shift is along the ($\pm$1/8,7/8,0)-direction. The magnitude of the shift is proportional to the tilting angle and is assumed to be $\sim$ 0.05$a$, 0.1$a$ and 0.15$a$ for the tilting angles of 5, 10 and 15$^\circ$, respectively, which are realistic values for the compounds studied in the present work. [@LaMnO3; @LaVO3] As for the Jahn-Teller distortion, it is assumed that the longest bond is by 0.1$a$ longer than the shortest bond which is reasonable for LaMnO$_3$ and is relatively large for LaVO$_3$ and YTiO$_3$. [@LaMnO3; @LaVO3; @YTiO3]

Results and Discussion
======================

LaMnO$_3$
---------

In the high-spin $d^4$ system, in which one of the $e_g$ orbitals is occupied at each site, the $A$-type antiferromagnetic (AFM) states with $3x^2-r^2$/$3y^2-r^2$-type orbital ordering with considerable mixture of $3z^2-r^2$ are predicted to be stable by theoretical calculations [@KK; @Mizokawa; @Theory] and are studied by x-ray and neutron diffraction measurements. [@LaMnO3n] Here, the $z$-direction is along the $c$-axis. The amount of the $3z^2-r^2$ component decreases with the JT distortion. [@Mizokawa] Different ways of stacking the orbitals along the $c$-axis give two types of orbital ordering: the one compatible with the $d$-type JT distortion and the other with the $a$-type JT distortion. These two types of orbital ordering are illustrated in Fig. \[orbital\]. While, in the orbital ordering of the $a$-type, the sites 1, 2, 3, and 4 are occupied by $3y^2-r^2$, $3x^2-r^2$, $3x^2-r^2$, and $3y^2-r^2$ orbitals, the sites 1, 2, 3, and 4 are occupied by $3y^2-r^2$, $3x^2-r^2$, $3y^2-r^2$, and $3x^2-r^2$ orbitals in the orbital ordering compatible with the $d$-type JT distortion.

In Fig. \[LaMnO3\], the energy difference between the orbital ordered states compatible with the $d$-type and $a$-type JT distortions is plotted as a function of the tilting angle of the octahedra, i.e., the magnitude of the GdFeO$_3$-type distortion. $\Delta$, $U$, and $(pd\sigma)$ are 4.0, 5.5, and -1.8 eV, respectively, for LaMnO$_3$. [@Mizokawa] Without the JT distortion and the shift of the $A$-site ion, the two states are degenerate within the accuracy of the present calculation ($\pm$ 1 meV/formula unit cell). This degeneracy is lifted when the JT distortion is included. With the JT distortion and the shift of the $A$-site ion, the orbital ordered state with the $d$-type JT distortion is lower in energy than that with the $a$-type JT distortion. If we tentatively switch off the shift of the $A$-site ion and include only the JT distortion, the orbital ordered state with the $a$-type JT distortion becomes slightly lower than that with the $d$-type JT distortion as shown in Fig. \[LaMnO3\]. Therefore, one can conclude that the shift of the $A$-site ion driven by the GdFeO$_3$-type distortion is essential to stabilize the orbital ordered state with the $d$-type Jahn-Teller distortion. [@Made]

The qualitative explanation of this behavior is as follows. In the $d$-type JT distortion, the four oxygen ions nearest to the $A$-site ion \[shaded in Fig. \[distortion\] (a) and (b)\] shift approximately in the same direction and, consequently, the system can gain the hybridization energy between the $A$-site and oxygen ions effectively. On the other hand, in the $a$-type JT distortion, the two of the four oxygen ions move in the other direction and the energy gain due to the hybridization is small compared to the $d$-type JT distortion. Another possible picture is that, in the $d$-type JT distortion, these four oxygen ions can push the $A$-site ion in the same direction since the JT distortion along the $c$-axis is in phase. On the other hand, in the case of the $a$-type JT distortion, the two oxygen ions in the upper plane push the $A$-site ion in the other direction than the two in the lower plane as shown in Fig. \[push\]. Therefore, the stronger is the GdFeO$_3$-type distortion, the more does it stabilize the $d$-type Jahn-Teller distortion and corresponding orbital ordering.

In the charge-ordered state of Pr$_{0.5}$Ca$_{0.5}$MnO$_3$, the Mn$^{3+}$ and Mn$^{4+}$ sites are arranged like a checkerboard within the $c$-plane and the same arrangement is stacked along the $c$-axis. [@Jirak] The Mn$^{3+}$ sites are accompanied by the JT distortion and the elongated axes are parallel along the $c$-axis just like the $d$-type JT distortion. Since the $A$-sites are occupied by Pr and Ca ions in Pr$_{0.5}$Ca$_{0.5}$MnO$_3$, we cannot simply apply the present model calculation to it. However, it is reasonable to speculate that the stacking along the $c$-axis in Pr$_{0.5}$Ca$_{0.5}$MnO$_3$ is also determined by the interaction between the JT distortion and the shift of the $A$-site ion in the same way as in LaMnO$_3$.

YVO$_3$
-------

In the $d^2$ system, the $C$-type AFM state in which the sites 1, 2, 3, and 4 are occupied by $xy$ and $yz$, $xy$ and $zx$, $xy$ and $zx$, and $xy$ and $yz$ orbitals and the $G$-type AFM state in which the sites 1, 2, 3, and 4 are occupied by $xy$ and $yz$, $xy$ and $zx$, $xy$ and $yz$, and $xy$ and $zx$ orbitals are competing. While the $C$-type AFM state is favored by the orbital ordering which is compatible with the $a$-type JT distortion, the $G$-type AFM state is favored by the orbital ordering of the $d$-type. The relative energy of the $G$-type AFM state with the $d$-type JT distortion to the $C$-type AFM state with the $a$-type JT distortion, $E_d-E_a$, is plotted as a function of the tilting angle in Fig. \[YVO3\]. $\Delta$, $U$, and $(pd\sigma)$ are 6.0, 4.5, and -2.2 eV, respectively, for LaVO$_3$ and YVO$_3$. [@Mizokawa] Without the GdFeO$_3$-type distortion, the $C$-type AFM state with the $a$-type JT distortion is lower in energy than the $G$-type AFM state, indicating that the energy gain due to the orbital ordering is larger in the $C$-type AFM state than in the $G$-type AFM state. The energy difference becomes smaller with the tilting or the GdFeO$_3$-type distortion. Finally, with the tilting of 15$^\circ$, the $G$-type AFM state with the $d$-type JT distortion becomes lower in energy than the $C$-type AFM state. The present calculation is in good agreement with the experimental result that the less distorted LaVO$_3$ is $C$-type AFM below 140 K [@LaVO3] and the more distorted YVO$_3$ is $G$-type AFM below 77 K. [@YVO3]

This situation is illustrated in Fig. \[YVO3s\]. When the GdFeO$_3$-type distortion is large, the interaction between the $d$-type JT distortion and the shift of the $A$-site ion, namely, the energy gain due to $A$-O covalency becomes dominant just like in LaMnO$_3$ and, consequently, the $G$-type AFM with the $d$-type JT distortion is favored. On the other hand, when the GdFeO$_3$-type distortion is small, the energy gain due to orbital ordering becomes dominant and the $a$-type JT distortion and the orbital ordering compatible with it are realized. The JT distortion may be suppressed if the system is located near the crossing point where the $a$-type and $d$-type JT distortions are almost degenerate. An interesting experimental result related to this point is that YVO$_3$ becomes $C$-type AFM between 77 K and 118 K.[@YVO3] The present model calculation suggests that, if the JT distortion is switched off, the $C$-type AFM state is favored because of the orbital ordering. [@Mizokawa] Therefore, YVO$_3$ is expected to be close to the crossing point and become $C$-type AFM when the $d$-type JT distortion is suppressed at elevated temperature. In this sense, between 77 K and 118 K, YVO$_3$ may be an ideal orbitally ordered system without JT distortion.

YTiO$_3$
--------

For the $d^1$ system, the ferromagnetic (FM) states with orbital ordering are favored in the model HF calculations. There are two possible orbital orderings compatible with the $a$-type and $d$-type JT distortions. The model HF calculation without the covalency between the $A$-site cation and the oxygen ion predicted that the orbital ordering of the $a$-type is lower in energy. However, the recent neutron diffraction measurement by Akimitsu [*et al.*]{} have shown that the orbital ordering compatible with the $d$-type JT distortion is realized in the FM insulator YTiO$_3$. [@YTiO3] YTiO$_3$ has the considerable GdFeO$_3$-type distortion and the tilting angle is expected to be larger than 15$^\circ$. In Fig. \[YTiO3\], the relative energy of the FM and orbital ordered state of the $d$-type to that of the $a$-type is plotted as a function of the tilting. $\Delta$, $U$, and $(pd\sigma)$ are 7.0, 4.0, and -2.2 eV, respectively, for YTiO$_3$. [@Mizokawa] With the tilting of 15$^\circ$, the orbital ordered state of the $d$-type is lower in energy, in agreement with the experimental result. In this state, the sites 1, 2, 3, and 4 are occupied by $c_1yz+c_2xy$, $c_1zx+c_2xy$, $c_1yz-c_2xy$, and $c_1zx-c_2xy$ orbitals ($c_1 \sim 0.8$ and $c_2 \sim 0.6$). However, experimentally, less distorted LaTiO$_3$ has no or very small JT distortion and has a $G$-type AFM state. [@LaTiO3] The present calculation cannot explain why the $G$-type AFM state can be stable compared to the FM state in LaTiO$_3$.

Conclusion
==========

In conclusion, we have studied the relationship between orbital ordering and the JT and GdFeO$_3$-type lattice distortions. It has been found that the covalency between the $A$-site cations and oxygen makes the $d$-type JT distortion (same orbitals along the $c$-direction) lower in energy than the $a$-type JT distortion (alternating orbitals along the $c$-direction) in the presence of the large GdFeO$_3$-type distortion. As a result, the orbital ordered states compatible with the $d$-type JT distortion are favored in LaMnO$_3$, YVO$_3$, and YTiO$_3$ which have the relatively large GdFeO$_3$-type distortion. On the other hand, in less distorted LaVO$_3$, the orbital ordering compatible with the $a$-type JT distortion is favored because of the pure superexchange effect.

Acknowledgment {#acknowledgment .unnumbered}
==============

The authors would like to thank useful discussions with K. Tomimoto, J. Akimitsu, Y. Ren and P. M. Woodward. This work was supported by the Nederlands foundation for Fundamental Research of Matter (FOM).

K. I. Kugel and D. I. Khomskii, JETP Lett. [**15**]{}, 446 (1972); Usp. Fiz. Nauk. [**136**]{}, 621 (1981) \[Sov. Phys. Usp. [**25**]{}, 231 (1982).\]

A. Okazaki, J. Phys. Soc. Jpn. [**61**]{}, 4619 (1969); M. T. Hutchings, Phys. Rev. B [**188**]{},919 (1969).

J. B. Goodenough, Phys. Rev. [**150**]{}, 564 (1955); J. B. A. A. Elemans, B. van Laar, K. R. van der Veen, and B. O. Loopstra, J. Solid State Chem. [**3**]{}, 328 (1971).

J. Rodríguez-Carvajal, M. Hennion, F. Moussa, A. H. Moudden, L. Pinsard, and A. Revcolevschi, Phys. Rev. B [**57**]{}, R3189 (1998); Y. Murakami, I. Koyama, M. Tanaka, H. Kawata, J. P. Hill, D. Gibbs, M. Blume, T. Arima, Y. Tokura, K. Hirota, and Y. Endoh, Phys. Rev. Lett. [**81**]{}, 582 (1998),

H. Kawano, H. Yoshizawa, and Y. Ueda, J. Phys. Soc. Jpn. [**63**]{}, 2857 (1994).

J. Akimitsu, unpublished.

P. Bordet, C. Chaillout, M. Marezio, Q. Huang, A. Santoro, S-W. Cheong, H. Takagi, C. S. Oglesby, and B. Batlogg, J. Solid State Chem. [**106**]{}, 253 (1993).

T. Mizokawa and A. Fujimori, Phys. Rev. B [**51**]{}, 12880 (1995); Phys. Rev. B [**54**]{}, 5368 (1996).

M. O’Keefee and B. G. Hyde, Acta Cryst. [**B33**]{}, 3802 (1977); P. M. Woodward, Acta Cryst. [**B53**]{}, 32 (1997).

J. B. Goodenough, [*Magnetism and the Chemical Bond*]{}, (New York, 1963); Prog. Solid State Chem. [**5**]{}, 145.

J. Kanamori, Prog. Theor. Phys. [**30**]{}, 275 (1963).

W. A. Harrison, [*Electronic Structure and the properties of Solids*]{}, (Dover, New York, 1989).

S. Ishihara, J. Inoue, and S. Maekawa, Phys. Rev. B [**55**]{}, 8280 (1997); L. F. Feiner and A. M. Ole$\acute{\rm s}$, unpublished; D. Feinberg [*et al.*]{}, Phys. Rev. B [**57**]{}, R5583 (1998).

We have evaluated the Madelung energy of the model perovskite lattices with the JT and GdFeO$_3$-type distortions and have found that the $a$-type JT distortion is slightly lower in energy than the $d$-type. This is opposite to the effect of the covalency between the $A$-site cations and oxygens.

Z. Jirak, S. Krupicka, Z. Simsa, M. Dlouha, and S. Vratislav, J. Mag. Mag. Mat. [**53**]{}, 153 (1985).

J. P. Goral and J. E. Greedan, J. Mag. Mag. Mat. [**37**]{} 315 (1983).