Optical excitations of transition-metal oxides under the orbital multiplicity effects

We investigated optical excitations of transition-metal (TM) oxides with metal oxygen octahedra taking account of the orbital multiplicity effects. We predicted excitation energies of intersite d–d transitions and p–d transitions of TM oxides. We compared the evaluated excitation energies with reported experimental data, and found that they are in good agreement with each other. Moreover, we could demonstrate possible answers for a few long-standing problems of the low-frequency spectral features in some early 3d TM oxides: (i) the broad and multi-peak structures of the d–d transitions, (ii) the low values (around 2 eV) of the d–d transition energies for some t2g1 and t2g2 systems, and (iii) the lack of the d–d transition below 4.0 eV region for LaCrO3, one of the t2g3 systems. These indicate that our approach considering the orbital multiplicity effects could provide good explanations of intriguing features in the optical spectra of some early TM oxides. In addition, we showed that optical spectroscopy can be useful as a powerful tool to investigate spin and/or orbital correlations in the TM ions. Finally, we discussed the implications of the orbital multiplicity in the Zannen–Sawatzky–Allen scheme, which has been used successfully to classify correlated electron systems.


Introduction
Recently, the orbital degree of freedom has emerged as an important physical term in understanding numerous intriguing phenomena occurring in condensed matter systems [1,2]. It has been found that intriguing physical properties of numerous transition-metal (TM) oxides, including manganites [1,2], titanates [3], vanadates [4,5], cobaltates [6], molybdates [7] and ruthenates [8], could be explained by taking account of the orbital degree of freedom. For instance, couplings between the orbital and the other degrees of freedom, such as charge, spin, and/or lattice, are considered as key ingredients in explaining novel phenomena observed in manganites, including colossal magnetoresistance [1], orbital ordering [2], and optical anisotropy [9]. Although the importance of the orbitals has been widely recognized recently, theoretical and experimental efforts to explain such effects are still quite scarce. The single-band Hubbard model, which has been most commonly used to explain the physical properties of strongly correlated electron systems, does not take account of the orbital degeneracy. In addition, there are few experimental techniques to probe this important degree of freedom [10].
Optical spectroscopy has been proven to be a powerful experimental tool to investigate the electronic structure of TM oxides [11]. Similar to experts in other condensed matter experimental fields, most optical spectroscopists have used the single-band Hubbard model as a starting point to explain optical spectra of TM oxides. Owing to this tradition, there have been important spectral features which are difficult to be understood in the single-band Hubbard picture. For example, numerous early transition oxides show absorption features, possibly due to optical 3 Institute of Physics ⌽ DEUTSCHE PHYSIKALISCHE GESELLSCHAFT transitions between the lower and the upper Hubbard bands; these features are usually very broad or have multiple peak structures, which cannot be explained using the single-band Hubbard model [12,13].
Quite recently, we demonstrated that optical spectroscopy could be a very useful and powerful method to investigate certain aspects related to the orbital degree of freedom, especially orbital and spin correlations [14,15]. Based on the orbitally degenerate Hubbard model, we evaluated the orbital correlations between the nearest-neighbour Ru ions from optical spectra of (Ca,Sr) 2 RuO 4 , and proposed a possible orbital ground state [14]. In a similar way, we also estimated the spin correlations between the Mo ions in RE 2 Mo 2 O 7 (RE = Y, Sm and Nd), and explained the spin ground states of the molybdates [15]. In these studies, we considered the intersite d-d transitions only in the systems of metal oxygen octahedra with two or four t 2g electrons on the TM ions.
It should be noted that the new approach considering the orbital multiplicity can be applied to a wealth of optical data for TM oxides, which have been accumulated over the decades. In particular, it can provide new insights on some optical spectral features, which could not have been explained in the single-band Hubbard picture. In this paper, we will take account of the orbital multiplicity effects in both interatomic d-d transitions and charge-transfer p-d excitations for all electronic configurations for systems of metal oxygen octahedra with t 2g electrons. We will evaluate all of the possible excitation energies for the d-d and the p-d transitions. We will compare them with reported experimental data and confirm that such an approach could provide a good basis to explain the unusual spectral features observed in early TM oxides. We will also demonstrate how to apply optical spectroscopy as a probing tool for nearest neighbouring spin and/or orbital correlations. Finally, we will discuss the implications of the orbital multiplicity to the Zaanen-Sawatzky-Allen scheme [16].

Orbitally degenerate Hubbard model
The single-band Hubbard model has been commonly used as a starting point to explain the physical properties of TM oxides. In this model, the Hamiltonian can be written as where n iσ ≡ c † iσ c iσ and c † iσ (c iσ ) is the creation (annihilation) operator for an electron on site i with spin σ(= ↑ or ↓). And, t ij is the transfer integral between sites i and j, which are located as nearest neighbours. U is the on-site Coulomb repulsion. In equation (1), the electrons at the oxygen sites are not included, and only the nearest-neighbouring hopping is included among numerous possible hoppings between the TM ions. In addition, the orbital degeneracies of electrons in TM ions are not included. To explain more realistic physical phenomena occurring in TM oxides, the single-band Hubbard model has been extended to various forms. Some workers considered the multi-band Hubbard model, which takes account of electrons in the oxygen 2p orbitals as well as in the TM d orbitals [17]. Also, many workers extended equation (1) to include hoppings between next-nearest neighbours (and/or third nearest neighbours) [18]. These approaches have been quite useful to explain magnetic and charge correlations, and the shapes of Fermi surfaces of some cuprates [18,19].
Numerous TM oxides have the simple perovskite structure. In this crystal structure, the d orbital states of TM ions are split into the triply degenerate t 2g (i.e., d xy , d yz and d zx ) orbital states and the doubly degenerate e g (i.e., d x 2 −y 2 and d 3z 2 −r 2 ) orbital states. Due to electrostatic interaction between the d electrons and the oxygen 2p electrons, the t 2g and the e g states will have the 'crystal field splitting', 10Dq, as schematically shown in figure 1(a). In most TM oxides, values of 10Dq are typically located between 1 and 4 eV [20]- [23]. For some perovskite oxides, a strong electronphonon coupling, called the Jahn-Teller distortion, or the GdFeO 3 -type distortion results in an additional splitting in the t 2g and/or the e g levels [24]. In most cases, values of the structural distortion energies are much smaller than 10Dq, so we do not include effects of this energy splitting in this paper.
Optical conductivity spectra of most TM oxides with the perovskite structure are composed of the interatomic transitions. Figure 1(b) shows a schematic diagram of optical transitions in a TM-oxygen-TM network, which is a basic structural unit of the perovskite structure. The oxygen 2p orbitals are fully occupied. As an example, we put only one electron in the t 2g orbitals at the TM site. The intra-atomic optical excitations between the oxygen 2p orbitals were not allowed, and those between the d orbitals of the TM ions are generally forbidden by the electric dipole selection rule. The interatomic optical excitations can occur from the occupied oxygen 2p orbitals to the unoccupied t 2g orbitals (i.e., the p-d transitions) or between the d orbitals (i.e., the d-d transitions). The transition energy of each excitation is given by the energy difference between the initial and the final states. For the p-d transition, its transition energy corresponds to the so-called charge-transfer energy, . And, for the d-d transition, its transition energy is related to the Coulomb repulsion energy, U. Zaannen, Sawatzky and Allen classified many strongly correlated electron systems in terms of and U [16], and their scheme has been widely used by many workers [25].
Note that these understandings are based on the single-band Hubbard model, which ignores the orbital multiplicity effects. In this simple model, the final d state is defined as a single upper Hubbard band, so energies of the p-d and the d-d transitions can be represented by single quantities, i.e., and U, respectively. When the orbital degeneracy of the involved electrons is included, each of the p-d and the d-d transitions should be understood as a transition from a given state to multiple final states. In the following two subsections, we consider how the p-d and the d-d transitions should be modified when we take into account the orbital degeneracy. Since the multiplicity effects can be experimentally recognized more easily for the d-d transitions of early TM oxides [12], we will look into systems only with n t 2g electrons (n = 1-5), called t n 2g systems. For systems with e g electrons, we will have a brief discussion in section 4.2.

Eigenstates and eigenenergies under the multiplicity effects
In order to provide a basis for later discussion, we will briefly explain eigenstates and their eigenenergy values of the multiplet states when the orbital degeneracy effects are considered. Let us give an example for the d 2 states, i.e., the t 2 2g state and the t 1 2g e 1 g state. Since electrons are elementary particles with spin 1/2, the total wavefunctions of the d 2 state should be antisymmetric about the exchange operation of the electrons. Therefore, the multiplet states should be constructed based on this important property.
When the multiplicity effects are considered appropriately, the t 2 2g configuration should have four eigenstates for two electrons, which can be written as 3 T 1 , 1 T 2 , 1 E and 1 A 1 in the irreducible representation [26,27]. The wavefunctions corresponding to each irreducible representation are shown in table 1. Similarly, for the t 1 2g e 1 g configuration, there are four multiplet states, represented as 3 T 2 , 3 T 1 , 1 T 2 , and 1 T 1 . The lower part of table 1 shows the wavefunctions of the multiplet states.
The corresponding eigenenergy values of each eigenstate are varied according to the spin configurations and the respective orbitals of each multiplet state. For both t 2g and e g electrons, their diagonal direct coupling U term is the same as U = A + 4B + 3C. Here, A, B and C are the Racah parameters [26,27]. 2 However, the exchange coupling terms can differ depending on the orbital wavefunctions involved. While the exchange coupling between t 2g orbitals is 3B + C, that between e g orbitals is 4B + C. When both t 2g and e g electrons participate, the exchange couplings should be given in more complex ways. That is, the exchange coupling between d x 2 −y 2 and d xy (d yz , d zx ) orbitals is C (3B + C, 3B + C), and the exchange couplings between d 3z 2 −r 2 and d xy (d yz , d zx ) orbitals is 4B + C (B + C, B + C). For clarity, let us define these exchange interactions as J 1 ≡ C, J 2 ≡ B + C, J 3 ≡ 3B + C and J 4 ≡ 4B + C. 2 Racah parameters = A, B and C can be written in terms of three Slater integrals, i.e., F 0 , F 2 and F 4 , as A = F 0 − 49F 4 , B = F 2 − 5F 4 and C = 35F 4 , and the Slater integrals are given as F 0 = F 0 , F 2 = 1 49 F 2 and Here, R d (r) is the radial part of the d orbital wavefunction, and r > and r < denote the larger and smaller of r 1 and r 2 , respectively. See [25,26].

7
Institute of Physics ⌽ DEUTSCHE PHYSIKALISCHE GESELLSCHAFT For the t 2 2g configuration, the eigenenergy values are given by A − 5B for the 3 T 1 state, A + B + 2C for the 1 T 2 state, A + B + 2C for the 1 E state and A + 10B + 5C for the 1 A 1 state [26,27]. Considering that U = A + 4B + 3C and J 3 = 3B + C, eigenenergy values for the t 2 2g states can be represented as U − 3J 3 for the 3 T 1 state, U − J 3 for the 1 T 2 state, U − J 3 for the 1 E state and U + 2J 3 for the 1 A 1 state. For the t 1 2g e 1 g configuration, the eigenenergy values are given by A − 8B for the 3 T 2 state, A + 4B for the 3 T 1 state, A + 2C for the 1 T 2 state and A + 4B + 2C for the 1 T 1 state. As we consider the composing orbital states of each t 1 2g e 1 g state, the corresponding eigenenergy values can be represented as U − 3J 4 for the 3 T 2 state, U − 3J 1 for the 3 T 1 state, U − J 4 for the 1 T 2 state and U − J 1 for the 1 T 1 state. Due to the crystal field splitting between the t 2g and the e g states, there should be an additional energy parameter 10Dq for the t 1 2g e 1 g configuration. In the second column of table 1, we show the eigenenergy values for both the t 2 2g and the t 1 2g e 1 g configurations. Since our main attention is focused on t 2g electrons, we define the exchange interaction between t 2g orbitals as J H , i.e., J H ≡ J 3 (= 3B + C).

p-d transitions using the simple electron configuration.
Before going into a detailed discussion of the optical excitations with consideration of the multiplicity effects, it would be useful to discuss p-d transitions using the simple electron configuration where we fill each orbital state with one or two electrons without antisymmetrizing the total wavefunction of multiple electrons. Figure 2 shows the d n+1 final states of the p-d transitions d n → Ld n+1 , where L indicates a ligand hole in the oxygen. The initial d n states are supposed to have the t n 2g configuration. The arrows with the solid triangles display the electrons occupied in the atom before the transition, and the arrows with open triangles indicate the transferred electrons.
In figure 2(a), we consider the p-d transitions of the d 1 system, i.e., d 1 → Ld 2 . There are five possible d 2 final states in the d 1 → Ld 2 excitation. We evaluate the excitation energies with the simple consideration of the orbital occupation, and display the results in the first line of each diagram. In figures 2(b)-(e), we give the same consideration to the charge-transfer excitations, d n → Ld n+1 with n = 2, 3, 4 and 5, respectively. We show possible d n+1 final states for each excitation and the corresponding excitation energy values. It should be noted that in the single-band Hubbard model such various kinds of p-d transitions for each electron configuration are not expected. In this respect, the approach based on this simple picture can provide at least a qualitatively good insight on how the optical excitations should be understood under consideration of the orbital degeneracy effects.

p-d transitions under
the orbital multiplicity effects. Now, we discuss the intersite p-d transitions under the consideration of orbital multiplicity effects. Let us begin with p-d transitions with one t 2g electron per metal-ion site, i.e., d 1 → Ld 2 . As final states, we have to consider t 1 2g e 1 g as well as t 2 2g configurations. For the t 2 2g configuration, there will be four final eigenstates, represented by 3 T 1 , 1 T 2 , 1 E and 1 A 1 in the irreducible representation, as shown in table 1 [26,27]. The S = 1 state, i.e., 3 T 1 , will have the lowest energy, so the excitation energy to the 3 T 1 final state is defined as 0 (t 1 2g ). Compared to the 3 T 1 state, three S = 0 states, i.e., 1 T 2 , 1 E and 1 A 1 , will have energies higher by 2J H , 2J H and 5J H , respectively. Then the excitation energies to the 1 T 2 , 1 E and 1 A 1 states can be written as 0 (t 1 2g ) + 2J H , 0 (t 1 2g ) + 2J H and 0 (t 1 2g ) + 5J H , respectively. For the t 1 2g e 1 g configuration, there will be four final eigenstates, represented by 3 T 2 , 3 T 1 , 1 T 2 and 1 T 1 . The excitation energies into these states can be evaluated in a similar manner. Table 2 summarizes all of the possible d 1 → Ld 2 transitions when the orbital multiplicity effects are fully taken into account. The first column shows all of the possible d 2 final states in the irreducible representation. The second and third columns show the eigenenergy values of the final d 2 configuration E(d 2 ) and the corresponding p-d transition energy costs E(d 1 → Ld 2 ), respectively. Both of them are represented in two different ways, i.e., in terms of Racah parameters, A, B and C, as well as in terms of U and J H .
In table 3, we show excitation energies of the possible p-d transitions, d n → Ld n+1 with n = 2, 3, 4 and 5. We assume that the ground states of the initial d n states have the lowspin configurations, i.e., t n 2g . The first column shows all of the possible d n+1 final states in the irreducible representation. And, the second column shows corresponding p-d transition energy costs E(d n → Ld n+1 ). The reference energies 0 (t n 2g ) are defined as the excitation energies to the t n+1 2g final states, shown in the second row of each table. Note that all of the excitation energies can be represented in terms of three energy parameters, 0 (t n 2g ), J H and 10Dq. In the above, we discussed the p-d transitions both using the simple electron configuration and considering the orbital multiplicity effects. Let us begin to compare the possible excitations Institute of Physics ⌽ DEUTSCHE PHYSIKALISCHE GESELLSCHAFT Table 3. Possible p-d transitions of the d n systems (n = 2-5), d n → Ld n+1 , considering the multiplicity effects. The first column shows the possible d n+1 final states, and the second column shows the corresponding energy cost of each excitation. Here, 0 (t n 2g ) are defined as the excitation energies to the t n+1 2g states shown in the second row of each table, where the energy eigenvalues of the t n+1 2g states are also shown. In this table also, we suppose that with results of the d 1 system. Among the five possible d 2 final states in the d 1 → Ld 2 excitation shown in figure 2(a), except for the state in the first diagram, the other four d 2 states are not eigenstates, but states composed of more than one eigenstate. For example, the second diagram corresponds to a state composed of 3 T 1 and 1 T 2 , whose wavefunctions are shown in table 1 as (t 2 2g Referring to the results shown in table 2, the excitation energies to the 3 T 1 and 1 T 2 states are 0 (t 1 2g ) and 0 (t 1 2g ) + 2J H , respectively. So, the p-d transition to the state shown in the second diagram of figure 2(a) should have two excitation energies, 0 (t 1 2g ) and 0 (t 1 2g ) + 2J H , whose averaged value happens to be the same as the excitation energy evaluated using the simple electron configuration.
In the second and subsequent lines above each diagram of figure 2, we display the eigenstates composing each final state and the corresponding energy costs with a parenthesis. Actually, most of the averaged values of all possible transition energies are quite similar to the values evaluated with the simple consideration of the orbital occupation, although there are some differences for the transition to the e g states. (Such differences originate mainly from the wide variation of the Institute of Physics ⌽ DEUTSCHE PHYSIKALISCHE GESELLSCHAFT Table 4. Possible d-d transitions considering the multiplicity effects. Among the final states of each optical process, we show the states having the multiplicity effect, such as the t 2 2g state for the t 1 2g system, the t 3 2g state for the t 2 2g system, and the t 2 2g and t 4 2g states for the t 3 2g system. Also, we display the corresponding energy costs in each optical process.

Optical process
Multiplet final states and energy costs exchange interaction values when both the e g and the t 2g orbitals are involved, as shown in section 3.1.) This indicates that the picture using the simple electron configuration can be more useful in understanding the optical excitations when only the t 2g -orbitals are concerned.
Since the photon absorption is a sudden transition process, the final state could be one of the states shown in the diagrams of figure 2. Note that this is composed of a linear combination of eigenstates, as explained already. In reality, the probability of entering one of its final eigenstates will be proportional to the square of the coefficient in the linear combination, and the required energy cost will be the difference between the final and the initial energy eigenvalues. Therefore, the simple schematic diagrams with proper energy costs, obtained by fully considering the orbital multiplicity effects, should be very useful to understand the optical excitations.

Intersite d-d transitions considering the orbital multiplicity effects.
As demonstrated in our previous papers [14,15], the orbitally degenerate Hubbard model, which takes into account the orbital degeneracy effects, can predict numerous intersite d-d transitions. For the t 1 2g system, the d-d transition happens to be t 1 2g t 1 2g → t 0 2g t 2 2g . Since the t 0 2g and the t 1 2g configurations contain zero and one d-electron, respectively, the Coulomb interaction between d electrons for these configurations should be zero. On the other hand, the t 2 2g configuration has four kinds of multiplet states, i.e., 3 T 1 , 1 E, 1 T 2 and 1 A 1 , whose eigenenergy values are U − 3J H , U − J H , U − J H and U + 2J H , respectively [26,27]. Therefore, the energy costs for the d-d transitions of the t 1 2g system should be U − 3J H , U − J H , U − J H and U + 2J H . Table 4 summarizes all of the possible optical d-d transitions of the t n 2g system (n = 1, 2 and 3). We display the multiplet final states of each transition: (i) for n = 1, 3 T 1 , 1 E, 1 T 2 and 1 A 1 states of the t 2 2g configuration, (ii) for n = 2, 4 A 2 , 2 E, 2 T 1 and 2 T 2 states of the t 3 2g configuration and (iii) for n = 3, 3 T 1 state of the t 2 2g and t 4 2g configurations, respectively. Note that the initial state of each d-d transition should be the ground state of each configuration, i.e., the 3 T 1 state for the t 2 2g system and the 4 A 2 state for t 3 2g system. We obtain the energy costs by evaluating the energy differences between the initial and the final states, and display the results at the lower parts of each multiplet final state. For the t 2 2g system, the excitation energies are U − 3J H , U and U + 2J H , which are similar to those for the t 1 2g system. Interestingly, for t 3 2g system, its d-d transition appears only at U + 2J H . d-d transitions for the t 4 2g and the t 5 2g systems are the same as those of the t 2 2g and the t 1 2g system, respectively, due to the electron-hole symmetry.

Intersite d-d transitions related to the spin and orbital correlations.
In this subsection, let us look into the possibility of probing spin and orbital correlations using the intersite d-d transitions. To begin with, it is useful to consider the selection rules for the intersite d-d transitions. In real oxides, the intersite d-d transitions could be allowed due to the hybridization between the O 2p bands and the TM d bands. Figure 3(a) shows an intersite d-d transition from one d xy orbital to other d xy orbitals in the nearest-neighbour TM ions. If their spins are aligned along the same direction, the intersite transition should be forbidden due to the Pauli exclusion principle. On the other hand, if their spins are in opposite directions, the intersite transition could be allowed, since the corresponding matrix element might not be zero due to the finite overlap of the d xy orbitals, hybridized with the O 2p y orbital. Figure 3(b) shows another intersite d-d transition from one d xy orbital to other d yz orbitals in the nearest-neighbour TM ions. In this case, even when their spins are pointing in opposite directions, the overlap between the d orbitals is negligibly small (even with any O 2p orbital), so the corresponding intersite transition should be almost forbidden. Let us apply this selection rule to numerous t n 2g t n 2g → t n−1 2g t n+1 2g transitions. It should be noted that, depending on the spin and orbital arrangements between the nearest-neighbour d metal ions, the allowed transitions could vary. As shown in figure 4, for each t n 2g t n 2g → t n−1 2g t n+1 2g transition, we can think of four spin/orbital configurations as initial states, i.e., FM/FO, FM/AFO, AFM/FO and AFM/AFO. Here, FM and AFM represent the ferromagnetic and the antiferromagnetic spin configurations between the d electrons, respectively. And, FO (AFO) represents the ferroorbital (antiferro-orbital) configuration, in which the same (different) t 2g orbitals are occupied in neighbouring sites. In each diagram, the allowed transitions are shown with the dotted arrows, and the corresponding excitation energy values are also written with the irreducible representations of the final states in parentheses.   4(b) and (c), respectively. For the t 3 2g system, we do not have to consider the AFO configuration, since all three t 2g orbitals are already occupied. Table 5 summarizes the relations between the intersite d-d transitions and the spin/orbital correlation for the t n 2g systems (n = 1 − 5). For n = 1, 2 and 3, the possible excitation energies are the same as those shown in table 4, but displayed in terms of the spin and orbital configurations. (As mentioned earlier, the results for n = 4 and 5 should be the same as those for n = 2 and 1, respectively, due to the electron-hole symmetry.) For the t 1 2g (or t 5 AFM/AFO configuration. Note that spectral weight of each d-d transition will critically depend on the overlap integral of the orbitals involved. Using the spectral-weight analysis, we can sometimes probe the spin and/or the orbital correlations between nearest-neighbour d ions. From an experimental optical spectrum, we could analyse its spectral features in terms of the allowed optical transition. For some spectral regions, most of such spectral features might come mainly from the intersite d-d transitions addressed in this section. In such a case, we might be able to analyse the spectrum in terms of the d-d transitions, displayed in figure 4. From the spectral weights estimated from such analysis, we can obtain information on spin/orbital correlations. For example, if we observe the d-d transition for the t 2 2g (or t 4 2g ) systems at U − 3J H and obtained the contribution from such transitions in the total spectral weights, we might be able to get information on the ratio of FM to AFM correlations [15].

Comparisons with earlier experimental data
4.1.1. Early 3d TM oxides LaMO 3 with a perovskite structure. Arima and Tokura have done pioneering works by investigating the optical conductivity spectra σ(ω) of 3d TM oxides, and showed that some of the early compounds have the Mott insulator character; namely, their optical gaps should be determined by the d-d transitions [12]. Figure 5 shows σ(ω) of some early 3d TM oxides LaMO 3 with a perovskite structure, where M = Sc, Ti, V and Cr, obtained by Arima and Tokura [12]. Note that each TM ion has zero to three t 2g electrons in order. All of the spectra show strong absorption edges around 4-6 eV, whose peaks correspond to the charge-transfer excitations. Accordingly, the spectral weight below such absorption edges should be attributed to the intersite d-d transitions. For LaScO 3 , the p-d transition appears from around 6 eV, but there is almost no spectral weight below 6 eV. In this d 0 band insulator compound, there is no d electron in the valence band, so the intersite d-d transition is not allowed.
Based on the d-d transitions discussed in section 3.3.2, we could provide a possible answer to the long-standing problems of the low-frequency spectral features in some early 3d TM oxide. Firstly, the broad and multi-peak structures of the d-d transitions could be understood in terms of the multiple intersite d-d transitions due to the orbital multiplicity effects. For LaTiO 3 , having the t 1 2g configuration of the Ti 3+ ion, the d-d transitions should appear at U − 3J H , U − J H , and U + 2J H and for LaVO 3 , having the t 2 2g configuration of the V 3+ ion, they should appear at U − 3J H , U and U + 2J H . Accordingly, their d-d transitions should spread over 5J H , which could be around 3-4 eV [28,29]. Note that these broad spectral features cannot be explained in terms of the single-band Hubbard model. Secondly, the low values of the d-d transition energies for LaTiO 3 and LaVO 3 can be explained from the fact that the lowest excitation energy of the d-d transitions is not U but U − 3J H for the t 1 2g and t 2 2g configurations. Since the U and J H values of these compounds could be assumed to be around 4.5 and 0.7 eV, respectively [28,29], U − 3J H should be around 2.4 eV. This value is quite consistent with the optically observed peak positions for both compounds [12]. Thirdly, the lack of a d-d transition below the 4.0 eV region for LaCrO 3 can be explained from the fact that the lowest excitation energy for the half-filled t 3 2g orbital configuration is U + 2J H . Note that our analysis, in section 3.3.2, showed that the d-d transition for the t 3 2g -compound should appear only at U + 2J H . Since the U and J H values of LaCrO 3 are similar to those of LaTiO 3 and LaVO 3 [28,29], the U + 2J H peak should be located above 6 eV. Consequently, the disappearance of the d-d transition around 2 eV can be understood in terms of the intrinsically forbidden U − 3J H peak due to the half-filled t 3 2g orbital configuration. Considering the orbital multiplicity effects, we could explain the intriguing low-frequency spectral features of some early 3d TM oxides successfully.

Probing spin and orbital correlations using intersite d-d transitions.
When we consider the orbital multiplicity effect to gain understanding of the optical excitations, one of the most important merits is that it can provide information on the spin and/or orbital correlations between the neighbouring TM ions in terms of the intersite d-d transitions. Very recently, there have been several studies addressing the spin/orbital correlations using the optical investigations for some t 2 2g systems and t 4 2g systems, such as LaVO 3 [32,33], YVO 3 [34], RE 2 Mo 2 O 7 [15] and Ca 2−x Sr x RuO 4 [14]. In these cases, the U − 3J H transition is directly coupled to the FM spin correlation, so its spectral-weight enhancements should be attributed to the increases of the FM correlation as the systems get into the FM state. 4 Similar to our approach, Khaliullin et al considered the orbital degeneracy to explain characteristic spectral features of LaVO 3 , i.e., the anisotropy in the optical absorption and its strong temperature dependence near the magnetic transitions [33]. In terms of the spin and orbital superexchange interactions, they demonstrated a theoretical evidence that the spin/orbital correlations play important roles in determining the optical spectral weights in the Mott insulators. 5 By the way, our systematic approaches on the intersite d-d transitions, demonstrated in section 3.3.2, could extend such spin and/or orbital 4 In our earlier papers on (Ca,Sr) 2 RuO 4 [14] and RE 2 Mo 2 O 7 [15], we considered diagrams of the t 4 2g t 4 2g → t 3 2g t 5 2g and t 2 2g t 2 2g → t 1 2g t 3 2g transitions, respectively, which are similar to figure 4(b). In those papers, we considered a simple argument, so we assigned only one excitation energy for the FO/AFM configuration. However, in this paper, we presented a more refined and correct picture, which takes account of the orbital degeneracy effects fully, and assigned two excitation energies to the configuration. Let us look into the conclusions of [14,15] based on the more refined picture. If the spectral weight of each peak is linearly proportional to configurations of the initial state, then the relative ratio between contributions from the AFO and the FO configurations to the U (or U + 2J H ) peak should be 2:1 (or 1:1). So, the spectral weight decrease of the U peak for the insulating (Ca,Sr) 2 RuO 4 , accompanying the spectral weight increase of the U + 2J H peak, could be related to the reduction of the AFO configuration. And, for the Mo-oxides, since the lowest U − 3J H excitation is solely attributed to the FM configuration in our modified picture also, its spectral weight enhancement could be attributed to the increase of the FM correlation. It should be noted that, although we adopt the more refined picture for the d-d transitions, the conclusions made in our previous papers should remain valid. 5 In [33] they emphasized the quantum effects beyond the mean-field theory on the optical spectral weights. They argued that even above the magnetic transition temperatures the optical spectral weights should show a finite temperature dependence due to the spin-orbital fluctuation effects. correlation studies to other t n 2g systems (n = 1 and 5). As far as we know, there have been few efforts to obtain spin/orbital correlations in the t 1 2g and t 5 2g systems using optical studies. Recently, Lukenheimer et al reported the temperature-dependent spectral-weight changes of LaTiO 3 , a t 1 2g system [35]. They showed that the spectral weight below 2 eV, which should correspond to the U − 3J H peak, increases slightly as temperature decreases. The ground state of this titanate is known to have a G-type AFM and orbital liquid state, and the magnetic transition occurs around 140 K [3]. Table 4 shows that the U − 3J H peak of the t 1 2g system should come from FM/AFO andAFM/AFO configurations. (Note that this is different from the t 2 2g system, where the U − 3J H peak should come from FM/AFO only.) From this table, it is clear that most of the lowest peak contributions should come from the AFO correlation. In addition, the weak temperature dependence of the peak should be due to the competition between contributions of FM/AFO and AFM/AFO configurations to the U − 3J H peak. A further spectroscopic investigation is highly desirable to obtain a more detailed understanding of the spin/orbital correlations for this compound.

Remaining problems in intersite d-d transitions.
Our new approach based on the consideration of orbital degeneracy provides a general framework to analyse the spectral features of numerous perovskite oxides. Most importantly, the approach allows us an important merit, i.e. to estimate the contribution of the spin and orbital degree of freedom in optical spectra. However, it should be noted that there remain some controversies and limitations in this approach.
Firstly, it could be controversial in determining the energy scale of U and J H . For example, Miyasaka et al investigated strongly temperature-dependent optical features in LaVO 3 andYVO 3 , and explained them in terms of the Mott-Hubbard gap transition due to spin and orbital ordering [32]. Especially, they observed that multiple peaks appear between 1.8 and 3.0 eV, and the splittings between the peaks are around 0.5 eV. They tried to explain this fine spectral splitting features in terms of different intersite d-d transitions, similar to this work but with a much more limited version [32]. However, such an explanation of the fine splitting may not be correct. Since the value of J H is known to be about 0.6 eV for both vanadates [28,29], the energy differences of the intersite d-d transition, based on our analysis, should be 3J H (i.e. 1.8 eV) between the U − 3J H and the U peaks. Moreover, while LaVO 3 and YVO 3 have different spin and orbital ground states, their temperature dependences closely resemble one another. Since the spectral weights of the d-d transition should be strongly affected by the spin and orbital correlations, the observed similar temperature dependences for different spin and orbital ground states cannot be explained in our picture by considering the orbital multiplicity effects. The spectral splitting should come from different physical origins, whose energy scale should be smaller than the exchange energy of the electrons. The origin and the temperature dependences of the spectral features of LaVO 3 and YVO 3 around 2.5 eV remain to be investigated further. 6 Secondly, our approach could become less realistic under the degeneracy lifting due to the local lattice distortion. Up to this point, we have considered the case when d orbitals have exactly the same energy. When there are physical mechanisms which will result in a higher energy splitting than the characteristic energy scale used in this study, i.e. U and J H , our analysis based on the orbital degeneracy cannot be applied. However, when the new interaction results in a smaller energy splitting, we should treat the new term as the perturbing Hamiltonian. For example, in the case of a system with a structural distortion, such as the Jahn-Teller distortion and the Institute of Physics ⌽ DEUTSCHE PHYSIKALISCHE GESELLSCHAFT Coulomb interaction and the charge transfer energy [16]. According to the scheme, while the optical conductivity spectra of the charge-transfer type insulator should have the p-d transition as their lowest optical excitation, that is, eff < U eff , σ(ω) of the Mott-Hubbard type insulator should have the d-d transition as its lowest excitation energy, that is, eff < U eff . Here, the effective charge-transfer energy eff and the effective Coulomb repulsion energy U eff correspond to the lowest excitation energies of the p-d transitions and the d-d transitions, respectively.
Previously, using photoemission studies, Fujimori et al [25] provided explicit arguments about the orbital multiplicity effects on the electronic structure considering the multiplet corrections to each energy state of the system. However, there have been few optical investigations which take account of such multiplicity effects. In the early works on the optical spectra of the 3d TM oxides, Arima and Torura et al [12] treated the optical gap without considering the orbital mulitiplicity effects explicitly and used the Zaanen-Sawatzky-Allen scheme. Here, we briefly address the estimated optical excitation energies considering the orbital multiplicity effects. The orbitally degenerate Hubbard model predicts that, except for the half-filled cases, the U eff values should be smaller by 3J 3 or 3J 4 than the U value. For the half-filled cases, such as t 3 2g , t 3 2g e 2 g and t 6 2g e 2 g systems, U eff should be larger by 2J 3 , ∼ 4J 3 and J 4 , respectively, than the U value. These differences should be explicitly included to use the Zaanen-Sawatzky-Allen scheme from the optically determined values of the related physical quantities.

Summary
We attempted to understand the optical excitations of the perovskite TM oxides with t 2g electrons by taking orbital multiplicity effects into consideration. We evaluated the excitation energies of the charge transfer p-d and the intersite d-d optical excitations, and compared them with the previously reported experimental results. We could provide a possible answer to a few intriguing long-standing problems in the optical conductivity spectra of some early 3d TM oxides. We applied these results to understand the optical spectral features of some TM oxides in terms of spin and/or orbital correlations. We also discussed some energy splitting features observed in the intersite d-d transition regions, which are not possible to explain within this model.