Singlet Ground State of the Localized d-Electrons Coupled with Conduction Electrons in Metals

Progress of Theoretical Physics, May 1973

The theory of the singlet ground state for the s-d exchange model is extended to the core of d-electrons with orbital degeneracy. The effective s-d Hamiltonian is derived from the extended Anderson Hamiltonian by the Schrieffer-Wolff canonical transformation. On the basis of the effective Hamiltonian, the ground-state wave function and the ground-state energy are calculated. The anomalous part of the ground-state energy is given by -Dexp[N/(2l+1)ρJ], independently of the d-electron number, when the Hund coupling is neglected compared with the effective s-d interaction. This binding energy is much larger than that for a localized s-electron because of an extra factor of 1/(2l+1) in the exponent. This large value is caused mainly by the orbital quenching. For a more realistic case in which the Hund coupling is larger than the s-d interaction, it reduces to a smaller value of -Dexp[(2l+1)N/ρJ] for a half-filled shell in which orbital exchange does not exist. This value is due purely to the spin quenching. Qualitative discussion is given about the spin quenching and orbital quenching on the effective Hamiltonian derived for n=2l, n being the number of d-electrons.

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Singlet Ground State of the Localized d-Electrons Coupled with Conduction Electrons in Metals

Progress of Theoretical Physics Singlet Ground State of the Localized d-Electrons Coupled with Conduction Electrons in Metals Isamu OKADA 0 Kei YOSIDA 0 0 Institute for Solid State Physics, University of Tokyo , Roppongi, Tokyo The theory of the singlet ground state for the s-d exchange model is extended to the core of d-electrons with orbital degeneracy. The effective s-d Hamiltonian is derived from the extended Anderson Hamiltonian by the Schrieffer-Wolff canonical transformation. On the basis of the effective Hamiltonian, the ground-state wave function and the ground-state energy are calculated. The anomalous part of the ground-state.energy is given by -Dexp[N/(2l+1)pJ], independently of the d-electron number, 'fhen the Hund coupling is neglected compared with the effective s-d interaction. This bindirlg energy is much larger than that for a localized selectron because of an extra factor of 1/(2l+1) in the exponent. This large value is caused mainly by the orbital quenching. For a more realistic case in which the Hund coupling is larger than the s-d interaction, it reduces to a smaller value of -Dexp[(2l+1)N/pJ] for a half-filled shell in which orbital exchange does not exist. This value is due purely to the spin quenching. Qualitative discussion is given about the spin quenching and orbital quenching on the effective Hamiltonian derived for n=2l, n being the number of d-electrons. - the impurity atom. at, and ak, are the creation and annihilation operators for a conduction electron with wave vector k and spin s. This form of the s-d exchange Hamiltonian describes s-wave scattering of the conduction electrons by the localized spin. Therefore, this Hamiltonian is a correct expression for the local s-orbital with spin the magnitude of which is 1/2. For local d-electrons, d-wave scattering IS important and 111 this case d-part of s-d exchange integral J(k', k) should be taken. In (1· 2), P 1 and Y1m are respectively Legendre polynomials and the spherical harmonics and !Jk represents the direction of wave vector k. If l-part of (1· 2) is used in place of J in (1·1), the s-d exchange Hamiltonian for d-wave scattering is obtained as -- ---2JN.t -2.l+1-1- -2R-3 "~".kk' "£...".J akt'Lms'tTs'saklms• S ., 3 kk' mss' where aLms represents the creation operator of the ·electron whose wave function is given by the spherical l, m-wave R being the radius of the spherical crystal, j 1 the spherical Bessel function, and it is related to at. (1·4) (1·5) This form of H.a expressed by (1· 3) may be used for discussing the Kondo effect of the localized d-orbitals of iron-group impurities. Since this Hamiltonian does not express the orbital states of d-electrons explicitly, some ambiguities remain except for a special case of the half-filled shell, for which (1· 3) repre­ sents the proper s-d Hamiltonian as will be shown in this paper. In order to derive the s-d effective Hamiltonian which describes the orbital states ford-electrons explicitly, one should start with the Anderson Hamiltonian9l extended to the case of local degenerate d-orbitals. This is done in the next section for d-orbitals by the use of the Schrieffer-Wolff transformation.10l In § 3 the starting ground-state wave function in perturbative approach is investigated for the case in which an impurity atom has one d-electrons. The scattering t-matrix for this effective Hamiltonian is calculated in the approximation that the most divergent terms are retained in § 4. In § 5 the energy and the wave function of the singlet ground state are calculated by the perturbational method developed so far for the usual s-d ·exchange Hamiltonian (1·1) .··· This calculation is extended to the general case in which an impurity atom has n d-electroris in § 6 without taking into account the Hund coupling and the case in which the Hund coupling is taken account of is discussed in § 7. (1·2) (1·3) § 2. Effective Hamiltoniap for local d-electrons Here H, and H"' represent respectively the energy of the conduction electrons and that of the localized d-electrons, and Hmix represents the mixing Hamiltonian expressing mutual transfer between conduction and localized d-electrons. at, and dJ.. 'are the creation operators for conduction and d-electrons, respectively. The second and the third terms in Hd. express the intra-Coulomb and exchange interac­ tions between d-electrons in the impurity atom. This form of interaction which satisfies rotational invariance in the real space and the spin space was adopted by Dworin/6l and it can also be expressed as where nd., Sd. are, respectively, the number and .spin operators of d-electrons. The matrix element Vkm in H,a is given ·by Vkm = Jv Se-tkryimp(r)Rd.(r)Y1m(SJ.),r 2drdSJ, where the impurity pQtential is assumed to be spherically symmetric, anq Rd.(r) · represents the radial part of the d-electron wave function. If one introduces the mixing matrix element between the sp~erical l, m-wave given by (1·4) and the localized d-orbital Vkm 1s given m terms of vkz as (2·1) (2·2) (2·3) (2·4) (2·5) (2·6) (2·7) I. Okada and 1(. Yosida Here and also in the following, q denotes the degree of degeneracy of d-electrons, namely, q=2(2l+1) =10. Now, the d-electron number in the most stable unperturbed state is assumed to be nd = n. Then we have JEn=sd+ nU>sF, where ep is the Fermi energy and is taken as the origin of energy. With the use of matrix S obtained by (2 ·12) ~ (2 ·15), the transformed Hamiltonian is given to second order in vk by H=H.+Hd+H.d, H ld-.- - -J ""-.J [ akt'm's'akms'dmtid.m/s'- -1ak/ mts a..kmsU-mIt/s'd m'l'J, 2N ••~,:;;,., q Here those parts which are constant and which change the number of d-electrons (2·9) (2·10) (2·11) (2·12) (2·13) (2·14) (2·15) (2·16) (2·-17) (2·18) Singlet Ground State of tAe Localized d-Electrons are omitted, and the ek-dependence of J is also neglected. The effective s-d interaction of (2·17) is chosen so as to exclude the part due to potential scatter­ ing which is diagonal with respect to the d-state by subtracting the second terms.lo), 16) When the intra-exchange integral Ji between d-electrons is taken into account, the calculation becomes somewhat complicated, though practicable, because the spin multiplicity should be taken care of. So, here, the effective Hamiltonian for the case in which the impurity atom has one d-electron in the unperturbed state is only written down: as , H=H.+Ha+H.a, H att-·:- "k-J {-Jakl'mt 's'akmsdmt•d m's' + -Ja2k'mt/sakmBdmt•'d m's' k' km'm•'• 2N 2N - _!i_al'ms'akmsdJ.,,dm'•'- _!i_al'msakmsdJ.,,,dm's'}, 2N 2N ~=.lvkFI2 2N · -J2 --1 VkF 12 2N U(ea+ U) -Ji' ea(ea+ U-Ji) (ea+ U+Ji) -Ji , (ea+ U-Ji) (ea+ U+Ji) <O' <O , _!i_ = _!_ (~- (2l-1) _!i_). 2N q 2N 2N (2·19) (2·20) The first term of the effective s-d interaction of (2·19) exchanges both orbital and spin states, the second only orbital states and the third only spin states. The second and third terms. seem to be important in distinguishing between orbital and spin states for the case in which the impurity atom has one localized d-electron, but, for simplicity, we shall neglect Ji compared with U in the s-d effective interaction and take account of it only in Ha in this paper. § 3. Zero-approximation of the ground-state wave function for the effective Hamiltonian In this section, we calculate the ground.,$tate wave function of zero-approxi­ mation for the effective Hamiltonian for one' d-electron given by (2 · 17), which will be taken as a starting wave function of the ground-state wave function by perturbative approach. The zero-approximate wave function denotes the ground­ state eigenfunction for the effective Hamiltonian in the subspace in which only extra one-electron is excited above the Fermi sea or extra one-hole is excited below the Fermi sea. In the present case, these two cases give different results as we. shall see later. First ~e shall consider the case in which one-hole is added to the Fermi sea. Since the effective Hamiltonian conserves the total angular momentum L. and spin S., we seek for the solution in the subspace of L.=O and S.=O. Such a wave function 1s written as where JF) represents the wave function of the Fermi sea and amplitude'S T~cms are determined by the Schrodinger equation Inserting (3 ·1) into (3 · 2) and cutting the states with electron-hole pair excitations besides one hole excitation, we obtain the following eigenvalue equation with reSpeCt tO amplitudeS TkmB; This solution expresses the bound state whose binding energy, is given by E= -Dexp [ 2N J. pJ(q''-:1/q) . . For this solution, amplitudes T~cms are i11dE)pendentof m and s ;mdso it repre~ents a singlet state with respect to both orbital and spin states, th~t is, in this state orbital and spin moments are quenched. *> p is connected with the state density Pp for plane waves by p= (3/2kF2R 2)Pp· On the other hand, lv~cFI 2 =ikF2R2(Jv~cp.ml 2) by (2·8). Therefore, the relation pJ=ppJp holds if one defines ·'p by replacing Jv~cFI 2 in the expression of J with (v~cFmJ2). · We assume that the state-density for l-wave conduction electrohs is a constant p*> between -D and D and zero outside, and introduce Gm, arid X by Then we have - -1~ 1 2N 1c - e~c- E ~ p 2N l o g-E-_- -1 . D X { - X - J)Gms+J~ Gm's'=O, \ q m'•' which leads to the following eigenvalue equation: Only the first factor of (3 · 7) gives rise to a negative-value solution for X, · (3·1) (3·2) (3·3) (3·4) (3·5) (3·6) (3·7) (3·8) (3·9) Singlet Ground State of .the Localized d-Electrons As shown above, W('! have obtained the singlet bound state by making an extra hole excited below the Fermi sea coupled with a localized d-electron, while the state with an extra electron added to the Fermi sea coupled with the lo~alized d-electron does not give any singlet bound state. For this case, the zero-approxi­ mate wave function is given by Inserting this wave function into the Schrodinger equation, we obtain the follow­ ing equation for amplitudes Fkms: Corresponding to (3·4) and (3·5) we introduce Gms and X in which the sum­ 'illation is now taken over the states above the Fermi energy. Then (3·11) gives which leads to the eigenvalue equation The bound-state solution is given by The amplitudes for this solution satisfy (x-: J)Gms+JG_m-•=0, (3·10) (3·11) (3·12) (3·13) (3·14) (3·15) This state has (2l+ 1)-fold degeneracy. If one introduces the intra-exchange Ji, this level splits into two levels having l-fold and (l+ 1)-fold degeneracies. These belong to the states of (S, L) = ( 1, 1 ), ( 1, 3 ) and those of (S, L) = (0, 0), ( 0, 2 ) and ( 0, 4 ) for l = 2. Thus: in the zero-approximation, the degeneracy of the ground state is not removed by the s-d interaction in contrast to the case in which a hole is coupled to the d~electron. This is quite natural, because our Hamiltonian has not particle-hole symmetry from the beginning for' the present case where one of ten local d-states is occupied. For the usual s-d exchange model, the fact is recognized that if the bound­ state solution in its zero-approximation is degenerate, this bound-state solution disappears in its final stage in which particle-hole excitations are taken into account by the perturbation calculation starting from the zero-approximate state.7h 17> For the present case, also, the bound state solutions having degeneracy in the starting approximation disappear as particle-hole excitations are taken into account and only the non-degenerate solution survives in the final stage. Before we show this fact by taking into account higher-order effects, we shall calculate the scat­ tering t-matrix for the present effective Hamiltonian (2 ·17) in the approximation in which the most divergent terms are taken into account. § 4. Scattering t·matrix In this section we calculate the scattering t-matrix for one occupied d-electron m the most divergent approximation after the Abrikosov theory18> for the usual s-d model. This calculation can be done in a way quite parallel to what Abrikosov has done for the s-d model and we have the following integral equation for the scattering matrix r (w): (m1s1o massir(t) imzsa, m,s,) J = - -tJm1miJmamaiJslsiJszsa +J- -1tJmlmz?Jmam,(Jslsztlsas, 2N 2N q - Jrotds :E {(m!Sb masair(s) im5ss, msSa)(msSs, mssair(s) lm2s2, m,s4) m 5m 8 B5Be In the matrix element (m1s1o mass Ir (t) Im2s2, m 4s,), m 1s1 and m2s2 denote orbital and spin states of the conduction electron and m 3s8 and m 4s4 denote those of the local d-electron. t is defined by p log(D/Iwl). The first and the second terms are nothing but the matrix element of the effective s-d interaction, and the third term consists of two parts AI and A2• AI represents the contributions from process (a) and A2 those from process (b) shown in Fig. 1. The matrix element of r can be put as (m1s1o masalrlm2sz, m4s4) (4·1) (4·2) ibecause it is expected that r has the same structure as the s-d interaction. If one inserts this form of r into (4 ·1), the following integral equations for r 1, and ---·-m,s. (b) Aa ro are obtained: Equation (4 · 3) can easily be solved and leads to rz.(t) = - -J+ q st dsr~.(s), 2N o ro(t) =J- -1 - 2N q dsd.(s). st o and rz.{w) = - -J {1 + qpJ- l o gD-} -1 2N 2N lwl ro (w) = - _!_rz. (w). q rz.(w) and r0 (w) have a pole at lwl =wk=D exp[2NjqpJ] =Tk. This value of Tk is very big compared with the value of D exp[N/pJ] on account of an extra factor q = 10 for l = 2 in the denominator in the exponent. With the use of these results A1 and A2 of the third term on the right-hand side of (4·1) are calculated as These two A's play the role of integration kernel of integral equation which the the amplitude for the ground-state wave function satisfies.19l § 5. The ground-state wave function. and its energy eigenvalue· for one local d-electron After the method developed for the s-d model, we expand the ground-state wave function in the perturbation series, starting with the zero-approximate wave function (3 ·1) or (3 ·10). For the case in which a hole is coupled with the = - (__!___) { pJ log _!!____/ (1 +.q pJ log _!!____)} 2N 2N. lwl 2N . lwl X { ('1 + q1')' iJm1m2iJmamiJs1slJsas•- -;;_iJm1m.iJm2maiJs1s.iJs2sa} , 2 = (_!_) {pJ log _!!____/ ( 1 + q pJ log _!!____J} 2N 2N lwl , 2N lwl' X { : 2 iJm1m2iJmam.iJs1s2iJsas• + (q- : ) iJm1m,iJm2maiJs1s,iJs2sa}. d-electron, this expansion becomes + ···. By inserting (5 ·1) into the Schrodinger equation, the following relations between the amplitudes r are obtained: (5·1) (5·2a) In (5 · 2a) and (5 · 2b), a single suffix m is used for ms and [ · · ·] m T indicates the antisymmetrized form: (5·2b) (5·3) From the set of equations (5 · 2a), (5 · 2b), ·· ·, we' derive the equation for the lowest amplitudes rkmB• This equation for the bound-state solution. in which Tkms=Tk holds is calculated as Singl~t Ground, State of the Localized d-Electrons T(ea) {,-ea-E+ (__{__) 2.E(_!_ - q )1- + ···} , 2N 4,o q D4,~6 ( -ek-E)Tkm= .E rk'm'{(m', miH.alm, m') 1 k'm' - (1 - -1) -1 (-qJ) 2p ""'-'rk' 1og ----e=k-----e-"k'-,----E- l q2 2N k' D x [1+ qpJ log· -ek-ek,-E + ( qpJ log -ek-ek'-E) 2 + ... J. 2N D 2N D If we here identify the series in pJ/2N in the integration kernel as the geometric series, (5 · 5) can be written as · (-ek-E)T(ek)=-( 1- 1, ) qJp so r(e')de'-(1- q12 ) 1qa(qJp)a , _q 2N -» 2N x f»r(e') {log -ek~'-E /(1- ~~- log -ek-~'-E)}ae'. E is defined by E=E-.JE' where LIE is called the normal part of the energy which is obtained by the usual perturbation calculation. I The integral equation (5 · 5) can be more generally written as - - 1-r. }l - -1-1-{-1-r1 - - -1r 2 D2,u · q Ds,2s Ds,1s D1,2s +···, where the suffixies k, are abbreviated simply to i, and D 1, 28 is used for energy denominator (e1 - e2 _: e8 - E). Carrying out the summations in (5 · 4) with l~garithmic accuracy or in the most divergent approxim~tion, we obtain (5·4) (5·5) (5·6) I. Okada and K. Yoshida where (mh m 8 l A21 m2, m4) is given by (4 · 8). With the use of (4 · 8) for A2 and relation (3. 15) which says rk-m±B= - TkmB = - rk> (5. 8) can be written as It is noted· that the binding energy IEl 1s enlarged by a factor (2l + 1) in the denominator of the exponent. we obtain a bound-state solution for (5 · 5) as G (e)= G( -D) [P(e) ]a-[P(O) ]<a-P>[P(e)]P. 1- ((3/a) [P(O)]<a-P> The eigenvalue E is determined by the relation qbtained by putting e= - D in (5·13)' This relation is satisfied by P(O) = 0, namely, 1= 1- [P(O)]<a-P> 1-((3/a) [P(O)]'a-P> E= -D exp[(2l+~)pJ]. This integral equation can also be derived by direct calculation. The integral equations (5 · 5) and (5 · 9) · can be solved with logarithmic accuracy by applying Yoshimori's method.6> Calculation is straightforward. Putting G(e)= fr(e')de', P(e) =1-~q-pJlog --e--E 2N D 1 a=1-~ q2 ' (5·9) (5·10) (5·11) (5·12) (5·13) (5 ·14) (5·15) On the other hand, for (5 · 9) we have no bound-state solution except for l = 0. Defining, as before, we have the relation, corresponding to (5 · 14), which determines the eigenvalue -E as Since a'- {3' = t >O for l = 0, the binding energy is obtained as but for l>1, a'- {3' becomes negative, and therefore, we have P' (0) = oo which means E= 0. Thus, we have no bound-state solution from (5 · 9). § 6. Ground state for n d-electrons In the preceding section we have derived the ground singlet wave function for one localized d-electron coupled with the conduction electrons by the s-d effective interaction. In this and the next sections we consider the case in which the impurity atom carries n dcelectrons (l<n<10). For this case, the intra­ exchange interaction between d-electrons may be neglected in the effective s-d Hamiltonian, but it may not be neglected in Ha. On account of this Hund coupling included in Ha, the situation becomes .. complicated. Therefore, to avoid this difficulty, we take two extreme cases; in one case the intra-exchange coupling J, is assumed to be zero, and in the other case J~, is much greater than H•a· We begin with the consideration for the first case. By inferring from the results obtained for one d-electron, a starting wave function for the ground singlet state may be put as (5·16) (5·17) (5·18) (6·1) where n is assumed n<2l+l. For n>2l+1, electrons and holes are exchanged. Inserting (6 ·1) into the Schrodinger equation and cutting the higher-order states with particle-hole excitations, we obtain the equation for amplitudes T as where the summation over ms is taken over the unoccupied states and misi. If rkms are independent of ms and satisfy the equation ( -e~~:- E)rk +L{q- (n-1) -~} :E r~~:,=O, n 2N q k' E .= -nD exp [ 2N {q- (n-1)-n/q}pJ Eq. (6·3) 1s satisfied by such T~~:. From (6·4) the energy eigenvalue is calculated as In order to take into account the higher-order effects due to particle-hole pair excitations, we expand the ground-state wave function as :E (mi'Bi')=F(mJBJ) 1=1,2,···'1-l,i+l,···n For the suffixes possessed by the second amplitudes, n+l n :E mi-mn+2=:E m/, l=l i=l n+l n :E Si- Sn+2 = :E S/ i=l i=l are satisfied. Also in the present case, we can derive an equation for the first amplitudes T[k1k2···k,] that are independent of (misi) by the same iterative procedure as has been done in § 5. To do this needs a somewhat lengthy calcu­ lation but its principle is so simple that we shall here write down only the (6·2) (6·3) (6·4) (6·5) where e£ is used for ek£ for simplicity, and a,. and /3,. are respectively defined by This equation (6 · 7) has been obtained within logarithmic accuracy. In deriving the integration kernel those scattering processes in which more-than-one holes participate have not been taken into account. Their contributions are always less-divergent. As has been shown before, the solution for the equation of zero approximation obtained by omitting the term with the integration kernel can be solved in a form of the products of one-particle amplitudes. ·For the complete equation of (6 · 7), a solution cannot be obtained in such a simple product form, but it can be obtained with logarithmic accuracy in the following form: result without describing detailed processes of calculation. where suffix n is attached to amplitude T to signify explicitly the number of d­ electrons. If we. introduce T,. of this form in (6·7), assume /(log(( -e-E)/D))/ (- e- E) for T 1(e; E), which will be confirmed later and neglect less-divergent quantities, then (6· 7) is reduced to the following one-particle equation: ( -e-E,~.)T1(e; EN.,.)= -a,.q-pJ- so T1(e'; E~.,.) de' -a,.f3n (-qp-J)2 . 2N -D 2N (6·7) (6·8) (6·9) x fJlog -e-~-En /(1- ~~log -e-~-En )}r1 (e';En)de'. A bound-state solution for this equation is given by (5 · 13) modified by replacing a and {3 by an and f3n, and En is given by E~n= -nDexp N (2l+ 1)pJ . We can see from this result that the singlet ground state for the n-d-electron system has a binding energy whose exponent is the same as that for one-d-electron system. § 7. Effect of a strong Hund coupling In the actual case, a strong Hund coupling will change the results obtained above. In the limit of a strong Hund coupling, the d-electrons are mutually coupled and they are in the state of the lowest multiplet specified by the total angular momentum L and the total spin S. For the half-filled shell with d-electron number n equal to 2l + 1, the effective s-d Hamiltonian can easily be written down by taking only the terms of m = m' in (2 · 17), because in this special case d-electrons cannot change their orbital states, and by projecting them into the subspace of S = 5/2, that is, by replacing d~,dms' by (2/q) :Em d~,dmr'· The result is This can be written in the following form: H.a = J L; at'ms'o-,,,. Sakms. 2 (2l + 1) N kk'm,.' This is nothing but the form given by {1· 3) and has first been used for discus­ sing the Kondo effect of degenerate d-orbitals by Schrieffer.14l However, this form is allowable only for the half-filled shell. For the s-d exchange Hamiltonian given by (7 · 2), 2l + 1 conduction electrons or holes can be coupled with the localized spin S, forming a singlet state on account of 2l + 1 d-wave components. The situation is quite similar to the case - for the s-d model with S = 1/2, and the energy lowering is given by This has a factor 2l + 1 in the numerator of the exponent, and is therefore very small compared with the value given by (6·11). This is due to the absence of (6·10) (6·11) (7 ·1) (7 ·2) (7 ·3) orbital exchange in this case. For the case in which d-electron number is less than the half-filled value of 2l +1, the lowest multiplet has an orbital angular momentum L as well as a spin momentum S. Therefore, for such cases, the s-d Hamiltonian should be projected onto the subspace made by L and S. This process can most easily be done for the case in which one electron is removed from the half-filled shell. For this case of n = 2l, there exists one multiplet with the maximum value of S. For this case of n = 2l, the effective Hamiltonian obtained by the projection process mentioned above can be written down as + ~~J (-1Yt ~ ( - 1)-M'-Makt-M'Bak'-Ms'T(M'-.(- M) 2N 2 kk~.'M,,M' X[&,,,+~ (s·S),.,]. (7·4) In this expression the state of the localized d-electrons is specified by z-components M and M, of the orbital and spin moments L and S whose magnitudes are both equal to l. T (M' ~ M) denotes the ope:t:ator of changing the orbital state of d­ electrons from M to M'. The second term of (7 · 4) exchanges spins between conduction electrons and the localized d-electrons, and the third term consists of the part which exchanges only orbital states and that which exchanges both orbitals and spins. For n = 1, the effective Hamiltonian (2 · 17) can be written in the represen­ tation in which the d-electron state is specified by M and M, as As shown in the previous sections, this Hamiltonian has an effect of quenching spin and orbital moments of the lodttlized d-electron simultaneously. In contrast to this special case, the effective Hamiltonian given by (7 · 4) has different effects on spin quenching and on orbital quenching through the process of projection to the L-S subspace. This can be seen from the fact that the Abrikosov-type integral equations which are satisfied by three parts of the scattering t-matrix, correspond­ ing to spin exchange, orbital exchange and orbital and spin exchange terms of (7 · 4) give rise to different w-dependence for these three parts. Therefore, in the following we shall give qualitative discussion on the quenching of spin and orbital moments on the basis of the effective Hamiltonian (7 · 4) for n = 2l. Hamiltonian (7 · 4) can also he written as + -2JN c-2w~ Tclc"£'.M..'JM's c- 1)-M'-Makt-M'sak'-M•TCM' ~ M )· +2JN- ( -18Y1'lcTMc"'£=M.F..MJ'M'B'B' ( - 1)-M'-Makt-M'Iak'-Ms' X T(M'~M) (s·S),,,. (7·6a) (7·6b) (7 · 6c) bound electron number 1-a ( -M, s), bound hole number a (m1 s) (m2 s) (ms s) · ·· (m2z s) lmt+-M, (7·8) where the number of bound holes a is given by the neutrality condition (1-a)=2la, namely, a=1/(2l+1). These numbers of bound holes and electrons are deter­ mined by the Anderson condition20> for the matrix elements of (7 · 6) between two different M-components not to vanish. The characteristic ··energy for this process is given by D exp [ (4/2l +1) (N/ pJ)]. The third term represents simultaneous exchange of spin' and orbital states between conduction electrons and the localized d-electrons and will link mutually independent spin exchange and orbital exchange given by (7 · 6a) and (7 · 6b). In the ground state for the whole s-d effective Hamiltonian (7 · 6), the local spin and orbital moments are both quenched. The ground-state wave function would be given by 1/Jcond(M=l, M,=S)~ i/Jcond(M=l-1, M,=S)~ electron . hole a (-(l-1)j)(-(l-2)t) .. ·((l-1)j)(lj), (1-a) ( -n) (- (l-1H) ... en), (-lt) electron (1-a) c-n) .. ·(l~), (-(l-1)j) a (-lj)(-(l-2)j)· .. (l-1j)(lj), hole etc. 1/Jground = :L; 1/Jcond (M, M,) ¢ (M, M,), M,M8 m which the wave function of conduction .electrons in each (M, M,)-component 1s characterized by the following nu mIbers of bound electrons and holes: a is given by (1-a)2(l+i)=2al, namely, a=(l+1)/(2l+1). Although in this paper we cannot derive the two characteristic energy values T Ks and T Ko, corresponding respectively to spin quenching and orbital quenching, the former value being much smaller than the latter, we can derive the phase shifts of the conduction electrons at the Fermi surface from the number of bound electrons and holes in each (M, M,)-component of the ground-state wave function by the use of the Friedel relation. The phase-shift values thus obtained im­ mediately lead us to the following residual resistivity for l = 2: p=5p0 sin2 (;0 n-), (n=4) (7 ·9) (7·11) (7·12) where p0 denotes the resistivity value of s-d model (S = 1/2). This result coincides with the result obtained by the Hartree-Fock approximation for the case in which local spin and orbital moment are quenched. It is expected that this coincidence is generally valid. It is easy to see that the same result as Hartree and Fock's is obtained for general values of n when the Hund coupling is neglected. At high temperatures at which only orbital moment is quenched, the resistivity value can be evaluated by (7 · 8) as p=25Po sm.2(Snn' ) . §8. Summary Starting from the Anderson Hamiltonian generalized to the case in which the orbital degree of freedom of the localized d-electron is taken into account, we have derived the effective s-d Hamiltonian for any number of localized d­ electrons in the limit of strong correlation, U'P J; On the basis of this effec­ tive Hamiltonian, the nature of the singlet ground state of the localized d-electrons has been studied in two limiting cases, one unrealistic in which the Hund cou­ pling is completely neglected and the other realistic in which the Hund coupling is so strong that the state of the local d-electrons is restricted into the subspace of the lowest multiplet of the isolated ion. In the former case, the spin and the orbital moment are simultaneously quenched and the energy gain due to the formation of the singlet bound state is given by D exp [ (NI pJ) (11 (2l + 1))]. This large value of the binding energy indicates that the orbital moments of the localized d-electrons are quenched at high-enough temperatures. The residual resistivity is equal to the value obtained by the Hartree-Fock approximation. For a strong Hund coupling, the effective Hamiltonian for n=2l+ 1 and n=2l has been derived. For the half-filled shell, the situation is so simple that the groundstate wave function and its energy are easily obtained by using the results for s-d exchange model. The binding energy or the Kondo temperature is given by D exp [ (NI pJ) 2S]. For general cases, further investigations are needed, but here a qualitative discussion has been given for the ground singlet state for n = 2l and it has been concluded with the aid of the Anderson theorem that the residual resistivity again coincides with the Hartree-Fock value. The authors would like to express their sincere thanks to Professor A. Yoshimori, Dr. A. Sakurai, Dr. H. Ishii and Mr. K. Yamada for frequent valuable discussions and comments. One of the authors (1.0.) is also much indebted to them for their advices and encouragements throughout the course of this work. 1) K. Yosida and A. Okiji , Prog. Theor. Phys . 34 ( 1965 ), 505 . 2) K. Yosida , Phys. Rev . 147 ( 1966 ), 223 . 3) A. Okiji , Prog. Theor. Phys . 36 ( 1966 ), 712 . 4) K. Yosida , Prog. Theor. Phys . 36 ( 1966 ), 875 . 5) K. Yosida , Phys. Rev . 164 ( 1967 ), 879 . 6) A. Yoshimori , Phys. Rev . 168 ( 1968 ), 493 . 7) A. Yoshimori and K. Yosida , Prog. Theor. Phys . 39 ( 1968 ), 1413 . 8) K. Yosida a,nd A. 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Isamu Okada, Kei Yosida. Singlet Ground State of the Localized d-Electrons Coupled with Conduction Electrons in Metals, Progress of Theoretical Physics, 1973, 1483-1502, DOI: 10.1143/PTP.49.1483