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New theory of the gamma-alpha phase transition in Ce: quadrupolar ordering

We present a theoretical model of the"isostructural"\gamma-\alpha phase transition in Ce which is based on quadrupolar interactions due to coupled charge density fluctuations of 4f electrons and of conduction electrons. Conduction electrons are treated in tight-binding approximation. The \gamma-\alpha transition is described as an orientational ordering of quadrupolar electronic densities in a Pa3 structure. The quadrupolar order of the conduction electron densities is complementary to the quadrupolar order of 4f electron densities. The inclusion of conduction electrons leads to an increase of the lattice contraction at the \gamma-\alpha transition in comparison to the sole effect of 4f electrons. We calculate the Bragg scattering law and suggest synchrotron radiation experiments in order to check the Pa3 structure. The theory is capable of accounting for transitions to phases of non-cubic symmetry, but it is not sufficient to describe the magnetic phenomena which we ascribe to the Kondo mechanism. We also present a microscopic derivation of multipolar interactions and discuss the crystal field of \gamma-Ce.


I. INTRODUCTION
The understanding of the nature of the α phase and the apparently isostructural transition between the cubic γ-and α-phases in cerium is a long-standing problem [1,2]. Numerous experimental data present an outstanding challenge for the explanation by theoretical models [3][4][5][6][7][8][9][10][11]. The most important question to be answered by theory is to find the driving force of the "isostructural" γ − α transition and to explain also the existence of the other non-cubic phases of Ce.
The theory should also describe the change of magnetic properties of Ce at the γ − α transition [12,1] which are reminiscent of an insulator-metal transition of 4f electrons [3]. Recently the Mott transition has been reconsidered by several authors using electronic band structure calculations with thermodynamic extensions [4][5][6] (see for a review Ref. [7]). There, the α phase is described as a regular band state formed by 6s, 5d and 4f electronic states while in γ-Ce different degrees of localization of 4f states are suggested and investigated. Two face centered cubic (fcc) phases of cerium are attributed to two local minima of free energy which develop for the same crystal structure (F m3m).
Assuming the localized nature of the 4f electrons throughout the γ − α phase transition, one can understand the magnetic properties on the basis of two principles. These are the singlet ground state of one 4f electron and the energy gap (characterized by the Kondo temperature T K ) separating the ground state from a manifold of excited magnetic states. So far these properties have been treated [10,11] on the basis of the Anderson impurity Hamiltonian [13] which implies an antiferromagnetic Kondo spin interaction. In order to describe the volume contraction [1] at the γ − α transition, the theoret-ical models [8,9,7] exploit the volume dependence of the Kondo temperature. Such a "volume collapse" leads to a phase instability without symmetry change and is interpreted as an isostructural transition. However within the Kondo theory scenario, the existence of the other, noncubic phases of Ce, remains unexplained. In the last few years the validity of this approach has been questioned by photoemission spectroscopy experiments [14][15][16][17][18] where the predicted temperature dependence of the intensity of the Kondo resonance at the Fermi level has not been observed. Recently it has appeared that the Kondo volume collapse model can not be applied to YbInCu 4 [19] which exhibits a 0.5% volume expansion during another isostructural phase transition [20] though Yb is the f-hole analogue of Ce. In addition, taking into account the thermal expansion of YbInCu 4 above the phase transition temperature T = 42 K one concludes that the Kondo temperature is not a unique function of cell volume [19]. An alternative theoretical model of the γ − α transition has been recently proposed by the present authors [21]. The theory is capable of accounting for transitions to phases of non-cubic symmetry. The quantum mechanical electric quadrupole interaction between 4f electrons on the fcc lattice is proposed as the driving mechanism of a phase transition. The γ-phase is characterized by the absence of spatial orientational order of the quadrupolar densities, the space group is F m3m. In the α-phase the quadrupolar densities order in a P a3 structure. Notice that this change from F m3m to P a3 conserves the fcc structure of the atomic center of mass points and is solely due to orientational order of the quadrupoles. This phase transition is accompanied by a contraction of the fcc lattice, however the theoretical estimation of these effects in [21] (in the following we will denote this reference by I), is about an order of magnitude smaller than the experimental result [1]. In addition the treatment of I does not indicate the existence of a critical end point of the phase separation line (γ − α) in the P − T (pressuretemperature) phase diagram.
In the present paper, we will extend the theoretical model of I by taking into account the polarization of (6s5d) 3 conduction band electrons. The polarization can be considered as a screening process of the quadrupolar density orientations of the 4f electrons and results in turn in a complementary ordered P a3 structure build up from conduction electron quadrupolar densities. The conduction electrons will be described within the formalism of tight binding approximation.
The paper comprises the following sections. We start (Sect. 2) with reconsidering 4f electrons and extending the treatment of I by taking into account the radial dependence of the 4f electron density. Next in section 3 we describe the conduction electrons in second quantization with basis functions in tight-binding approximation. We derive the multipolar interactions among conduction electrons as well as interactions with 4f electrons. In considering conduction electrons we have to distinguish between on-site and inter-site interactions. Section 4 is devoted to a study of the crystal field which acts on the individual 4f electron. It is found that the refinement by the radial density dependence of 4f electrons does not improve but rather spoils the agreement between experiment and theory. On the other hand the inclusion of conduction electrons improves in turn the situation. In Sect. 5 we study the quadrupolar ordering of the coupled system of 4f electrons and conduction electrons. We conclude that a state of lower free energy (in comparison with the disordered state) can be achieved by a complementary ordering of 4f electron and conduction electron quadrupolar densities. On a same atomic lattice site, high charge density regions of the 4f electron correspond to low charge density of the conduction electrons and vice versa. Finally we present numerical estimates that the inclusion of the conduction electron quadrupolar order improves the magnitude of the lattice contraction at the F m3m → P a3 transition. In order to propose an unambiguous experimental proof of the present theoretical model, we calculate the Bragg scattering law and suggest synchrotron radiation experiments for the P a3 structure (These calculations will be presented in the full original version to be published in Eur. Phys. J. B, 2000). In Sect. 6 (Discussion) we recall the salient features of the present theory and situate it with respect to the conventional approaches that are based on the Kondo theory concepts. While we do not adopt the Kondo volume collapse models, we discuss the relevance of the Friedel-Anderson hybridization mechanism for the explanation of the magnetic anomalies in Ce and suggest a link with our theory of the electronic charge degrees of freedom driven structural phase transition.

II. RADIAL DEPENDENCE OF 4F ELECTRONS
In our previous paper [21] the quadrupolar coupling between 4f electrons has been calculated by assuming that the electron on each lattice site n is localized on a sphere with a fixed radius r f = 1.378 a.u. Here we want to extend the previous calculation by taking into account the radial dependence of the 4f electron wave functions. Such an extension is necessary if we want to study the interaction with the conduction electrons (see next sections). Furthermore it is useful in assessing the validity of our previous calculations.
We consider a face centered cubic crystal of N Ce atoms. Each atomic core possesses one 4f electron. In the γ-phase the 4f electron densities are orientationally disordered. The space group of the crystal is F m3m. The Coulomb interaction between two 4f electrons (charge |e|=1) at positions R( n) and R ′ ( n ′ ) near the lattice sites n and n ′ reads The position vector R( n) is given by Here X( n) is the lattice vector which specifies the centers of the atoms on a rigid fcc lattice, while r( n) is the radius vector of the 4f electron; in spherical coordinates r( n) = (r( n), Ω( n)), where Ω = (Θ, φ). We perform a multipole expansion of V by using site symmetry adapted functions (SAF's) [25] which transform as irreducible representations of the cubic site point group O h : The SAF's S Λ (n),n ≡ Ω( n) are linear combinations of spherical harmonics Y m l [25]. The index Λ stands for (l, τ ), with τ = (Γ, µ, k). Here l accounts for the angular dependence of the multipolar expansion, Γ denotes an irreducible representation (in the present case of the group O h ), µ labels the representations that occur more than once and k denotes the rows of a given representation. Expansion (2.3b) still depends on the instantaneous radii r( n) and r ′ ( n ′ ). In Ref. and 6s electrons of an isolated Ce atom. Valence density is superposition of (6s) 2 and 5d; rc is the close contact radius of γ-Ce. function, with s = ± for the spin projections s z = ±1/2 on the z-axis respectively. (The consideration of spin orbitals is necessary since we calculate the eigenvalues ε i of the cubic crystal field in presence of spin-orbit coupling.) Since at present we take into account the radial dependence of the orbitals, we consider basis functions where we have assumed that the function R f (r) is the same for all i. The real radial function R f (r) is obtained from a DFT (density functional theory) calculation of an atom of Ce within LDA (local density approximation) for J = 5/2 states. In Fig. 1 we plot the radial density for the outer electrons. For a non-relativistic hydrogen-like atom R f (r) would correspond to the Laguerre function R n=4 l=3 (r). The matrix elements of the interaction (2.3a) are obtained as accounts for the average radial dependence and where We use the superscript F in order to indicate that we have transitions between two 4f states, i.e. F ≡ (f, f ). By summing V ( R( n), R ′ ( n ′ )) over all pairs of lattice sites n, n ′ , the total Coulomb interaction operator is then obtained as Introducing Fourier transforms where q is the wave vector, we get The multipolar interaction (2.8) or equivalently (2.11) can be separated into two parts. We recall that Λ ≡ (l, τ ).
Firstly we consider the case where l = 0 and l ′ = 0. In I it has been shown that some of the coefficients c F Λ (ij) ≡ c τ l (ij) for l = 2 and τ = (T 2g , µ = 1, k = 1 − 3) are different from zero. (We recall that i and j refer to quantum states of the 4f electron). Therefrom we have inferred the existence of quadrupolar (l = 2) density fluctuations caused by transitions among 4f electron states. In labeling the quadrupolar T 2g functions, we recall that the functions S k 2 ≡ S (l=2,T2g ,k) are proportional to the Cartesian components yz, zx and xy for k = 1, 2 and 3 respectively. In the basis of real spherical harmonics Y 0 l , Y m,c l and Y m,s l of Ref. [25] (see also Eqs. (2.1a)-(2.1c) of I), these functions correspond to Y 1,s 2 , Y 1,c 2 and Y 2,s 2 . Writing only the index k for Λ = (l = 2, k), we denote the quadrupolar density operator by The interaction between quadrupolar 4f electron densities becomes The explicit form of v F F kk ′ ( q) is discussed in Appendix A.
Secondly we have the case where l = 0 and l ′ = 0 or vice versa. This means that we consider a multipole l on a given lattice site while the surrounding multipoles on neighboring fcc lattice sites are taken in spherical approximation. This interaction contributes to the crystal field. The crystal field has unit cubic symmetry, the lowest nonzero value of l is 4 and τ = (A 1g , 1), where A 1g is the unit representation of the cubic site group O h . The crystal field contribution from 4f electrons at site n is then given by (2.14) Here we have restricted ourselves to the 12 nearest neighbors on the fcc lattice, and l = 4, 6, ... . The leading contributions to U f f are then given by In the following of this section we will discuss the physical consequences of the term U f f QQ , expression (2.13), also called orientational pair quadrupolar interaction. We will give a discussion of the crystal field in Sect. 4.
Previously (I) it has been found that the quadrupolequadrupole interaction matrix v F F ( q) becomes diagonal and has a twofold degenerate negative eigenvalue called λ X + 5 at the X point of Brillouin zone (BZ) of the fcc lattice (see also Appendix A). This attractive interaction induces an orientational ordering of the quadrupolar densities in a P a3 structure. A condensation scheme for the phase transition F m3m → P a3 reads: Here the bar stands for a thermal expectation value, whileρ F is the order parameter amplitude. The above condensation scheme corresponds to one of eight possible domains of P a3. The wave vectors q X x , q X y and q X z stand for (2π/a)(1, 0, 0), (2π/a)(0, 1, 0) and (2π/a)(0, 0, 1) respectively, where a is the cubic lattice constant. For each arm of the star of * q X = { q X x , q X y , q X z } there are two basis functionsρ F k . Hence the functions ρ F k ( q X ) of the condensation scheme (2.16a,b) form a basis of the six dimensional irreducible representation X + 5 of the space group F m3m. In real space the ordering implies four sublattices of simple cubic structure as shown in figure 3 of Ref. I.
DFT calculation with LDA. Although the atomic 4f electron density is small beyond the close contact radius r c = a/(2 √ 2) = 3.448 a.u., the opposite holds for the inter-site interaction potential v ΛΛ ′ ( n, n ′ ; r, r ′ ) and its first derivative (see below), which increase substantially when r → r c and r ′ → r c (see Fig. 2). We have investigated several models for the radial integral (2.6). In model 1 we consider the radial integration in the range 0 < r < r c , that is without any overlap of the atomic 4f electronic densities of neighboring atoms, and we obtain λ X + 5 = −2121 K, next (model 2) we have allowed for an overlap between neighboring sites and extended the integration over the range 0 ≤ r ≤ ∞. We then obtain λ X + 5 = −2478 K. Finally (model 3) we assume again the integration range 0 ≤ r ≤ r c but renormalize the 4f electronic density to unity. The result for λ X + 5 is -2682 K.
Comparing these values with λ X + 5 = −3491 K obtained for the calculation with the characteristic radius r f , we conclude that a refinement of the theory in smearing out the radial extension of the 4f electron distribution does not increase the strength of the quadrupole-quadrupole interaction and consequently does not increase the transition temperature.
So far we have considered multipolar interactions on a rigid lattice. In order to account for the lattice contraction at the γ − α phase transition, we have to include lattice displacements. In I we have shown that the intersite quadrupolar interaction is modified by lattice displacements u ν ( n). The correction to the potential reads Here S k 2 are SAF's with l = 2, T 2g , k = 1 − 3. The coupling coefficients v ′ ν kk ′ are given by the derivative of the quadrupole-quadrupole interaction with respect to lattice displacements: Previously (I) this expression was considered for 4f electrons on a shell with the (fixed) characteristic radius we now consider matrix elements with basis functions (2.4) and obtain The correction to the quadrupolar interaction between 4f electrons then becomes with summation over repeated indices k, k ′ , ν. We introduce the Fourier expansion where m is the Ce mass. Using definition (2.10a), we rewrite expression (2.22) in Fourier space. In the long wavelength limit q → 0 and taking p close to the star of p X , we obtain Here the sum ′ refers to ν = x, y for k=3, to ν = z, x for k=2 and to ν = z, y for k=1. The coupling matrix is obtained as , where ν = x or y and X( n) − X( n ′ ) = (a/2)(1, 1, 0) on the fcc lattice (for more details, see Ref. I). The other elements of v ′ ν F F kk ( q, p) follow by symmetry considerations and permutation of indices.
We consider expression (2.24) in the P a3 ordered phase by using the condensation scheme (2.16a). The lattice displacements are taken in the long wavelength limit where they are related to the homogeneous strains. Symmetry implies that only longitudinal strains occur: which corresponds to a coupling of ordered quadrupoles (quadratic) to longitudinal strains. Notice that the sign of Λ F F is negative, this is a consequence of the repulsive nature of quadrupole-quadrupole interaction. The strains give rise to an elastic energy of the cubic lattice: Here c 0 11 and c 0 12 are the bare elastic constants. The interplay of quadrupolar order and lattice displacements follows from the interaction Hamiltonian Minimizing U [ρ F , ǫ νν ] with respect to the strains for a given P a3 ordered structure, we obtain ǫ νν , while the change of the lattice constant is given by △a = ǫ xx a. Here κ L = (c 0 11 + 2c 0 12 ) −1 is the linear compressibility. Hence the present theory leads unambiguously to a lattice contraction. In order to provide a numerical estimate of the lattice contraction, one has to calculate Λ F F , to estimate κ L from experimental results and to calculate by the methods of statistical mechanics the discontinuity of the order parameter at the γ − α phase transition. In I (fixed radius r f ) we did obtain Λ F F = −445 K/a.u. (−841 K/Å) [22]. Now, for the models 1), 2), and 3) of the spatial radial integrals we obtain Λ F F =−498, −581 and −659 K/Å respectively. We conclude that the present refinement in calculating Λ F F is not helpful in view of obtaining a larger value of Λ F F and hence of the lattice contraction △a. Since our theoretical value of △a is more than one order of magnitude too small to account for the ∼ 15% volume contraction of cerium, we conclude that a significant process has so far been omitted in our treatment. In the following section we will study the effect of conduction electrons on quadrupolar interactions.

III. MULTIPOLAR INTERACTIONS
Here we will investigate about the existence of multipolar interactions between the localized 4f electrons and the (6s5d) 3 conduction electrons U f c as well as multipolar interactions among conduction electrons U cc .
As a consequence of Bloch's theorem the conduction electron states are classified according to the irreducible representations of the translational symmetry group of the crystal. Nevertheless, in the proximity of the nuclei the corresponding wave functions can be expanded in terms of spherical harmonics. This fact reflects the importance of Coulomb singularities associated with the nuclei. In the following we will focus on the interactions inside the "muffin-tin" or touching spheres centered on the nuclei and for the description of itinerant states adopt the tight-binding approximation. In absence of a static magnetic field, conduction electronic states with spin projections s z = ±1/2 are degenerate. Hence we will omit the spin dependence of the wave function. The wave function of a conduction electron with wave vector k and band index α is then written as a linear combination of local atomic wave functions where the position vector R is given by Eq. (2.2). The atomic wave functions are given by φ lm ( r) = R l (r)Y m l (Ω). The expansion coefficients γ lm ( k, α) and corresponding eigenvalues E( k, α) are obtained by solving the secular equation (see e.g. [48]) Here H( k) and S( k) are matrices of single particle Hamiltonian and overlap, respectively. Here and in the following λ = (l, m), δ λλ ′ = δ ll ′ δ mm ′ . The eigenvalues E( k, α) refer to the energy band spectrum of the conduction electrons. Notice that the wave function ψ k,α ( R) satisfies the Bloch condition. In the following we shall use an idealized basis set of orthogonal Wannier functions φ λ ( r( n)) = R|λ n without overlap, The coefficients γ λ satisfy the relation The functions ψ k,α ( R) are then normalized: The corresponding electronic Hamiltonian in second quantization is where a † kα and a kα are creation and annihilation operators for one electron in state ( k, α).
In case of cerium, we will restrict ourselves to conduction electrons of d type (l = 2, m=-2,...,+2, with principal quantum number n=5) and of s type (l=0, m=0, with n=6). Fig. 1 shows the radial density of the valence electrons of a Ce atom, it should also be a guide of the relative spatial extension of the localized 4f and the conduction electrons in the crystal.
Although the derivation of multipolar interactions presented later in this Section is general in the following we imply the high temperature γ phase of cerium. There the total electronic density associated with the conduction electrons has the symmetry of the space group F m3m [21]. At each lattice site the band electronic density has on average the unit symmetry (A 1g ) of the cubic point group O h . It is important to realize that the onsite quadrupolar charge density fluctuations of T 2g and E g symmetry do not interact with the totally symmetric charge distribution (A 1g ) representing the ground state of cerium (see also Eq. (3.11)). On one hand this allows us to use the Landau concept of Fermi liquid and to consider the conduction electrons described by Eq. (3.1) as quasiparticles where the ground state on-site interactions have already been taken into account through the coefficients γ lm ( k, α). On the other hand we can focus only on the quadrupolar interactions where conduction electrons are involved and consider these interactions separately from the ground state.
Interactions involving conduction electrons will be described within the formalism of second quantization. We introduce field operators The operators Ψ and Ψ † satisfy the usual anticommutation relations for fermion fields. Although the derivation of multipolar expansions for these interactions is analogous, in particular as far as symmetry is concerned, to the procedure of Sect. 2, there is an essential difference. Since the conduction electrons are not localized, they give rise to quadrupolar pair interactions on a same site (intra) and to interactions between different sites (inter). The Coulomb interaction between localized 4f electrons at sites { n} and the conduction electrons is given by We observe that d R ′ → n ′ d r ′ , where the integral extends over the volume of the cell n ′ . In expression (3.8) we perform a multipole expansion of V similar to Eqs. (2.3a,b) and then calculate matrix elements In the following we will introduce a single index L for . We notice that the index l 1 (l 2 ) takes the values 0 and 2 corresponding to s and d electrons. We recall that c F Λ (ij), referring to the 4f electron transitions, is given by expression (2.7).
Since the conduction electrons are delocalized, we will have to distinguish between interactions where n = n ′ (inter-site), and where n = n ′ (on-site). In the first case v ΛΛ ′ ( n, n ′ ; r, r ′ ) is given by an expression of type (2.3b), in the second case we have which is independent of the site, as is also the case for Λ ′ ( n − n ′ ) still depends on the distance | X( n) − X( n ′ )|, as follows from the translational invariance of the lattice.
In addition to the multipole density of 4f electrons ρ F Λ ( q), Eq. (2.10a), we define the multipole density of conduction electrons The interaction for the inter-site contribution of U f c is then given by The on-site part of U f c is obtained as Here we have also used the orthogonality relation (3.11). In expressions (3.13a) and (3.14a), summation is understood over indices Λ, Λ ′ , L (l 1 , l 2 ). In a similar way we treat the multipolar interactions between conduction electrons. We now start from the expression in operator representation We have to consider matrix elements Again we distinguish inter-site and intra-site interactions. We obtain for the inter-site contribution In Eq. (3.18a) summation is understood over the indices α..., L(l 1 , l 2 )... , Λ... . Expression (3.18a) can be rewritten as where ρ L Λ ( q) is given by Eq. (3.12b) and where η is the normally ordered product operator such that all the a † are placed to the left and all a to the right in the product.
The on-site contribution is given by So far the present formalism is general as far as multipoles are concerned. In the following of this section we will study the interaction between quadrupoles (l = 2, l ′ = 2). In Sect. 4 we will study the crystal field (l = 4, l ′ = 0). We consider the three quadrupolar components of T 2g symmetry and write the index k for Λ = (l = 2, k), with k = 1 − 3. The quadrupolar density of conduction electrons becomes We observe that c k L ( p, α; p − q, β) comprises the factor c k (l 1 m 1 , l 2 m 2 ) which is defined (compare with expressions (3.10b) and (3.10c)) by The quadrupolar contributions from the interaction potentials (3.13a), (3.14a) between 4f electrons and conduction electrons then read Calculated coefficients c k (l1m1, l2m2), k = 1 − 3, l1 = l2 = 2; mi (i = 1, 2) stands for the indices (mi, c) or (mi, s) of real spherical harmonics [25] Those functions which are not quoted here give zero contributions.
The contribution of the quadrupolar pair interaction potential to expression (3.8) is then given by The quadrupolar interactions between conduction electrons follow from Eqs. (3.19) and (3.20a): The contribution to expression (3.15) is given by and zero otherwise. The transition elements between 5d states are quoted in Table 1.  Since the coefficients c k (l 1 m 1 , l 2 m 2 ) always occur in conjunction with interaction matrix elements v l1l2 , we can immediately select the relevant matrix elements. We observe that the indices l 1 and l 2 in the interaction matrix elements then directly refer to s or d electrons and we will adopt the notation v sd k dd . The indices f , s and d of these interaction matrix elements refer only to the radial dependence, they are irrelevant for symmetry considerations. The structure of the intersite quadrupole-quadrupole interaction matrices are investigated in Appendix A. From Eq. (3.11) it follows that the on-site elements C F k L k ′ are diagonal in k, k ′ and equal for k=1,2,3; the same holds for C L k L ′ k ′ . In the following we omit the indices k, k ′ and write C f f sd , C sd sd etc.. Numerical values are given in Table 2. By introducing quadrupolar pair interactions on a same site we generalize the concept of spherically symmetric onsite electron-electron repulsions which is a characteristic property of strongly correlated electron systems.
In order to treat the effect of conduction electrons on the lattice contraction, we have studied the coupling of quadrupole-quadrupole interactions with lattice displacements within the tight-binding approach. We start from expression U QQT , Eq. (2.17), and remind that u ν ( n) and u ν ( n ′ ) refer to lattice displacements at different sites. We first consider matrix elements of V ′ ν , expression (2.19), between 4f electron and conduction electron states. We proceed in analogy with Eqs. (3.8)-(3.13b) but retain only inter-site contributions. We then find (compare with Eq. (5.13) of I) in the long wavelength limit q → 0, and p close to the star of p X : (3.28) Here the sum ′ refers to ν = x, y for k=3; to ν = z, x for k = 2 and to ν = z, y for k = 1. The coupling matrix is obtained as with Λ F L given by v ′ ν F k L k ( n− n ′ ), where ν = x or y, k = 3 and X( n)− X( n ′ ) = (a/2)(1, 1, 0) on the fcc lattice. Here with v ′ ν kk ′ ( n − n ′ ; r, r ′ ) given by Eq. (2.18). The matrix elements of V ′ ν between conduction electron states are treated in analogy with Eqs. (3.15)-(3.19). Now only inter-site terms occur. In the limit q → 0 and p close to the star of p X we find (3.31) Here again we have the same relation between the indices ν and k as was the case for Eq. (3.28). The coupling matrix reads where we use expression (2.18). Numerical values of Λ L L ′ are given in Table 2.
Taking into account the contributions due to conduction electrons, we see that the interaction Hamiltonian (2.29) has to be replaced by where

35b)
Before studying the quadrupolar ordering and the accompanying lattice contraction (Sect. V) during the γ → α phase transition, we will next investigate the crystal field of γ-Ce in the presence of conduction electrons.

IV. CRYSTAL FIELD OF γ-CE
In the disordered γ phase there are only charge density fluctuations of quadrupolar type and the quadrupolar Hamiltonian (3.34) (or (2.29)) averages to zero. The first nontrivial orientational interaction then corresponds to a crystal field Hamiltonian. In Ref. I we have defined the crystal field of γ-Ce as the potential experienced by a single 4f electron at a site n when spherically symmetric contributions (l ′ =0) from nuclei, core electronic shells, conduction electrons, 4f electrons at the twelve neighboring sites n ′ on the fcc lattice and similar terms from the homogeneous electronic density in the interstitial regions are taken into account. In I we have shown that crystal field effects are reduced to a single particle term to which we have added a spin-orbit coupling for the 4f electron. We first generalize the results of I by taking into account the radial dependence of the 4f electron density. We start from Eq. (2.8) with Λ = (l = 4, A 1g ) ≡ Λ 1 and Λ ′ = (l = 0, A 1g ) ≡ 0. Since we are dealing with a Coulomb potential and a spherically symmetric charge distribution, the coupling function v Λ1 0 ( n, n ′ ; r, r ′ ), Eq. (2.3b), does not depend on r ′ and we observe that Eq. (2.6) can be written as is the same for all 12 neighbors. We obtain v F Λ1 • 0 < 0. The charge in units e of the 4f electron at a neighboring site is given by In our model Q f = 1. However, if we distinguish two regions in the crystal, the first inside muffin-tin (MT) spheres and the second in the interstices, then in the MT-region Q f < 1. The other contributions to the crystal field are dealt with similarly. Since the interaction parameter v F Λ1 • 0 remains the same it is only the charges Q i (i stands for core, nucleus, conduction electrons and interstitial contributions) which we shall take care of. Collecting the contributions from the various charges Q i together with Q f , we obtain for the crystal field at site n In I we have calculated the crystal field coefficient B f (we used the notation Λ for B f ) for a fixed radius r f . The obtained value B f =346 K corresponds to r f = 1.156 a.u. and not to r f = 1.378 a.u. as was quoted erroneously in I. With a radial distribution as specified in models 1), 2), and 3) of Sect. II, we obtain B f =970, 1403 and 1104 K, respectively. This implies that the strength of the crystalline electric field of the 4f electron should be 2.8-4 times larger than calculated in Ref. I and therefore by the same factor (2.8-4) larger than the crystal field measured experimentally [23,24]. However, expression (4.2a) is implicitly based on the approximation that the distribution of conduction electrons at site n is spherically symmetric and hence the on-site interaction from l = 4 multipoles is ignored. In the following we will show that the effective crystal field decreases if we take into account the average on-site cubic distribution (l = 4) of conduction electrons.
From Eqs. (3.14a,b) for the on-site interactions we observe that the 4f electronic density ρ F Λ1 couples with the conduction electron density of the same symmetry (Γ = A 1g , l = 4). In real space Eq. (3.14a) with Λ = Λ ′ = (l = 4, A 1g ) ≡ Λ 1 reads We obtain the crystal field potential due to ρ L Λ1 ( n), Eq. (3.12a), by replacing this quantity by its thermal average: Here n α k δ q,0 δ αβ = a † kα a k− q β is the Fermi distribution and l 1 = 2, l 2 = 2 takes into account the fact that only d electrons contribute to the matrix element c Λ1 l1l2 . We then write ρ c Λ1 = ρ d d Λ1 . The corresponding crystal field at site n reads Collecting the contributions (4.2a) and (4.5a) we obtain the effective single particle potential acting on the 4f electron at site n: We recall that B f > 0. The sign of B f c depends on ρ c Λ1 . We next investigate other interactions that affect ρ c Λ1 . We consider the crystal field acting on ρ c Λ1 ( n). By studying the inter-site interactions we find that the matrix elements v ΛΛ ′ ( n, n ′ ; r, r ′ ), Eq. (2.3b), are negligibly small for l = l ′ = 4. On the other hand terms with l = 4, l ′ = 0 are significant and we retain the interactions which involve v dd where v dd Λ1 • 0 < 0 (compare with Eq. (4.1b)). Here again v dd Λ1 • 0 is independent of r ′ . Proceeding in analogy with the calculation of V f CF , we calculate the field due to the surrounding 4f electrons, core electrons, nuclei, conduction electrons and interstitial contributions at site n, thereby obtaining with B c = 6093 K > 0. The positive sign of the field B c implies that ρ c Λ1 , calculated with the crystal field V c CF ( n), is negative. Hence B f c , Eq. (4.5b), is negative, which leads to a reduction of the effective crystal field of a 4f electron in Eq. (4.6).
The previous considerations indicate that the inclusion of conduction electrons leads to a reduction of the crystal field experienced by a 4f electron. However, a more rigorous approach should start with the crystal field Hamiltonian of γ-Ce, Here U c 0 is the "bare" electronic term (3.6), which includes the kinetic energy and the spherically symmetric part of the electronic potential of conduction electrons, U so = n V so ( n) stands for the spin-orbit couplings of localized 4f electrons (see Appendix A of I for details). U CF is the crystal field comprising 4f electrons and conduction electrons: is the on-site l = 4, l ′ = 4 interaction between conduction electrons. In a mean-field approximation U CF leads to self-consistent crystal field potentialsṼ c CF ( n) andṼ f CF ( n) for conduction and 4f electrons, respectively. A quantitative calculation of these effects is beyond the scope of the present work.

V. QUADRUPOLAR ORDERING
In Section III we have found that the system of localized 4f electrons and of conduction electrons are coupled by means of the intra-site and inter-site quadrupolar potentials (3.23b) and (3.23a). Such a bilinear coupling suggests that an ordering of 4f electron density quadrupoles should imply an ordering of conduction electron quadrupoles and vice versa. In Appendix A we have investigated the wave vector dependence of the quadrupole-quadrupole interaction matrices and the condition for quadrupolar order. The present section is divided into two parts. In part A we study the interplay of quadrupolar ordering of 4f electrons and conduction electrons on a rigid cubic lattice; in part B, where we consider a deformable lattice, we show that quadrupolar ordering implies a lattice contraction.
A. We are investigating the possibility of a condensation of quadrupolar densities in a P a3 structure. At the X-point of the BZ, the matrices of inter-site in- ) become diagonal in k, k ′ and have two degenerate negative eigenvalues. We then are led in analogy with the condensation scheme (2.16a,b) for 4f electrons to suggest the condensation scheme F m3m → P a3 for the quadrupole densities of conduction electrons: where L = (sd), (ds), (dd). We disentangle the various contributions to U QQ , Eq. (3.35a), in the ordered α phase where we assume a simultaneous condensation of quadrupolar densities of 4f electrons and conduction electrons. Taking into account the condensation scheme (2.16a,b), we obtain from Eq. (2.13) where λ F F stands for λ X + 5 , the twofold degenerate negative eigenvalue of v F F ( q X ) (see Appendix A). Similarly, using in addition the condensation scheme (5.1a,b) We observe that all coefficients C L L ′ are positive while λ L L ′ are negative (see Table 2). Obviously the intersite interaction (5.4b) favors quadrupolar order while the intra-site coupling (5.4a) disfavors quadrupolar order of conduction electrons. The leading quadrupolar interaction in the ordered α phase, is a quadratic form inρ F ,ρ sd andρ dd , in particular the terms (5.3a) and (5.3b) represent a bilinear coupling between the localized 4f quadrupolar densityρ F and the conduction electron quadrupolar densitiesρ dd andρ sd . Since in tight-binding the (6s5d) 3 electrons are hybridized, we introduce the total conduction electron densityρ Defining average interaction coefficients we approximate U QQ by an effective interaction and A cc = C cc + λ cc . (5.9b) We observe that A f c > 0, A cc > 0. For a fixed value of ρ F , we minimize U QQ with respect toρ c and obtain We see that quadrupolar orderρ F of 4f electrons produces (as a type of mirror image) a quadrupolar order of conduction electronsρ c of opposite sign. Herewith we associate the pictorial representation of Fig. 3 and 4.
Regions of an excess density (+) of the 4f electron distribution overlap with a depletion (−) in density of the conduction electrons and vice versa (Fig. 4). Substituting the right hand side of Eq. (5.10) into (5.8) we get where we use the fact that λ F F < 0. The bilinear coupling between quadrupolar densities of 4f electrons and conduction electrons leads to an increase of the attractive interaction between 4f quadrupolar densities. The results of the last two sections can be summarized by suggesting an effective Hamiltonian in the disordered γ phase, where U 0 is the crystal field Hamiltonian, (4.9), and Here where ρ f k ( q), k = 1 − 3, is the quadrupolar density operator (2.10a) of 4f electrons while ρ c k ( q) is the quadrupolar density of conduction electrons (3.21) with ρ c k = ρ dd k + 2ρ sd k . The first term, Eq. (5.14a), represents an effective attraction at the X point of the Brillouin zone. The other terms in (5.14a) (c.p.) follow by cyclic permutation of indices, i.e. when q X x → q X y , k changes 2 → 3 and 3 → 1, etc. The direct lattice is face centered cubic. In that case the condensation schemes (2.16a,b) and (5.1a,b) correspond to a structural phase transition where the space group changes from F m3m to P a3.
Starting from the interaction (5.8) we construct a Landau free energy in the condensed phase. We takeρ 1 ≡ρ F andρ 2 ≡ρ c as components of a two dimensional vector ρ = (ρ 1 ,ρ 2 ) and define the matrix Inspired from the theory of orientational order in molecular crystals we write Here F 0 is the free energy in the disordered phase: where U 0 is the crystal field, Eq. (4.9). The quantity is the inverse susceptibility matrix, T is the temperature and Here ... 0 are single particle thermal expectation values that have to be calculated by means of U 0 , Eq. (4.9). Cubic symmetry implies (see Appendix B for details) that the three quadrupolar components are equal and for expectation values of conduction electrons we obtain: Here δ L L ′ = δ l1 l ′ 1 δ l2 l ′ 2 . The contributions F (3) and F (4) stand for the third and fourth order terms inρ. Symmetry of the order parameter components (see condensation schemes (2.16a,b) and (5.1a,b)) implies that there exists a non-zero third order cubic invariant F (3) and hence the transition F m3m → P a3 is of first order. As a first approximation we investigate the possibility of a second order phase transition. Neglecting F (3) and F (4) , we minimize F with respect toρ F and toρ c . We obtain two coupled homogeneous equations from which it follows again thatρ F andρ c have to be of opposite sign. The compatibility condition leads to the transition temperature where We observe that T C > J f f and the inclusion of conduction electrons leads to an increase of transition temperature. The calculation of the first order transition temperature T 1 , (T 1 > T C ), and of the accompanying discontinuitiesρ c (T 1 ) andρ F (T 1 ) in the coupled order parameters would require the evaluation of the higher order terms F (3) and F (4) . Such an endeavor poses not only analytical but also numerical problems that are beyond the scope of the present work. We therefore will adopt below (Sect. 6) an alternative point of view and suggest experiments in order to check qualitatively the predictions of the theory.
B. We next investigate the lattice strains at the transition to the P a3 phase. We recall the result U f f QQT , Eq. (2.27), for the coupling of strains to 4f electrons with quadrupolar order. The same procedure, as outlined in Sect. 2, is now applied to derive the coupling of longitudinal strains to conduction electrons. Using the condensation schemes (2.16a,b) and (5.1a,b) and taking into account the definitions of interaction parameters Λ (Table 2), we obtain from Eqs. (3.28) and (3.29) 1 The coefficients Λ are quoted in Table 2. We define the total interaction potential of ordered quadrupolar electron densities coupled to longitudinal strains by Here the right hand side terms are given by Eqs. (2.27), (5.23) and (5.24), respectively. In analogy with expressions (5.7b,d) we define and use again definition (5.6). Then we approximate U QQT /N by an effective interaction This relation allows us to express the longitudinal strains as function of the order parameters. The contributions U T T and U QQT lead to a supplementary term in the free energy (compare with Eq. (5.16)): (There are still additional contributions from thermal lattice vibrations, but these are irrelevant here.) For a given quadrupolar order, i.e.ρ F andρ c fixed, we minimize F QQT [ρ F ,ρ c , ǫ νν ] with respect to the strains ǫ xx , ǫ yy and ǫ zz and obtain where a is the cubic lattice constant and where κ L is the bare linear compressibility. From the numerical values of Table 2 we obtain Λ F F = −498, Λ f c = −1205, Λ cc = −2966 (units K/Å). Sinceρ F andρ c are of opposite sign, the f c contribution on the right hand side of expression (5.29) leads to an expansion of the lattice while the f f and cc contributions lead to a contraction. From Table 2 we also find that A f c = 30516 K, A cc = 24033 K, and hence from Eq. (5.10) we deduce that |ρ c | > |ρ F |. Hence we conclude that inclusion of the conduction electrons in expression (5.29) leads to an increase of the lattice contraction by a factor 4.4 in comparison with the effect of the 4f electrons. A numerical calculation of the lattice contraction △a = aǫ xx at the first order phase transition would require the knowledge of the discontinuitiesρ c (T 1 ) andρ F (T 1 ). Experimentally [1] one finds in the pressure (P ) -temperature phase diagram of the γ − α transition in Ce indications of a critical end point where the lattice contraction vanishes. Although the transition F m3m → P a3 is always of first order and hence leads to finite discontinuities ofρ c (T 1 ) andρ F (T 1 ), there exists the possibility that the relative importance of the coefficients Λ f c and Λ cc , which have opposite sign, changes as a function of (P, T ) and hence the expression within square brackets on the right hand side of Eq. (5.29) could vanish or even change sign without the requirement thatρ c (T 1 ) = 0 andρ F (T 1 ) = 0. This opens up a possibility of a lattice expansion during the discussed first order phase transition. Recent experiments on YbInCu 4 which exhibits an isostructural phase transition similar to the γ − α change observed in cerium (see [20] for references) indicate a 0.5% volume expansion at the transition. Within the present theory such behavior can be understood if for Yb in YbInCu 4 the f c term responsible for expansion prevails over f f and cc contributions, Eq. (5.29).

VI. DISCUSSION
The present paper is an extension of our previous model of the γ − α phase transition in Ce (Ref. [21] or I). We predict that the γ − α transition is accompanied by a symmetry change F m3m → P a3 in the electronic structure. The idea of such a proposal is borrowed from the theory of molecular crystals, where P a3 structures due to orientational order of molecular mass distribution are not unusual. For example, such crystal symmetry occurs in NaO 2 [29], N 2 [30] and in solid C 60 [31]. The conventional characterization of the γ − α transition in Ce as a phase transition "without change of symmetry" is based on several X-ray diffraction experiments [1]. It is possible that domain formation in the α-phase has precluded an identification of this phase as a P a3 structure.
If new experiments are done, particular attention should be given to the possible coexistence of domains [32].
In comparison with our previous work, Ref. I, we have extended the model in two respects. First, we have included quadrupolar interactions between 4f and conduction electrons and secondly, we have calculated the relevant parameters of interactions (Table 2 and Sect. 2) by using the radial dependences of valence electrons obtained from a DFT-LDA calculation of a cerium atom. To our knowledge, in the literature there exists no microscopic derivation of multipolar interactions between conduction and localized electrons in solids although the concept of quadrupolar moment of a 4f electron shell is well established [33,34]. Therefore in Sect. 3 we have presented a detailed calculation of multipolar interactions treating band conduction electrons in second quantization with wave functions in tight-binding approximation. Since the intersite quadrupole-quadrupole interaction is short ranged and anisotropic, special attention has been given to lattice site symmetry. While we have restricted ourselves here (see in particular Appendix A) to the case of an fcc lattice (γ → α Ce), our procedure is general and can easily be extended to other structures.
Our results can be understood on the basis of the following generalized Hamiltonian which we ascribe to the γ phase: Here U 0 is the crystal field Hamiltonian (4.9). The term U QQ represents the quadrupole-quadrupole interaction comprising contributions from 4f and conduction electrons (Eq. (3.35a)). The quadrupolar interaction between localized electrons is due only to intersite contributions on the fcc lattice, see Eq. (2.13) in Fourier space. The presence of conduction electrons leads to two types of contributions: inter-site terms, given by (3.23a) and (3.25a) for U f c QQ and U cc QQ , respectively, and on-site terms, given by (3.23b) and (3.25b), correspondingly. Collecting inter-site and on-site terms separately, we write where and the summation over L(L ′ ) = (sd), (ds), (dd) is implied. The structure of the interaction matrices v AB ( q), A(B) = F or L, is discussed in Appendix A, the relevant eigenvalues λ AB at q X are quoted in Table 2. The intrasite contributions are given by where the calculated parameters C AB are quoted in Table  2. Next in Eq. (6.1), U QQT , given by (3.35b), is a correction to U QQ for a deformable lattice, while U T T is the elastic energy of the cubic crystal in harmonic approximation. In terms of homogeneous strains, U T T is given by Eq. (2.28) and U QQT by Eq. (5.25). In principle a bilinear coupling term U QT between quadrupolar electronic and displacive degrees of freedom [35] should be included in expression (7.1). This term which is known from Jahn-Teller phase transitions [36] can be essential if we want to describe transitions from a cubic phase with quadrupolar disorder to non cubic phases with ferro-quadrupolar order [37]. However for the transition to the P a3 phase, which we identify with α-Ce, the atomic center of mass positions still occupy a face centered cubic lattice and the term U QT is irrelevant. Indeed, the driving force for the transition F m3m → P a3 is the quadrupolequadrupole interaction which becomes attractive (in reciprocal space) at the X point of the Brillouin zone. This fact leads to an orientational order of quadrupolar electronic densities on four different sublattices (Figs. 3, 4). The term U QQT then prompts a lattice contraction at the first order phase transition. Notice that the term U QT is found to vanish for a wave vector q at the X point of the BZ. We insist on these facts since within our view, the electronic charge degrees of freedom, together with the lattice displacements, are the driving forces of structural phase transitions in Ce and related compounds. The Hamiltonian (7.1) is not sufficient to describe the magnetic phenomena that occur at the γ − α transition [12,1]. In accordance with Kramers' theorem the quadrupolar ordering, as described by Eq. (7.1) is not accompanied by a magnetic ordering. We then conclude that the addition of an Anderson Hamiltonian term H cf which takes into account the Friedel-Anderson hybridization between conduction electrons and 4f electrons as well as the repulsive energy among the 4f electrons on a same site is necessary (for a review see [38]). Such a Hamiltonian leads to the disappearance of local magnetic moments below a characteristic temperature T K . The Kondo temperature T K increases with increasing hybridization matrix element V . We then conclude that the lattice contraction accompanying the P a3 quadrupolar ordering or the quadrupolar order enhances the hybridization V and hence increases T K . This can lead to a situation where the structural γ → α transition and the demagnetization of the 4f state occur at a same temperature T 1 . Notice however that within this scenario the process is driven by the structural (quadrupolar) transition at T 1 and not by the Kondo volume collapse [8,9]. In case where the enhancement of T K is insufficient, the quadrupolar order and the concomitant lattice contraction would occur at T 1 without Kondo anomaly (disap-pearance of magnetic moment). The condition T K < T 1 does not ensure that the Kondo anomaly actually occurs at lower T . Quadrupolar ordering has been observed in a number of Ce, Pr, Tm and U based compounds [33]. A remarkable example is the magnetic semiconductor TmTe with 4f 13 ( 2 F 7/2 ) electronic configuration and Néel temperature T N =0.43 K [39]. Although TmTe had been extensively studied before 1995, the phenomenon of quadrupole ordering below T Q =1.8 K was completely overlooked [40,41]. Numerous data on such compounds (CeAg [34], CeB 6 [42], TmTe [40], TmAu 2 [43], DyB 2 C 2 [44] etc.) with quadrupole phase transitions show that the magnetic ordering occurs at lower T . A transition to an ordered magnetic phase in these rare-earth intermetallic compounds indicates that the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction between localized magnetic moments prevails over the the Kondo transition mechanism (for a review, see [45]). In zero magnetic field the magnetic susceptibility shows no anomaly at T Q for TmTe [40], CeAg [34] and a very small anomaly for DyB 2 C 2 [44], TmAu 2 [43]. Within the present work we come to the conclusion that quadrupolar order, as an electronic charge degrees of freedom driven process on one hand, and magnetic properties (Kondo anomaly, magnetic order) on the other hand are related indirectly via their coupling to lattice displacements. An open question is a possible relation between quadrupolar order and hybridization. In our opinion a microscopic derivation of the Anderson hybridization Hamiltonian which should include details about the symmetry of the lattice site and the electronic orbitals constitutes a challenge for further work. VII. ERRATUM FOR REFERENCE [21] In Table 3 −β (last column, third row) should be replaced with +β. In the last column of Table 5 −λ (first row) should be replaced with +λ, −µ (second row) -with ν and −ν (third row) -with µ. In Table 2 i = 7, 8 correspond to Γ 7 , 2 (third row, second column) while i = 13, 14 -to Γ 6 (fifth row, second column). In equation (4.7d) c    The Fourier transforms of the intersite electronic quadrupole-quadrupole interactions are 3 × 3 matrices where the rows and columns are labeled by the indices of the T 2g functions S k 2 , k = 1, 2, 3. The structure of these matrices depends on the symmetry of the lattice and of the T 2g functions. The magnitude of the elements depends on the nature of the electrons (localized 4f , or conduction 6s, 5d) and we abbreviate the electronic indices f or s, d by the label A, Here v A k B k ′ ( h) refers to the elements in real space with h = n ′ − n. Taking the site n as origin on a fcc lattice, the index n ′ (or h) labels the twelve neighbors. Compare with expressions (2.6), (3.10a) and (3.17) for the case Λ = (T 2g , k), Λ ′ = (T 2g , k ′ ). Performing the lattice sums and using the symmetries of the elements where C ij = cos(q i a/2) cos(q j a/2), and S ij = sin(q i a/2) × sin(q j a/2). Here i, j stands for the Cartesian indices x, y, z, and a is the cubic lattice constant. Additional information on coupling matrices can be found in Ref. [47] where a problem with similar symmetries was considered for a molecular crystal. The quantities γ AB , α AB and β AB stand for the matrix elements v A k B k ′ ( h) for h = a(1/2, 1/2, 0) with (k = 3, k ′ = 3), (k = 1, k ′ = 1) and (k = 1, k ′ = 2), respectively. The interaction matrix v AB ( q) has the largest negative twofold degenerate eigenvalue at the X-point of the Brillouin zone. For instance for q X x = (2π/a)(1, 0, 0) λ AB Hence the quadrupolar interaction parameters λ AB X + 5 are completely specified, their numerical values are given in Table 2, fourth column.
The Bloch functions can be rewritten as basis functions of a d-dimensional irreducible representation of the space group F m3m which is induced by a small representation Γ .) The basis functions R| k i , s p , α , where s p = 1 − t and k i ∈ * k 1 , all have the same energy. We can use the small representation index p to label the energies. Hence we will write for the band index α the composite index (p, ν), where ν labels the small representations p that occur more than once. We consider (see also Appendix B of Ref. for fixed p and k i ∈ * k 1 . By the generalized Unsöld theorem [49] this expression is an invariant of F m3m for any r n ′ . This means that the site symmetry is O h and the density k φ * l (Γµk) ( r n ′ )φ l (Γµk) ( r n ′ ) stays invariant for any representation Γ and index µ of O h (Unsöld theorem). We therefore obtain the condition t sp=1 q i=1 γ * lτ ( k i , s p , α) γ l ′ τ ′ ( k i , s p , α) = δ ll ′ δ τ τ ′ f k 1 α; Γ, µ , where k 1 belongs to a basis domain Φ [25] and where f is a function of k 1 and depends on the labels Γ, µ, α. Now we calculate the single particle expectation values (ρ l1l2 k ( n)) 2 0 , Eq. (5.20), for conduction electrons. From Eqs. (3.12a,b) we have Here summation is understood over the repeated band indices α, α ′ , β, β ′ . The thermal expectation value is calculated with the single electron Hamiltonian U c 0 , Eq. (3.6). We use standard Green's function (GF) techniques [50] and evaluate first the susceptibility (retarded GF) in the static frequency limit (z=0) Here n pα is the Fermi distribution function. Therefrom we get the identity ρ l1l2 k ( n)ρ l ′ 1 l ′ 2 k ′ ( n) 0 = (ρ l1l2 k ( n)) 2 0 δ l1 l ′ 1 δ l2 l ′ 2 δ k k ′ , (B12) where (ρ l1l2 k ( n)) 2 0 = α β h1∈Φ p1∈Φ s h i sp j c k l1l2 ( h i , s h , α; p j , s p , β) 2 P ( h 1 α; p 1 β).