A. EFFECT OF INFERATOMIC INTERACTIONS ON FHE ZERQ- BAI‘IDWIDTH KUBBARD HAMILTGNM 8. THEORY OF SUPERHGHANGE INWIOIIS III: MNEI’EC IWMTGRS Dissertation for the We of Pm D. MICHIIEAN STATE UNIVERSITY REM SING W I 9 7.5 This is to certify that the thesis entitled A. Effect of Interatomic Interactions on the Zero-Bandwidth Hubbard Hamiltonian. B. Theory of Superexchange Interactions In Magnetic Insulators. presented by Rem Sing Tu has been accepted towards fulfillment of the requirements for EMF—degree in .Bhysics_ 4km... A. ‘Ca‘ugam Major professor Date August 28, 1975 0-7639 mfiw LIBRARY amncns ‘ granny, maul; lll ;- I Cl C:‘\ \f\. Jx ABSTRACT A. EFFECT OF INTERATOMIC INTERACTIONS ON THE ZERO-BANDWIDTH HUBBARD HAMILTONIAN B. THEORY OF SUPEREXCHANGE INTERACTIONS IN MAGNETIC INSULATORS BY Rem Sing Tu The works of theoretical solid state physics can be divided roughly into two types of problems. The first type is to find the thermodynamic properties from a given model Hamiltonian. The second type is to find out an appropriate model Hamiltonian for a given problem or system. Part A is of the first type, and part B is of the second type. Therefore, the two main subjects of this thesis, unrelated as they may seem, can be regarded from a general theoretical point of view as being two different aspects of the same branch of physics. In part A, we consider the linear-chain zero- bandwidth Hubbard Hamiltonian with added nearest-neighbor interaction, with a magnetic field present. By the trans- fer matrix method, exact expressions for thermodynamic Rem Sing Tu quantities are obtained in simple closed form for the half-filled band. Recently, the half-filled band Hubbard model for the linear chain was proposed to explain the properties of the organic salt NMP-TCNQ. It was shown that the susceptibility X versus temperature T obtained from the Hubbard model disagreed in an essential way with the experiment. The experimental susceptibility rapidly becomes too small with increasing T, showing in particular what appears as a Curie-Weiss law with a moment appre- ciably reduced from the theoretical value. Since the nearest-neighbor Coulomb interaction causes a transition to a ground state of zero magnetic moment if large enough, it seemed possible that adding the Coulomb interaction might reduce the discrepancy between the experiment and theory. The answer we find is unfortunately negative. In part B, we study Anderson's theory of super- exchange. It is thought that the exchange interaction between magnetic ions in a magnetic insulator is des- cribed essentially by the Heisenberg Hamiltonian; also the exchange parameter J is of 4th order in the overlap between para- and dia-magnetic ions. However, the Wannier functions are not uniquely defined in the superexchange problem. Therefore, if one uses an arbitrary set to cal- culate J, one has to go to the 4th order perturbation theory in order to exhaust all the terms of the 4th order Rem Sing Tu in overlap. Anderson suggested that there exists "the exact" Wannier function which makes the perturbation theory converge rapidly. By using this set, the exchange parameter was presumed to come mainly from lst and 2nd order perturbation terms, the 3rd and 4th order perturba- tion term being negligible. He proposed to use the Hartree-Fock method which put all the electrons in the magnetic ions spin parallel and doubly occupy the diamag- netic-ion orbitals to find the Wannier functions. However, his Hartree-Fock leads to a magnetic solution, that is, the spatial function of the spin-orbital depends on the spin. This is inconsistent with Anderson's requirement that they be nonmagnetic. In this work, we use a different variational approach, namely the thermal single determi- nantal approximation (TSDA) to substitute for his Hartree- Fock method. We first investigate a 3-site 4-electron linear cluster, and then generalize to a crystal. We find that there exists nonmagnetic solutions which make 3rd and 4th order perturbation term vanish in both cases. The exchange parameter therefore comes only from lst and 2nd order perturbation terms. Hence Anderson's idea is ful- filled. His "exact Wannier function" turn out to be the TSDA solution. In the 3-site case, we also show the size of the contribution to J from each order in perturbation Rem Sing Tu theory is very sensitive to the choice of Wannier func- tions. The generalization of this type of consideration to more realistic model containing more than one electron on a magnetic site is important and interesting. A. EFFECT OF INTERATOMIC INTERACTIONS ON THE ZERO-BANDWIDTH HUBBARD HAMILTONIAN B. THEORY OF SUPEREXCHANGE INTERACTIONS IN MAGNETIC INSULATORS BY Rem Sing Tu A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Physics 1975 ACKNOWLEDGMENTS It is a great pleasure to express my gratitude to Professor T. A. Kaplan for having given me the opportunity to collaborate closely with him in physics research. This collaboration has been a continuous stimulation for me to learn, especially because of Professor Kaplan's deep in— sight into the problems of concern and his way of enjoying teaching and doing research. This thesis would not have been possible without his continuous friendship, his time and effort. I also learned a lot of solid state from other people in Michigan State. I would especially like to acknowledge many interesting and instructive hours spent in the company of Professor S. D. Mahanti and Dr. M. A. Barma. I am greatly indebted to Professor T. O. Woodruff and the Physics Department of Michigan State University for providing me with a research assistantship. I also appre- ciate the encouragement from the committee and the fellow students at Michigan State. ii TABLE OF CONTENTS ACKNOWLEDGMENTS LIST OF FIGURES APPENDICES PART A I. INTRODUCTION II. SOLUTION OF THE PROBLEM PART B II. III. IV. VI. REFERENCES INTRODUCTION THE 3-SITE MODEL THE ONE-ELECTRON SITES IN THE THERMAL SINGLE DETERMINANT APPROXIMATION RESULTS OF THE 3-SITE MODEL GENERALIZATION OF THE 3-SITE RESULTS TO A MANY-ATOM LATTICE SUMMARY AND DISCUSSION REFERENCES iii Page ii iv 17 20 26 35 43 45 53 S4 Figure A1. A2. B1. B2. B3. LIST OF FIGURES Page Inverse susceptibility xnl versus temperature T 14 Specific heat Cv versus temperature T 16 Three-site model 27‘ Linear-chain model 47 Peroskite Structure ABF3, showing - only the B-ions (.) and the F-ion (0) ' 47 iv APPENDICES APPENDIX Page A. DIAGONALIZATION OF THE TRANSFER MATRIX 56 B. WAVEFUNCTIONS OF 3-SITE 4-ELECTRON LINEAR CLUSTER 59 C. MATRIX ELEMENT OF THE HAMILTONIAN OPERATOR WITH RESPECT TO DETERMINANTAL WAVEFUNCTIONS 61 D. VARIOUS ORDERS OF ENERGY CORRECTION ON THE PERTURBATION THEORY 65 E. THE MATRIX ELEMENT FOR NEAREST-NEIGHBOR HOPPING OF ONE ELECTRON 67 F. DEGENERATE PERTURBATION APPROACH VIA EFFECTIVE HAMILTONIAN 68 G. THE EFFECTIVE HAMILTONIAN IN SECOND ORDER PERTURBATION THEORY 71 PART A EFFECT OF INTERATOMIC INTERACTIONS ON THE ZERO-BANDWIDTH HUBBARD HAMILTONIAN I. INTRODUCTION It is well known that in a crystal the energy levels of the electrons are grouped in bands. We consider the case of a crystal of N atoms and an average of N electrons filling exactly half of one nondegenerate band; and we disregard the presence of all the other bands. To do so, we define an orthonormal complete set of N Wannier functions for this band. The Wannier functions are local- ized at the lattice sites, i.e., each of them is appre- ciably different from zero only in the neighborhood of a lattice site. We then define operators C10 and cio which respectively create and destroy an electron in the Wannier function at site i with spin 0. The cio's and cio's satisfy the usual fermion anticommutation relations and ni0 = Ciocio is the number operator of site i and spin 0. The Hubbard Hamiltonian1 is written in terms of these Operators as = 2 t H ijo ij Ciocjo + U i “1+“iI (1) * The bij( = bji) and U are constant parameters and have precise physical meaning. For simplicity we take bij = b for i and j nearest neighbors, and zero otherwise. b is called the transfer or hopping integral. U is the intra- site Coulomb repulsion energy. The Hubbard model has found wide use in the theoretical description of electro- nic states in magnetic insulators. It was studied earlier than Hubbard did; e.g., des Cloizeaux2 discussed it in the late 50's. Hubbard and also Gutzwiller3 reintroduced it in 1963. Presumably Hubbard's name is attached because he was the only one who tried to give a serious derivation. His derivation leads to completely unsatisfactory behavior of the bij as a function of distance between sites i and j as shown by N. Silva and T. A. Kaplan4. They present an essentially different theory which yields a satisfactory result. Although the derivation aspect is an important one, one can take this Hamiltonian phenomenologically as a model and study its physical predictions. That is, H is given, and is to be studied as a function of the para- meters bij and U, as well as temperature. Some exact 50- 5-9. This is an interesting problem lutions can be found essentially because it is probably the simplest model such that special cases yield pure band-like behavior and atomic-like behavior; and of course the question of how electrons go from one type to the other has been of in- terest in solid state physics for many decades. When U = 0, we get a very familiar simple example of noninteracting fermions. H can be written in terms of Bloch function occupation numbers as ck nk0 (2) . + + n£0 are defined as nkg - a&o 859’ where 21120 and a5? are respectively the creation and annihilation operators for an electron in the Bloch function with crystal momentum k and . + . . - . . sp1n o. ak0 15 related to the Wannier function creation operators by -l/2 ik ° R = N 2 e c. (3) 8k are the one-electron energies of the band in question whose width is prOportional to bi in general 15 . Bi . c = X b.. e 3 . (4) J' E. 13 Thus the energy eigenfunctions are single Slater-determi- nants with Bloch functions occupied in all possible ways. The calculation of all physical prOperties is tractable in the manner discussed in any elementary solid state textbook. When bij = 0, H becomes u 2 NM NH (5) Two electrons with opposite spins occupying the same site interact with an energy U. They do not interact if they are on different sites. The complete set of eigenstates in this limit is given by the set of Slater determinants obtained by occupying Wannier functions in all possible ways. This was pointed out by Kaplan10 in 1968 and Kaplan 11 in 1970. and Argyres The half-filled band Hubbard model for the linear chain was recently proposed to explain the properties of the organic salt NMP-TCNQ. These organic solids are com- posed of two types of molecules, a donor and acceptor giving rise to the presence of unpaired electrons in the crystal. In this material the TCNQ molecules are presumed to be simple minus ions with the extra electron per TCNQ being the source of the observed electronic properties of the system. These molecules are large and flat, and stacked in linear arrays. These salts are highly aniso- tropic displaying a very pronounced one-dimensional beha- vior, the unpaired electron moving along the chains made up of the acceptor molecules. The one-dimensionality is clearly displayed by the conductivity measurements by ShchegolevlS. It was shown14 that the susceptibility x versus temperature T obtained from the Hubbard model with T independent parameters, b and U, disagreed in an essen- tial way with the experimentlz. On the other hand, the introduction of a T-dependence into b, phenomenologically, such that b increases appreciably with T (with U = constant) can correct x versus T. Since the physical interpretation becomes drastically modifiedls, it was concluded that under- standing the physics beyond the Hubbard model is the essence of understanding NMP-TCNQ. The difficulty with the constant b and U is that when one uses values which fit x to experi- ment at T=0 the experimental susceptibility rapidly becomes too small with increasing T, showing in particular what appears as a Curie-Weiss law with a moment appreciably re- duced from the theoretical value14. Since large enough nearest-neighbor Coulomb interaction V(V > g) actually causes16 a transition to a ground state of zero magnetic moment, it seemed possible that values of V smaller than this critical value but still appreciable might importantly reduce the discrepancy. In this part, we consider a zero bandwidth modi- fied Hubbard Hamiltonian in the half-filled chain. A sum- 17,18 mary of the results appeared earlier Our model Hamiltonian is in the form H = U E n + V Z n.n (6) i i i+ni+ 1 1+1 The second term of the above Hamiltonian represents the intermolecular electron repulsion. It may be considered as a first step towards taking into account the long range character of the electron-electron interaction. This Hamiltonian is related to Hubbard's by putting bij = O in Hubbard case and adding the interatomic interaction. In the half-filled case, as we show below, the ground state configuration consists of one electron per site if V < U/Z, but it consists of alternating pairs if V > U/Z. This effect was first pointed out by Bari16 who investigated the role of electron-lattice interactions in a very narrow half-filled band. His Hamiltonian which incorporates elec- tron-electron and electron-lattice interactions can be decoupled via a canonical transformation, and in one dimen- sion reduces to the above equation. II. SOLUTION OF THE PROBLEM In the grand canonical case, one must consider H-UN = U PM :3 :3 1+ 1+ + V i(“1++“1+)(“1+1++“1+1+) ' “§(“1++n1+) (1) 1 Here n is the chemical potential and we assume periodic boundary conditions such that nN+10 = n10 (0 equals either spin direction). Since we are only interested in the case Z(ni++ni+) = N, u can be set by finding the average number i of particles. This Hamiltonian is rewritten by Kaplan and Argyres11 in terms of "spins" S1 5 n1+*“1+‘1 (2) We find that19 _ u 2 H’“N ‘ 7 E S1 + V F S1 S1+1 1 l u c - 7 - 2V) 1; 81 + N(U-u) (3) 1 We drOp the last constant (Si-independent) term and add the magnetic field energy. We obtain, for a linear chain in the magnetic field, H—uN = N|C N h is tu x the magnetic field. This Hamiltonian is quite analogous to a spin-1 Ising Hamiltonian in a magnetic field. However, one must be careful since the "spin value" Si=0 can occur in two ways. This arises from the fact that a singly occupied electron site is two-fold spin degenerate. This means that, if we treat the thermodynamics of H from equation (5) of this section, we must take special care in counting the states. From this, the grand canonical partition function is the following: 1 1 N Z = Z Z ... exp [- Z (x- S. S. _ _ ._ 1 1+1 nif—o ni+-o 1—1 + ys + 28 2 - uM-ll (5) i i 1 where x=BV y=8(%+2v-u) 2:12] .11 _ _ (6) “ ' 87 M1 ‘ “1+ "1+ Let us introduce the notation £1 = l, 2, 3, 4 corres- ponding respectively to (n1+ ni+) = (1,1), (1,0), (0,1), (0,0) and note that we can write the partition function 10 as following with the periodic boundary condition 4 4 g g ... r r' ... r 51—1 52-1 a z = , ENEI where + T5152 = exp {-[x 5152 § (51 + 52) + 5 (52 + $2 2 1 2) (”1+M2)]} = T gzgl. (8) N|C Let a 4 x 4 matrix T be so defined that its matrix ele- ments are given by TEE' <€|TI€'> = exp {-[xSS' + %(S + S') 2 + g (52 + s' ) - %-(M + M')]}. (9) Thus an explicit representation for T is 71 e'(X+Y+Z) e-%(y+2'u) e- %(Y+Z+U) ex_z _ e-%(y+z-u) eu ‘ 1 e-%(‘Y+z-u) T = (10) e-%(y+z+u) 1 e'u e-%(-Y+z+u) ex-z e-kC-y+z-u) e-%(-y+z+u) e-(x_y+z) k ,I From (T), - —N _ N N N N Z - Tr T - A0 + A1 + A2 + A3 (11) where Ai are the eigenvalues of T. T is called a "transfer matrix”20 11 In the following, we limit ourselves to the half— filled band case. This condition on the number gives <§ s.> a 3 log 2 = 0. i 88p Referring to eqs. (3) and (6), one sees that y is formally an effective magnetic field acting on the 81' Therefore, + V. (12) The transfer matrix becomes (e-(x+z) e-%(z-U) e-%(z+u) ex-z \I e-’(z-u) eu l e-%(z-u) T = (13) e‘;fiZ+tfl 1. e.u e-%(z+u) ex-z e-%(z-u) e-%(z+ifl e-(x+z) K /‘ Immediately, one sees that an eigenvector of (13) is the transpose of (l, 0, 0, -l) with eigenvalue -2e'Z sin hx(#0); furthermore, it can be seen that det T=0 so that the eigenvalues are obtained in simple closed form. For N + w, the free energy per particle is f = -kT log Am, where Am is the eigenvalue of maximum magnitude, which is ..Z 'Z A e cosh x + cosh u + [(e cosh x - cosh u)2 + 4e.Z cosh u] . (l4) 12 All these results are shown in Appendix A. ‘From this, the zero-field magnetic susceptibility is Z 1 3A 2A = -(E_§ = kT[-;7 ( 3m )2 + %_ 3 m] 3h h=0 m m 3h h=0 = B [1 + Ze-z-e-Z cosh x+1] e-2 cosh x+l + /K /K where (e.Z cosh x-l)2 + 4e-z. D m Similarly, the specific heat is given by C 3A 2 2 2 V _ 8 m 2 B 3 Am 2 d log A (16) —-(————)+—-——7=s-—— m —E Am 88 Am BB d82 To understand these results we investigate the ground state. Without the magnetic field, the electronic Hamiltonian in the case of the half-filled band can be rewritten in the form _ U 2 He — 7 2 Si + V E S S1+1 + NV 1 1 _ U _ 2 V 2 - (7 V) i Si + 7 §(Si + Si+l) + NV (17) The variable 81 can take on the values -1, 0, l and each summation in the above equation is a positive quantity. We first consider U > 2V; the minimum energy is obtained by simultaneously minimizing each summation in this case. This is obtained by requiring Si2 = 0 and Si + Sj = 0 for 13 all 1. Thus for U > 2V, a minimum energy eigenstate V1 has each site singly occupied. For the case U < 2V one must maximize ; Si2 while minimizing ; (si + sj)2. This leads to the cindition Si2 = 1 and (8: + Sj)2 = 0 for all i and j. These conditions imply S1 = l for all i on sub- lattice A and Si = -l for all i on sublattice B. Thus, ni++ni+ = 2 for 1 on sublattice A and nip-n1+ = 0 for 1 on sublattice B. A ground state W2 consists in this case of alternating empty and doubly occupied sites. For U > 2V the ground-state energy is NV and for U < 2V, it is %NU. This difference shows up strikingly in the zero-field susceptibility x as shown in Fig. A1. As is seen, for V/U 5 0.5, x is very similar to the atomic limit of the Hubbard model (V/U = 0). But for V/U > 0.5, a marked change occurs at low T, since in this case X + 0 (rather than 00) as T=0, due to the fact that 0 (whereas is not zero for many of the 2N degenerate state V1). In fact, it is easy to see from (14) that asymptoti- cally at low T U x = 48 exp [-s(2v - 7)], 2v > u (18) so that X is exponentially small at low T when 2V > U. In the other case, equation (14) gives the Curie law x z 8 2V S U (19) 14 1 54 o V/U = 0.75 —-—0-—_ 0.55 —-— H I gfi 0.50 __ _— mm o :2 0.00 \\ D 55 1.0‘ 0. Figure Al. 0.12 0.24 Inverse susceptibility x-1 temperature T. versus 0. 36 .48 15 In Fig. A2, we see that when V turns on, the peak of the specific heat becomes narrower and it moves to lower T; the lowest value occurs when 2V = U. As V continues to increase, the peak moves back to higher T, with a consi- derable additional sharpening. Clearly, the area under the large -V/U peak is appreciably greater than that for the small -V/U peaks; this is consistent with the easily proven facts, IA C 10g 2 V 00 V _ Io ET dT “{2 log 2 v U/Z U/Z (20) 'V It is interesting to note that the correct low-T behavior, equation (18) is rather different from what one might have guessed from continuity given equation (19), namely suscep- tibility ~B exp [- ecv - U/2)] for 2v 2 U. Finally, it is the insensitivity of x vs T for V < U/2 that shows that the addition of V to the zero-bandwidth Hubbard model will not significantly improve the theory in connection with the experimental x found for NMP-TCNQ. That is, the experimen- tal x-l vs. T is nearly straight over T ~ 30°tho T ~ 200°K with a slope corresponding to ~ 3kT rather than either the RT we find at kT + 0 or the maximum of 2 ZkT at higher T. Furthermore, the insensitivity of the theoretical x to the addition of h0pping terms to the zero-bandwidth Hubbard model in the Curie Weiss region14 suggests that perhaps adding b and V also will not help. , 1.5 ‘ 1.0‘ 0.5' Cv/k 16 o V/U = 0.75 —0—— 0.50 ———-— 0.25 - - ° 0.00 kT/U Figure A2. ‘ 0.24 0.36 0.48 Specific heat Cv versus temperature T ...: 10. 11. 12. 13. REFERENCES J. Hubbard, Proc. Roy. Soc. (London) A276, 238 (1963); 5311, 237 (1964); A281, 401 (1964). J. des Cloizeaux, J. Phys. Radium 20, 606 (1959). M. C. Gutzwiller, Phys. Rev. Letters 10, 159 (1963). N. P. Silva and T. A. Kaplan, Bull. Amer. Phys. Soc. 18, 450 (1973); N. P. Silva and T. A. Kaplan, AIP Conf. Proc. No. 1g, Magnetism and Magnetic Materials (1973) (p. 656) E. H. Lieb and F. Y. Wu, Phys. Rev. Letters 20, 1445 (1968). C. N. Yang, Phys. Rev. Letters 19, 1312 (1967) M. Takahashi, Progr. Theoret. Phys. 43, 1619 (1970). A. A. Orchinnikov. Sov. Phys. JETP 39, 1160 (1970). H. Shiba and P. A. Pincus, Phys. Rev. B5 1966 (1972). T. A. Kaplan, Bull. Amer. Phys. Soc. 13, 386 (1968). T. A. Kaplan and P. N. Argyres, Phys. Rev. B1, 2457 (1970). A. J. Epstein, S. Etmemad, A. F. Garito and A. J. Heeger, Phys. Rev. B5, 952 (1972). I. F. Shchegoler, Phys. Stat. Sol. (a) 12) 9 (1972). 17 14. 15. 16. 17. 18. 19. 20.° 18 D. Cabib and T. A. Kaplan, AIP Conf Proc. No. 10, Magnetism and Magnetic Materials (1972). T. A. Kaplan, Bull. Amer. Phys. Soc. 18, 399 (1973). R. A. Bari, Phys. Rev. B3, 2662 (1971) R. 8. Tu and T. A. Kaplan, Phys. Stat. 801. (b) 63, 659 (1974). Essentially similar results were found independently by G. Beni and P. Pincus, Phys. Rev. B9, 2963 (1974). This differs from the first equation which appears in ref. 17 because that equation is derived from ref. 11 while it is derived from ref. 16 here. The dif- ference only comes from a different choice of bii’ which merely causes a constant shift in the chemical potential. See, e.g. K. Huang, Statistical Mechanics, New York, Wiley, 1963. PART B THEORY OF SUPEREXCHANGE INTERACTIONS IN MAGNETIC INSULATORS 19 I. INTRODUCTION There is a large class of materials called mag- netic insulators; some examples: MnO, EuO, Man, KMnF3, KNiFS, Fe203, Cr203. They have an extremely low electri- cal conductivity; they are presumed to have localized electronic magnetic moments on the metal ions, the nonmetal ions being diamagnetic. The magnetism in these crystals arises either from incomplete 3d- or 4f— electron shells. The outer s-electrons are always importantly involved in the binding energy of the system. The s—electrons from the metal atom are pictured as being transferred to fluorine or oxygen atoms; e.g. in Man, the manganese (neutral Mn is 3d54sz) are considered Mn2+ while each fluorine is F'; in Mn0 we presumably have Mn2+ and 02- ions. Furthermore, the 2 closed shell ion F- or 0"(both ls 2522p6) are pictured as having much larger ionic radii than the positively charged cation. Hence, the 3d electrons on the Mn2+ are prevented from overlapping very strongly with their neighboring Mn 3d-e1ectrons, and therefore one might treat the overlap as small. At high temperature, the atomic moments behave paramagnetically. But at lower temperature, they undergo a phase transition to a magnetically ordered phase. The 20 21 critical temperatures TC range from ~1°K to =1000°K. TC has a different name for different magnetic ordering. For ferromagnetic crystals, the magnetic moments of the consti— tuent magnetic ion align parallel to one another, and TC is called the Curie temperature. Substances of these kind are Fe, Co, Cr02, EuO, GdBr3 (Fe and Co being metals, however). In an antiferromagnet the spins are ordered in an anti- parallel arrangement with zero net moment at temperature below the ordering or Neel temperature. For example, Mn0, FeO, and Cr are antiferromagnets (Cr being a metal). If however, one of the magnetizations is stronger than the other, it is to be expected that the difference between the two magnetizations will give rise to a strong magnetism. These substances are called ferrimagnetic, such as Fe304. Other types of ordering are also observed, e.g. spiral or helical ordering.1’2 Clearly the existence of such a Tc implies inter- actions between the atomic moments and it is the purpose of this thesis to contribute to the theory of these interactions. If TC > 1°K then the electron-electron and electron-core Coulomb interaction plus the electronic kinetic energy are generally accepted as giving the important source in the Hamiltonian of these interactions. Also essential is the fermion nature of electrons. The effective interactions that arise in this way are called exchange interactions. 22 They are generally thought to be essentially of the form + + -.z Jij 31-5]. (1) " th where Si is the spin of the i ion and Jij is the exchange parameter for ion i and j. If the interaction Jij involves only the overlap of free-ion 3d-states associated respec- tively with ions i and j, it is called direct exchange. If the exchange couplings exist between ions separated by one or several diamagnetic groups, Kramers3 pointed out that the magnetic ions could cause spin-dependent perturbations in the wave functions of intervening ions thereby transmitting the exchange effect over large distances. The latter effect is called superexchange. One of the stumbling blocks in the theory of superexchange is the derivation of the Heisenberg HamiltonianCILThe first formulation is in terms of the non- orthogonal atomic orbitals. The second formulation uses orthogonal "atomic” orbitals, namely Wannier functions. Because of the considerable mathematical advantage when dealing with a macroscopic system, we follow Anderson's approach which uses Wannier functions. However, the Wannier functions are not uniquely defined in the superexchange problem. Therefore, the convergence of the perturbation series will depend on the choice of the Wannier functions. A major presumption of Anderson's theory of superexchange4 is that the perturbation theory defined in terms of 23 "the exact" Wannier functions would be rapidly convergent. (The small perturbation parameter is the nearest neighbor overlap A). His Hartree-Fock (HF) definition of these exact Wannier states was shown by Silva and Kaplan5 to be unsatis- factory; in particular these states do not satisfy his re- quirement that they be nonmagnetic (nonmagnetic wavefunc- tions are by definition products W(;)o of spatial and spin functions in which the spatial functions are independent of spin 0). In the present paper, we nevertheless investigate the presumption of rapidity of convergence using a differ- ent variational definition, namely the nonmagnetic local- ized solutions in the thermal single determinant approxi- mation (TSDA).6’7’8 Our investigation is made first within a pre- viously studied 4-electron 3-orbital 3-site9 model of superexchange. Then we generalize to a 3-dimensional crystal. The function space is defined to have atomic functions ai centered on magnetic atoms and bi centered on diamagnetic atoms (with one orbital at each atom); a1 and bi are real; since they are presumed to be free-atom (or ion) states, they are not orthogonal (interatomic overlap integrals are nonzero). The states ai are related, for the crystal, by a lattice translation Operation L, and similarly, for the bi' We assume that each a1 and bi is inversion invariant about its respective atomic site. We also assume nearest neighbor overlap, the next nearest 24 neighbor and more distant overlap being taken to be zero. In case of-3 sites, we have the inversion operation I through the central site instead of the lattice translation operation L. This situation is shown in Fig. Bl. Wannier functions Ai and B1 are constructed in this space. We re- quire the constructed Wannier function to be real, ortho- gonal and to satisfy A1 + a B. + bi as the overlap A 4 0. i’ 1 The Wannier functions so constructed turn out to be not unique. Adding the spin to these Wannier functions, we obtain our complete set of orthonormal one—electron states. Occupying these one—electron states with electrons, we get the many-electron states, namely the Slater determinants. The model Hamiltonian is defined as the projection onto these many-electron states of the usual Hamiltonian containing electronic kinetic energy, electron-electron, and electron- nucleon Coulomb interactions. The Hamiltonian is divided into two parts. HO, the unperturbed part is diagonal in the basis defined in the previous paragraph, and is of zero order in the overlap A. The perturbation part is V E H-Ho which is of first order in the overlap. In the unperturbed ground state, there is by definition one electron on each magnetic site and two elec- trons on each diagmatic site. Because the Wannier functions as defined above are not unique, this division of HO and V is correspondingly not unique. As is well known, for the three-site model, the exchange, i.e., the splitting between 25 the lowest singlet and triplet is 0(A4). Therefore, it has contributions through 4th order perturbation theory. Each order is very sensitive to the choice of Wannier functions, as will be shown. Although the sum of all perturbation terms through fourth order is independent of this choice. We show however, that with the TSDA choice, the tetal energy to 0(A4) is given exactly by the lSt and 2nd order perturbation theory10 both in the 3-site case and in a 3-dimensional crystal. Previous attempts at calculating the exchange pa- rameter J within the general low-order perturbation approach have failed to give agreement with the experiment. In the most recent and elaborate attempt (for KMnFB), Fuchikami's11 straightforward perturbation calculation led to a J which is an order of magnitude too smalllz. But her11 Wannier func- tions were apparently chosen arbitrarily and she considered perturbation theory only through second order. Hence, our finding of extraordinary sensitivity to this choice the re- lative size of lSt through 4th order perturbation terms (we give simple example where the exchange constant comes only from fourth order perturbation theory!) suggest that she might not have obtained all the leading contributions to J. II. THE 3-SITE MODEL The system we are considering in this section is a single linear cluster with a diamagnetic atom in the center and magnetic atoms on each end. It is a four-electron sys- tem. The Hamiltonian of this system is the usual kinetic energy of electrons, the Coulomb interaction between elec- tron and electron, and between electron and nuclei: 4 1 4 H= z h(i)+-2— z v(i,j) (I) :1 j i 1 Here hi is the kinetic energy of the "i" electron plus its interaction with the atomic cores, and v(i,j) is the Coulomb interaction between electrons i and j. Imagine three atomic orbitals a ho, a1 localized 0’ at three atomic sites as indicated in Fig. Bl. We assume that the central one,bO is invariant under I, inversion through the central site Ibo=bo, and Iao=a]. Among these orbitals, we assume they are real, and have only nearest neighbor overlap A, i.e. f aobodv = A, aoa1 = 0. (2) From these one can construct the following set of orthogonal functions: 26 27 Howoz ouwm-oohcb o a Q .Hm ensued 28 A2 = C [a0 + ubo +va1] E9 = 0 [b0 + Y(ao+a])] (3) A _ 1 - C [a1 + ubo + vao] Here C and D are normalization factors, while u, v, y are parameters assumed to be real and of 0(A). A0, 80’ A1 satisfy the same symmetry properties as a0, bo These "Wannier functions" will have ’ a], i.e., I 52 = 52’ I 52 = A . —J * I a0, 59 + bo’ pressions for u, v, Y dictated by orthogonality are some- the properties A0 A1 + a1 as A + 0., The ex- what complicated; to leading order in A they give IJ2 V = - (f- + HA) Y=-(u+A). (4) A0, 80’ A1 are not uniquely defined because the set 2 I A A =-——-——— (A +AB -'—— A ) ° 1+A2/2 —9 A —3- 2 —1 1-122 1 B=——2-L-(B-——2-—(A+A)] (5) °1+1/2 —‘-’- l-A/Z —° —1 A=—-]———(A+).B-)‘2A) o 2_ o I 1+12/2 J- —- -— are orthonormal to 0(A) provided A + 0 as A + 0, and still satisfy the other symmetry conditions. After adding the spin to these "Wannier function", we have a six dimensional space. 29 Occupying the six spin orbitals Aoo, Boo, A10 (o=++), one obtains 15 four-electron determinants. From these we constructed those linear combinations 9]... 915 which are eigenfunctions of the total spin operators 92, S2 and the inversion I. The Hamiltonian matrix <¢i|H|¢j> is then reduced into block form. This is shown in Appendix B. For small enough A, the lowest singlet can be shown to come from a 4 x 4 submatrix connecting the states '9' II I —— [IAotBotBOIA 1 /§ +> - |A0+80+30+A1+>J I '9' ll 1 f [IAO+A0+BO+A]+> - IAO+A0+BO+A1+> + |A0+Bo+A1+A1+> - IAOIBO+A1+A1+>J ‘(6) '6' II I 3 IE [IAO+AO+BO+BO+> + |B0+BO+A1+A1+>J 9 +A1+> . 4 |AO+AO+A1 Similarly, the lowest triplet comes from a 2 x 2 submatrix connecting the states _ I 05 - 5E [IAO+80+80+A]+> + IAO+BO+BO+A1+>J -1 Q6 - 2 [|A0+AO+BO+A,+> + |AO+A0+BO+A1+> + |AO+BO+A1+A1+> + |AO+BO+A1+A1+>J . (7) Here | ... > is the normalized antisymmetrized product of 30 spin orbitals indicated. The matrix elements H = <91|H|oj> can be classified according to their order ii of magnitude as follows: (I) 0(A°) ”n 22 - - I - +A 1 1 |H|AO+BO+A +>+ + +A1+|H|AO+AO+A +A +> (8) I I I + + + + (2) 0(4) 12 3 (3) 0(A uses the matrix ofIH - /2[ - J /2[ + J 2 (9) 11> + ] 2 /Z (10) 1+A1~I> The detailed calculation of each matrix element technique given in Appendix C. Each diagonal sub- j is found to have nondegeneracy among the diaggonal elements. Because of our requirement that the grollnd state must be singly occupied on the magnetic sites 32 in the zero order overlap limit, we obtain the inequalities E °-E ° = - > 0 3 I o o o o o l o 1 E2 -E1 = g + - + - >0. ’ (ll) 00 00 54°-E,° = 2[ - ] + 2 - + 3 - 4>0 where * 3 = [a (1) h(l) b(l) d v] = fa(l)* b(2)* v(l,2) f(l) g(2) dv1 dv2 . (12) Hence, we apply nondegenerate perturbation theory up to 4th order to calculate the singlet-triplet energy differ- ence keeping all terms in the energy through 0(A4). We get 4 4 AE = Z E St - (i) - 1 AEst " 02(1) - 5(1)) . ~ (13) l 1=l where i is the orderor perturbation theory. The formulas for various order corrections to the energy from perturba- tion theory are displayed in Appendix 0. Up to 0(A4), each AEél) can be written as 33 .(I) _ ( . 2 2 2 2 E(2) = _( "13 + “14 ) _ ( ”12 _ ”56 5t ”33'”11 H44'HH H22'HH “66'H55) E(3) = 2“]2 (_121511 . ”24“]4 , (14) 5t “22'”11 ”33'”11 “44'”11 2 2 (4) , ”12 2 H23 H 24 E , ' '(H -H ) H -H + H -H st 22 11 33 11 44 11 Each Eéé) is a function of A because of (5) but the total energy difference to 0(A4) is independent of A. 4 = —o—o —o—l z 2 AEst 2<floflilvlfliflo> + w ' ‘2"P 21 w 21 4 2 2 2r 2 - ———-(x + —£9) - ———-(y + ——9) (15) "31 "21 "41 "21 = (0) - (0) - . (0) (0) . where Wm" Em in 15 obtained from Em - En given in (II) with the replacements A0 + 50’ B + Bo, A1 + A], and p ‘ (flolhlfif * (flofiolvlfioflfi + (flofiolvlgogc? T (AOAHVIEOB. > q = + <_A_O_B_0|v|_B_oB > + 2 r = + Amway + Moslvléoav (is) 34 x = <£o|h|fli> + Atlases + “4.2905193 - —o—o —o—l Y = ’ <50§01V1§051> z = 2 + 4 + 2 -—1 —o—o —1—o —o—o -1—o “2 - 2 The A independence is expected because changing A simply amounts to changing the basis set. In the following two sections we examine aspects of the behavior of the four individual terms Eéi), i=1, 2, 3, 4 under different choices of the Wannier func- tions A0, B A1, i.e., of A. We find, in particular, 0’ an a priori way of determining A which makes AEEE) = AEéfi) = 0; we also show that these individual terms AEgi) depend very sensitively on A. The utility of these findings, while not apparent from the present simple example (where the complete answer to 0(A4), (15) has been obtained), will be seen in connection with the generalization to a crystal. III. THE ONE-ELECTRON STATES IN THE THERMAL SINGLE DETERMINANT APPROXIMATION The well-known method of determining one-electron states is the Hartree-Fock approximation. As we noted earlier, this was Anderson's approach and it fails to give one-electron states that satisfy the requirement that they be nonmagnetic. This requirement is clearly impor- tant, since the magnetism is to be predicted by the result (effective Hamiltonian = -Z JijSi-Sj) of our perturbation theory. We therefore turn to a recently introduced devel- Opmenté, the thermal single-determinant approximation (TSDA), which is closely related to the Hartree-Fock ap- proximation in that it also determines "best" one-electron states. But in the variational context, it is better (more precisely, it is never worse) than the Hartree-Fock approximation.4 The idea of TSDA is as follows. We consider the minimum (or variational) principle of quantum statistical mechanics for.a system with a Hamiltonian Operator H and number of particle operator N, l Ftp) 2 TriptH-uN+B' log 0)] 2 Fe (1) 35 36 Here 9 is an arbitrary Hermitian and nonnegative Operator with unit trace, (i.e. a density operator) 8 = 1/kT, u is the chemical potential, and Fe is the exact grand-canoni- cal free energy for the system, namely -1 -B(H-UN) ; (2) '1'] ll F[pe] = 8 log Tr e De = e’BcH'UN)/Tr e-B(H-uN) (3) is the exact grand-canonical density matrix. For a system of interacting fermions, we have _ . . i 1 .. i i H - Z<1lh|J>CiCj + 7 Z X <13|Vlk1>ci cj c Ck (4) ij k2 9' where Ii>, lj> ... is any complete and orthonormal set of one- particle state,and C1 are the corresponding Fermion destruc- tion operators. The free energy in the TSDA is the minimum of F(p) for a trial density matrix of the form = e-B(H-HN)/Tr e ‘B(fi'UN) (5) where ~ - 1 A H ‘ z hii n1 T 2 1? Vij,ji ninj (a) is a function of the occupation-number operators ni cor- responding to a complete orthonomal set of one-particle states |i>. Here 37 hij = (ilhlj> Vij,k1 = <1j|v|kl> (7) Vij,kl = - Requiring stationarity Of F(p) under arbitrary variations of the states |i>, we find that the one-particle states |i> are determined by the system of TSDA equations, <> hij + v. . <<(ni-nj)nk>>=0 (8) i ik,jk plus the condition that they form a complete orthonormal set. The double brackets in the TSDA equations denote the average over the trial density matrix, <<0>> = Tr[p0] (9) where p is given by equations (5) and (6). Thus the equa- tions Obtained by first putting 0=ni into (9) and then 0=ninj form, together with (8), a set that must be solved self-consistently: Given the average in (8), solution of these equations yields a set of one-electron state |i> which then determines H and p and therefore the average occurring in (8). And the latter must match those "given" values that started the process. In our 3-site problem i, j, k run over 6 values corresponding to our 6-dimensiona1 single-electron function 38 space. Although this was derived from the grand canonical ensemble, these equations are applicable to our present considerations. Because we are interested only in T << U, the smallest unperturbed excitation energy from the ground state, the fluctuations in the number of particles is negligible. The reason for this restriction to low T is that we are after the best one-electron states which will define our perturbation expansion for the low-lying many- electron states. Let nio be the occupation number for site i with spin 0. Then equation (6) becomes ~- 1 A H - Z hii nio + 2 2 viojo',iojo'nionjo' (10) and (8) becomes <> hij + £§,Viokogjo£o'<<(niO-njo)n£o'>>=0 (11) Now for the 3-site case, isl, 2, 3 correspond to A B A1 respectively. For i=Ao, j=A o’ 0, we get 1 <> vA A A A + n ‘n n >> h <<( A01r AH) Ao+ o o’ 1 o '11 Act Alt AOA1 AOAO’AlAO +<<(n -n )n “ AO+ Alt Ao+>>v + <<(nA0+-nA1+) (12) A -n )n >>v Aot Alt Alt AOA1,A1A1 nA +>>VA A1,A A T <<(“ 1 o o o + <<(n ~n ) >>v + <<(n >> A0+ A1+ 9%1 AOB 0+ A v = AOBO,AlBo 0 B -n )n 051 O Ao+ Al B 39 we see that from the symmetry v = v and AOAO,A1Ao A0A1,A1A1 the left-hand side of the above equation vanishes identi- cally. Therefore, for i=AO, j=A1 the TSDA equation is satisfied by symmetry. For i=A, j=B, we get the TSDA equation <>hAoBo + <<(nAOI-nBOI)nAo+>>VAOAo’BOAO + <<(nAOI-nBo+)nBo+>>vAoBo’BOBO + <<(nAo+-HBO+)nA1+>> VA0A1,BOA1 +<<(nAo+-nBO+)nA1+>>vAOA1’BOA1 = <<(DAOI-nBo+)nA1I>>vAoA1’A1B0° (13) If we substitute equation (5) of section I into this equa- tion and use (5) and (6),it becomes a transcendental equa- tion for A thus obtaining A0, BO,A1. For kT << U we can neglect terms of 0(exp-BU/2) so that = 1, 0 ~ _ % = 0 at T + 0. The above TSDA equa- o o < > “A01 tion then becomes +++=0 (14) 1 to the leading order in A. To the leading order the fol- lowing matrix elements are vA A A t: O O’BO 0 VA B B B 2 o O’ O o v AOAI’BOAI Substituting A we get the lap A AT where W21 is given in (16) in Section II. overlap A v + A(v -v ) (15) AOEO’EOEO EOEO’EOEO flo§0250§0 v + A(v v 505143-051 AOEO’A B A 414 Al) the above equation into (14) and solving for TSDA value of A to leading order in the over- : -_E_ (16) w21 defined below (15) in section II, and p is To the leading order of A0 2 a0 + le0 E0 = bO - (A+u) (ao+a1) (17) 51 2 a1 + “be Substituting this into (16), we get a useful relation AT + u = 6 (18) where -A e O O o o o (19) w21 and e 2 h ; h+jd3¥' [a02(?')+b02(¥')+a12(¥')] (20) 41 o . . . . w21 is Obtained from w21 Simply by letting A0 + a0, BO 0, A1 + 31. Although AT is linearly dependent on p, 0 is independent of p. The solution of TSDA, A0, B0’ A1, should be independent of the choice of AC, BO, Al for given nonor- thogonal atomic orbitals a0, b0, 31 or independent of u from equation (3) of Section II (since A0, B0, A1 is simply a basis set). We can verify that A0, B A1 do satisfy this 0, by the useful relation (18). Take the TSDA of A and express it in terms of the nonorthogonal atomic orbitals. That is, substituting (3) into (S) in Section II we get _ 1 _ A - ———7—— { [c ATD(A+p)]ao + (Cp+ATD)bo 1+AT/2 Hz 2 -[C(—7— +pA) + DAT(A+p) + C AT/2]a1} (21) with C2 = l-ZpA-uz and 02=1. The coefficient of a0 to 0(A2) is C-ATD(A+p) 2 A2 92 = 1-..? him.) - ,1 = 1-9:. (22) 2 1 + AT/Z The coefficient Of a1 to 0(A2) is Z 2 2 -[ + uA-+ (e-u)(A+u) + £9%El—l = “(0A + %—) . (23) N": 42 To the 0(A2), we thus Obtain 02 92 A = (1- 0A- 7—) a0 + ebo - (9A + 7—)a1 (24) Similarly, we can prove that B0 is independent of p. Therefore this yields the Wannier functions in TSDA. To 0(A2) these are13 62 02 (1- 0A- -2—) [a0 + ObO - (9A + 7—)31] A B = (1- 02 + A2) [b0 - (A + 0) (a0 + a1)]. (25) IV. RESULTS OF THE 3-SITE MODEL To the leading order, eq. (9) Of Section II yield H = H 12 56 = + + + This is the left hand side of the TSDA equation (14) in the last Section. Therefore, from equation (8), (9), (10) and (14) of Section II, we immediately have to 0(A4) and zeroth order in exp (-8U) (3) = (4) = ABSt (AT) 0 , AEst (AT) 0 2 2 AE(2)(A ) H13 H14 (1) 5‘ T H33‘H11 H44’H11 ° nd Therefore using the TSDA states, the 2 order perturbation theory exhausts to 0(A4), the total energy splitting: AEst = AE§I)(AT) + AB§E)(AT) ° (2) To show how sensitive is the dependence of the individual perturbation terms on the choice of A, we con- sider the following simple example. Choose 43 44 A0 = a0 B0 = b - A(a +a1) (3) A1 = 31 instead of the TSDA states. Then dropping all intersite Coulomb matrix elements, one can see very easily that (1) = (2) _ (3) _ AEst 0, AEst - 0, AEst - 0 4 __4I| (1+2) AB = ABéfi) w 21 w31 w41 st (4) That is, the total splitting to 0(A4) comes entirely from the fourth order perturbation rather than from first and seCond order as is the case where TSDA orbitals are used. In summary, equation (1) or (2) shows that, within the model considered here, Anderson's hope that there exist "exact Wannier functions" which would lead to rapid convergence of the perturbation series has been fulfilled, the exact Wannier functions being the TSDA localized states given by (16) of Section III or alter- nately by (25) of Section III. In fact, the rapidity of convergence is probably better than expected, in that the 3rd and 4th order perturbation terms vanish identically to 0(A4) using the TSDA states. V. GENERALIZATION OF THE 3-SITE RESULTS TO A MANY-ATOM LATTICE Similarly to the 3-site case, the function space is defined to have atomic functions ai centered on magnetic atoms and bi centered on diamagnetic atoms. These func- tions are real; the ai are connected by lattice transla- tions, and similarly for the bi' We also assume nearest neighbor cation-anion overlap of these functions; the next nearest neighbor and more distant overlaps are assumed to be zero. The Wannier functions 51’ Bi are constructed in this space. We require the constructed Wannier functions to be orthogonal, and to approach the atomic functions ai and bi as the overlap approach zero. We also take them to satisfy "maximum similarity to the ai (bi)" i.e., we require Ai(Bi) to transform like ai(bi) under all lattice symmetry Operations that leave the point Ri unchanged. For simplicity we take the 31 to be s—functions; then we require the 51 to be real and invariant under the symmetry group of rotations that leave the point Ri fixed. Despite these restrictions, the Wannier functions so constructed are not uniquely defined (as is also true for the 3-site model). 45 46 For illustrative purposes, suppose we have a periodic linear chain with 2N+l unit cells,with one a-atom and one b-atom in each cell, as shown in Fig. BZ. The Wannier functions in cell "0" on atom a and b respectively will be AC a0 + A1(b_1+bo) +A2(b_z+b + ... + Ai(b_i+b. + 1) 1-1) T AN(b-N+bN-l) T AN+1bN T V1(a-1T31) T ”2(32Ta-2) T 4. vi(a_i+ai) + ... + vN(a_N+aN) (1) _ ' t 1 Bo — b0 + A1(ao+al) + A2(a_1+a2) + ... +Ai(a_i+1+ai) + ... I g I T ANCa-N+1TaN) T ”1(b-1Tb1) T “2(b-2Tb2) T + vi(b_i+bi) + ... + vN(b_N+bN) (2) There are 2N+l parameters (A1,A2, ... AN, AN+lvl...vN) in o o . ' AC; and Similarly 2N+l parameters in BO (41, 1%, 0 1 AN 9 t 1 ... 9N). A 3—dimensional example for which our assump- I ’ A N+1’ tions would reasonably apply is indicated in Fig. B3. There the dots represent cation (magnetic-ion) sites with wave-functions ai, the open circles stand for axion (diamagnetic-ion) sites with wave-functions bi' This is (III III, III. ‘I ...)I ' 47 Figure B2. Linear-chain model Figure B3. Peroskite structure ABFS, showing only the B-ions (.) and the F-ion (0). . 48 pertinent to magnetic materials of the perovskite struc- ture, e.g. KMnFS, KNiF314. In the latter example, - = N12+, 0 = F'; the potassium ions, which are not thought to con- tribute appreciably to the superexchange, are not shown. The model Hamiltonian is written in terms Of these Wannier functions as follows: 4.. . c. 10 30 + + H = Z X h..c Vij,k£Ciono'C£o'Cko (3) 1 .. 1] +72: 2:2 13 0 ij k2 oo' ‘10 are the destruction Operators corresponding respectively to the various Wannier functions. The hamiltonian is divi- ded into two parts, H = HO + V where NIP-i (4) H = Z h..n. + v.. .. n.n. o i 11.1 13,1] 1 j E ii which is clearly of zero order in the overlap A; here ni = 2 111 The perturbation part V = H-HO is lSt order 0 in the overlap. In the unperturbed ground state there is o' by definition one electron on each magnetic site g ac and the electrons on each diamagnetic site 2 nb0 = Our purpose is to derive for smalT overlap the appropriate spin Hamiltonian which will describe the low- lying magnetic states (including the magnetic ordering and thermal magnetic prOperties). We therefore look for our unperturbed states in a temperature region in which there is no magnetic ordering, i.e. T >> the magnetic ordering temperature. Similarly, we want our states to relate to 49 the physical situation where the number of electrons on a magnetic ion is approximately 1, hence we consider kT < z % z 1 aio i (na tna I) z 0 2 T (S) i i i j l (n n ,> 2: .... aio bjo 2 The TSDA equations between different magnetic sites are satisfied exactly by symmetry, a similar result being valid for different diamagnetic sites. The only nonzero TSDA equation are between the magnetic and diamagnetic sites. In the case of a linear chain, we have N+1 equations, namely between a0 and bi’ i=0, 1, ... N. From equation (11) of Section III let i=ao, j=bj and substitute the con- ditions (5). We get + X + 2 Z 31 1 i J + = 1- OJ J J 2 2 aifao J + bifbj (6) Besides N+1 of these equations, there are orthogonality 50 conditions of the following types: = 2(Ai+Ai+l)A + Zvi i = 1, 2, ... N = v v v (bo'31T T (Vi+vi+Vi-l+vi-l )A T (Ti+Ti) = v I <3 |A_N> 2(vN+uN)A + AN+1+AN+1 (7) ____ v r I 2(Ai+Ai+l)A + 291 Therefore, we have totally 4N+2 equations to determine 4N+2 parameters. To the leading order of overlap, we find f - Ae' f ‘ 43 A1 = _—__€'-€ Xi = _E'€' (8) where f = c' = c = and :‘z n 2 3 e 2 2 2 h + f d r —————— [z a +b +2 b ] IT'T'I 9, R, O £§O 2 The effective Hamiltonian HS defined in Appendix F can be expanded as follows: 1 HS = P[H0 + V-VQ H_E QV]P (9) = P[H + v-v 1 V+V 1 v 1 v ]p o H‘TE" fi‘TE— H -E °°' 0 0 O 0 O 0 where P is the projection operator, defined in Appendix F, 51 which projects onto the ground state manifold. Looking at the 3rd order perturbation term, we should begin with the ground state and come back to the ground state. Each term in the expansion will be prOportional to the product Of three matrix elements of V, and has the form where lg'>, |g> are in the ground- state manifold and |x>, Iy> are excited states. For the factor , either one electron or two electrons can be hopped from the ground state. If two electrons are hopped, this factor is of 0(A2). If only one electron is hopped, and the electron goes from one a-site to another a-site, this factor is also of 0(A2). If the electron goes from a b-site to an a-site, this factor is the following (see Appendix E): + 3 < 9° ¢°> = ... + 0 A 10 CaOCbO gIHI g ( ) ( ) where ... is the expression on the left hand side of TSDA equation (6). The rest Of the terms are higher order in the overlap. Therefore, if we use the TSDA basis, this factor is Of order A3 instead of A. The total product is th therefore higher than 0(A4). For the 4 order perturbation term, we can apply the same analysis. Therefore, both the 3rd and 4th order perturbation terms are of higher order as long as we use the TSDA states as basis-functions. Using nd this basis, we therefore need apply only through 2 order 52 perturbation theory to give all the leading-order contri- butions (0(A4)) to the effective Hamiltonian “5' As shown in Appendix G, this gives the Heisenberg spin Hamiltonian where “5 = -E. Jij gi'gj (11) J _ P k N where Jij - Jij+ Jij+ Jij Z 2|t..| J..k = - ———ll——, JP: v.. .. 13 1).. 1J 1J.J1 1) 2 J..N = _ z IV1j,2i' 13 2 (12) . .. 2'5 (on b-51te) 13’ here = 1,1 ti) hi) T i 2V12.j£ T Vii.jj T i _ Vik.jk V - .. - - _ 9 lJ’TT jo to 10 to 0 u. = - c (13) ij 1 j 0 ' e = E0, i,j are on a-sites, 2 is on a b-site,and |v> is an unperturbed ground state. VI. SUMMARY AND DISCUSSION In principle, one has to go through fourth order perturbation theory in order to exhaust all terms of 0(A4), a very complex task for realistic models of magnetic insu- lators. We showed that the nonmagnetic Wannier function rd and 4th solutions Of the TSDA equations make the 3 order perturbation terms vanish, for the simple models considered. This result proves Anderson's idea that there exist "exact Wannier functions" which make 3rd and 4th order perturba- tion terms negligible, the main contribution to the exchange parameter coming from lSt and 2nd order perturbation terms. The generalization of this type of consideration to more realistic models (containing more than one electron on a magnetic site and describing a crystal) is felt to be of considerable importance.’ The reason is that, in our opinion, a conclusive evaluation of Anderson's general perturbation theoretical approach as a practical means of calculating spin-Hamiltonian parameters is impossible with- out such considerations. That is to say, one must correctly evaluate all terms to 0(A4); hopefully an appropriate choice of Wannier functions will greatly simplify that task. 53 10. 11. REFERENCES S. Chikazumi and S. H. Charap, Physics of Magnetism (John Wiley 6 Sons, Inc., New York) p. 440. A. H. Morrish, The Physical Principles of Magnetism, (John Wiley 6 Sons, Inc., New York) p. 478. H. A. Kramers, Physica l, 182 (1934). P. A. Anderson, Solid State Physics 14, 99 (1963). N. P. Silva and T. A. Kaplan, AIP Conf. Proc. NO. 18, Magnetism and Magnetic Mat'ls. (1973), p.656. T. A. Kaplan and Petros N. Argyres, Phys. Rev. 33, 2457 (1970). In Ref. (5) another variational approach was studied. However, the present one is simpler and adequate to our more limited purpose. T. A. Kaplan, Bull. Am. Phys. Soc. 13, 386 (1968). Ken-Ichiro Gondaira, Tukito Tanabe, J. Phys. Soc. Japan 33, 1527 (1966), referred to as GT. GT also carried out perturbation calculations using these basis functions, but they did not discuss the corresponding 3rd and 4th order perturbation- theoretic terms. N. Fuchikami, J. Phys. Soc., Japan 33 871 (1970). 54 12. 13. 14. 15. 16. 17. 55 In Ref. 11, agreement with experiment is claimed. But to obtain that,the carefully calculated energy denominator U was reduced by a factor Of crudely 2; the physical origin of this reduction, if it pro- perly exists, is outside the simple low-order ( ~fourth-order) perturbation theory. These TSDA Wannier functions are the same as those derived by GT by a different method. Their method also differs from Anderson's HF theory although they use the term "Hartree-Fock" to describe their approach. However, if there is more than one electron on each magnetic site, their "Hartree-Fock" approach does not take care of Hand's rule while TSDA does. Therefore, it is expected that the TSDA will have different solutions in such cases. D. E. Rimmer, J. Phys. C (Solid St. Phys.) 3 329,(1969). R. D. Mattuck, A Guide to Feynman Diagrams in the Many- Body Problem, (McGraw-Hill Book Company, New York). L. I. Schiff, Quantum Mechanics, (McGraw-Hill Book Company, New York). T. A. Kaplan, unpublished lecture notes for a course in Magnetism given at Michigan State University in 1971 and 1972. APPENDICES APPENDIX A DIAGONALIZATION OF THE TRANSFER MATRIX The transfer matrix T is in the following form: rA B c 07 A = e'(XTz) 1 . - - (2+U) B e u 1 B B = e 2 T = where -%(z-u) (A1) 0 1 euC C=e = x-z gD B C A’ D e If we make a similarity transformation on T with the unitary matrix W (.1 0 0 _1_ f2 f2 0 1 0 0 S = (A2) 0 0 l 0 .1. o o ._1 /2 (TI \ the T is transformed into the following form: 56 57 (A+D /2B J20 0 \) /2B e.u l 0 5‘1 T s = (A3) /2c 1 e'1 0 0 0 O A-D k J Immediately, one Of the eigenvalues is A = A-D = -2 e‘2 sinhx. (A4) The other three eigenvalues are found from the following characteristic determinant (A+D)-A /2B /2C /2B e'u-A 1 = 0 . (A5) By using A+D = Ze-z coshx B2+C2 = 2e.Z coshu one can easily prove A+D /2B 2C /28 e'u 1 = 0. (A6) /2C 1 eu 58 Therefore, another eigenvalue is A1 = 0. The rest of the eigenvalues are found from the quadratic equation A2 - 2A(e'z coshx + coshu) + 4e'Z coshu (coshx-l) = 0. (A7) The solutions are: A 2 3 = (e'2 coshx + coshu) : [(e"z coshx - coshu)2 ’ + 4e'z coshu]1/2 (A8) AZ > A3 and AZ > A1 are Obvious. .Also A2 -|A O| > [(e"2 coshx + coshu) + (e'Z coshx - coshu)] - Ze'z sinhx = 2e'Z coshx - 2e'Z sinhx = e'CZTX)> 0 Hence 42 > |on (A9) Therefore, A2 is the eigenvalue of maximum.magnitude. APPENDIX B WAVEFUNCTIONS OF 3-SITE 4-ELECTRON LINEAR CLUSTER Let "t" denote the space orbital which is occu- pied by a spin up electron and "+" by a spin down electron, "1" denote the space orbital which is occupied by two elec- trons with the order of spin up followed by spin down. If the space orbit isn't occupied, we denote it by "0" and the order of space orbit is A0, 30, A1. With these conven- tions we get the following 15 4-e1ectron wavefunCtions on three sites: 1 a = —— [|+1+> - |+:+>] 1 x: 92 = % [II++> - IS++> + |++3> - |++1>] 4’3 = i [|110> + |0n>1 /2 94 = lsoz> 05 =._1 [|+t+> + |+:+>] /2 <16 = J; [|:++> + |¢++> + |++z> + |++¢>] 97 = % I|¢t+> + |3++> - |++t> - |++t>] g8 = I+t+> 59 60 99 = —1 [|:++> + |++:>] 2 v = _1 [|:++> — |+oz>1 10 111 = |+t+> Y =-—1 [|:++> + |++:>] 12 ,7 v = —1 [|z++> - |++z>] 13 ,7 v = —1-[|::0> - |0u>1 14 /7 v = 1 [|z++> - |x++> - |++s> + |++¢>] 15 2 These are eigenstates of 52, S2 and I with eigenvalues given as follows: Y. 1 2 3 4 S 6 7 8 9 10 11 12 13 14 15 From the above table, we see that the 15 x 15 matrix fac- tors into one 4 x 4 submatrix from the singlets, four 2 x 2 submatrices, corresponding respectively to (52. s I) = (2. o, -1). (2.1. -1). (2. -1, -1). z, (0, 0, -l), and three 1x1 matrices. APPENDIX C MATRIX ELEMENTS OF THE HAMILTONIAN OPERATOR WITH RESPECT TO DETERMINANTAL WAVEFUNCTIONS The Hamiltonian Operator for an n-electron system is taken to have the form H = Z h(i) + Z v(i,j) C1) 1 i = { (C4) 0 otherwise (case 2,3,4) The matrix element of H can be broken up into two pieces, one from the one-electron part, the other from the two- electron part. The matrix element Of the one-electron part is the following: 63 1 = 1 = _1 z (-1)p p[¢(1)¢(2) ...)* z h(i) n! p 1 3(4)Q QI9'(1) ¢'(2) ...) d1(1) dr(2) at I (¢1(1) 42(2) ...) 2 h(i) 3(4)Q 1 Q[9i(1) 95(2) ...] d1(1) dT(2) (CS) I (41(1) 92(2) ...1* I h(i) [9i(1) 45(2) ...] dT(1) dr(2) The 4th line follows from the fact that each of the permu- tations P merely affects the labeling of the variables of integration. The 61h line follows from the fact that Xh(i) is a sum of one-electron operators; any nontrivial permutation P produces two noncoincidences of one-electron states one of which integrates to zero because of the orthogonality. We then obtain immediately 2 <9i|h(l)|¢i> for case 1 i I = <¢n|h(l)|¢p> for case 2 (C6) 0 for case 3 and 4 64 For the matrix element of the two-electron part, the reduction proceeds similarly although now of all the permutations Q we must retain for each term v(i,j) the per- mutation that interchanges electrons i and j II 5 <9 | Z v(i,j)l9'> i ’ <¢i¢i|v(l,2)|¢j¢i>] for case 1 II at 2 [<¢i¢m|v(1.2)l¢i¢p> - 1 <¢i¢m|v(l,2)|¢p¢i>] for case 2 <¢m¢n|v(l,2)|¢p¢q>-<¢m¢n|v(l,2)|¢q¢p> for case 3 L0 for case 4 (C8) APPENDIX D VARIOUS ORDERS OF ENERGY CORRECTION ON THE PERTURBATION THEORY Suppose the total Hamiltonian H is divided into unperturbed part H0 and perturbed part V: H = H + AV 0 and H9 =E9 n n n 0= 00 Ho9n En9n therefore E = E° + AE(1) + 125(2) + n n n n If E; is nondegenerate we have the formula up to fourthls’16 order, as follows: Let E(11 = <9 °|vl9 °> n n n = o o vij <9i |v|9j > then Iv. '2 BOTT'Z ()1?) n . O O i#n Ei -En v v v v [2 E(3) = z z ni ij jn _ V in .. - >3 , n ifn an (B§O)'E£b))(E§O)-E£01) “n i#n (5§03-5E0))2 65 66 v. [2 2 BIT) = Z lvinl z -——l£“—2’ “ ifn E£°)—E§01 ' an (Hg-E3) v nivijvjn nign jgn (E(O) _E(o))2(E(0)_ E(O)) +Vn v .v..v. n1 1) in Z Z n 12m jxn (B§°)-E§°))(E§°)-B§°1) +V 2 Z vnivij v jk vkn i#n jfn kfin (BIO) BI°))(BI°)-BI°1)(BI°1-EI°1) 2 Ivin|2 Z " V APPENDIX B THE MATRIX ELEMENT FOR NEAREST-NEIGHBOR HOPPING OF ONE ELECTRON + o O - - <... AotAOIBO+...|H| ... A0+B0+BO+..> lo¥Bo A = + 22 O’ A jo#Bo+ = + z + 2 io i jo + I + 2 z - i0 -j§ + I + Zjio i#o + - .z j#0 0] JO 67 APPENDIX F DEGENERATE PERTURBATION APPROACH VIA EFFECTIVE HAMILTONIAN A Hamiltonian can be partitioned into an unper- turbed part Ho and a perturbation part V H=H+v (Fl) We want to solve H9n = En9n (F2) and we know 0 = O O H09n En9n (F3) we can write = o = 0 9n 9n + ”n , En En + An. (F4) Assume (F3) has a degenerate ground level Ei=E§=E§= ... §=e; E; > e for n > g. We also assume H0 is hermi- tean so that we can take 9; such that O O - <9n | 9m > - anm (F5) It is natural to call the degenerate ground state manifold (9°, 9;, ...9;) = subspace G, and all the excited states (9;,1 ...) = subspace X. From now on, 9; stands for the projection of 9D onto G, this projection being normalized to unity. Thus "n is in X and <9n | 9n> = 1 + . (F6) 68 69 That is, 9n is not normalized to unity; however, we can always normalize 9n by dividing by the square root of (F6) Of course. Substituting (F4) into (Fl) gives (HO+V) (93+nn) = (E;+An) (9; + ”n) or (H-En)nn = (En-H)9; = (An-V)9; (F7) or (Ho-ESMn + (V-An)nn = -(V-An)9; . (F8) If one defines idempotent Operators P and Q such that PW belongs to G and Q9 belongs to X, then we get P+Q=1, P2=P, Q2=Q, and PQ=QP=0. Clearly we also have Pnn=0, enn=nn, P93=9fi , and Q9fi=0, for n 5 g. Since H0 is diago- nal in {9;} , equation (F7) can be split into two equations with these relations by applying the Operators P and Q on both sides. pvonn = P(V-An)P9; (F9a) Q(H-En)an = -QVP9; (F9b) (F9b) gives Qn = -_____l__. QVP9° (F10) “ Q(H-Bn)Q n and (F9a) becomes 1 [PVP-PVQ-———————— QCH-En)Q QVP] 9n = AnP9n . (911) 70 By adding PHOP to the left hand side and En to the right we have Hs9n = En9n (F12) where the "effective Hamiltonian” H5 is 1 H s P[H-V ————————— Q(H-En)Q S v19 . (F13) This is of course a gxg matrix. If V is "small", we expand 1 1 m ' - —— = . .23 (-1)1[ —i—— Q(V-An)Q11 (F14) Q(H-En)Q Q(Ho-En)Q 1-0 Q(H-Bn)Q Up to the 4th order, the effective Hamiltonian become HS = P{Ho + v-v 1 V+V 1 v 1 v HO-EO H0.130 HO-EO - v 1 v 1 v 1 v }P (F15) One might note17 that this perturbation theory differs in an essential way from the "standard" textbook degenerate perturbation theory 1? in that here the "proper zero-order states" 9; are allowed to change as higher-order correc- tions are added, whereas in the standard method these 9; are fixed by the lowest order correction that removes the degeneracy. This difference can drastically affect the rate of convergence Of the expansion. APPENDIX G THE EFFECTIVE HAMILTONIAN IN SECOND ORDER PERTURBATION THEORY From equation (3) Of Section V + l H = Z 2 h.. c. c. + Z Z 2 v.. ii 0 11 1° 1“ ij ki 00' 13’k1 + + (:10. CjU'CRO'CkO’ (G1) H0 is defined as all terms Of 0(A°): + l H = Z 2 h.. c. c. + — E v.. .. o i o 11 io 10 2 ij 00' 13,1] + + C10)C O'Cj o'cio = 2 h.. n. + 2 v.. .. n. n. . 11 1 11,11 1+ 1+ 1 i + l 2' v. . n n (G2) 2 ij ij,ij i j The perturbation term is defined as V = H-Ho. The pro- jection operator P projects on those states with 1 if i on a s-site n. = { (G3) i on b-site 71 72 If Dv are single Slater determinants with Wannier functions singly occupied on magnetic sites, and doubly occupied on diamagnetic sites, then we have < > = O 6 E 6 DvIHOIDu Ev v“ e VU (34) E 0: E = = N(h +h +V ) + l}: v + 2 V + 2 v aa':aa' éb' bb',bb' ai ab,ab (GS) If there are N a-sites (magnetic sites), the degeneracy of B3 will be 2N. In order to apply the perturbation theory, the one-electron and two-electron interaction terms are analyzed as follows: ... ,‘~ H =2 Zh..c.c. = .+i j. (G6) 1 ij 0 13 10 30 Q1 Qpi refers to the terms for which i=j and i/ ‘j refers to terms with ifj. l + + = _ z . . . ' 1 + + l = ... 2 v.. .. c. c. 'C- c. 2' Z X 2 ij 11,3] 10 10 J0: J0 j2 i 00' v+ c+ c+ c c + l 2' Z 2 'v ii,j£ i0 10’ “20’ jo Z ik j 00' ik,jj + + 1 C. c c. c- + - 2' 2' 2 v. . 10 RC. J0, JO 2 ik 3'2, 00,! 1k,J£ + + io ko'cio' 30 = A + B + C + D (G7) 73 where 6 = -o, and A, B, C, D stand respectively for the four sums of the previous expression, taken in the same order as they appear. They may be represented by differ- ent types of graphs, in which each line corresponds to a C+C pair with the same spin index. G t J J R J k I +D2. l 1 +d A: f \ r i=j 1 o W A0 + A2 1<:::::> 3 o '3 O s I k J o =° r ~ B. r 2 1 2 ‘ I o 1 B1 '6' + + B2 1 O \l o‘ _ 3 ° 3', N x j <£:; j=i J (68) C: k + k=j r ‘ k 0 C1 _ o o + C2 j o . . . 3=1 1 _ . 1 O J . - _ o D. f O 1 N r k-£O k_..‘__2a ‘ o 0’ Do I D1 “TL 0 Oi=j 0’ + Y [i=g<::>j=k] J'J 74 Let us first consider the leading term PHP in the effective Hamiltonian, eq. (F15). The projection operators demand that only matrix elements of H among the G-states occur. Clearly CT C 10' j i=j; thus only the diagramo contributes from H. As to 06 is in G if and only if the two-electron interaction, we see similarly that only A0, DO and Y contribute. Hence, H (1) = PHP = P2 cf c. p+ l P z 2 v.. S 10 2 1 1‘ “i n' 'P id 1 ij GUI J: J O J0 + + v.. .. c. c. ,c. v 139J1 10 30’ 10,chP (G9) The first two terms of the above equation only give a con- stant, while the last term will give a Heisenberg Hamil- tonian, partially from the flip of a part of the electron spins on different a-sites. We can see this from the following: Here i,j indicate different a-sites, and we use Pni P = % + 812 , Pn P = % - Siz PCI+C1+P = 81+ , PCI+C1+P = 51- . (G10) Clearly % Ej go, Vij,ji Ciocjo'cio'cjo = ‘ i ?. z Vij,ji [nionjo+ciociécjo Cjo] = ‘ l .¥ Vij,ji 13 o 2 13 75 _1 [ni+nj++ni+nj++si+sj-+Si-Sj+] - 7 Ej Vij,ji 1.. [(7+s lz)(7+sJ ) + (7- sz)(7-sjz) + si.sj- +S.S]=-—1-)3v (1+2§o's’) (Gll) 1-j+ 2ij ij,ji I i j The contribution from 2nd theory has the form order perturbation (2) = _ 1 But H = H +A+Bl+B2+C1+C2+DO+D1+D2+d2+Y, and we see from the diagram that QAP=0. QYP=0, QB2P=0, and QD0P=0. Therefore QHP = Q(H1+B1+C1+D1+C2+D2+d2)P (613) _ + = PHQ - [QHP] P(H1+C1+B +D1+B2+D2+d2)Q (614) For the d2 graph, spin on different a-sites. a. a 0I the following operation will flip the 76 Although this operation gives a contribution to the Heisenberg Hamiltonian, we see immediately that the over- lap is higher order than 0(A4) no matter whether we consi- der 3 linear chain or perovskite lattice. Therefore, we can drop d2. Let R E H1+B1+C1+D1+D2. Aside from numerical factors, the effect of operator R is + . . equal to CiOCkO where 1 has to be on an a-51te, and k can be on an a- or b-site. Therefore QHP Q(R+C2)P PHQ P(R+BZ)Q (615) Because we start from a ground state, and must come back to a ground state, it is very easy to see from the graph that 1 _ 1 _ PRQ ——————— QCZP - PBZQ ———————-QRP — 0. (616) QCH-€)Q QCH-€)Q Therefore “5(2) arises from two kinds of term. The first one is called the Nesbet term; it comes from the "double hopping" i.e. 1 ~PB2Q —————7—— QC P QCHo-€)Q 77 I2 + + + + _ z 7 z 'Vij,22 _ _ _ Cfio'cfio'cjo'Cio'ciocjocfiocko ' ~i ' - 1, 1 oo Uij,£ c Here i,j have to be on different a-sites but 1 is on one . + + - .. = < .- . - b Slte’ and ”11,2 CJchacloczo vIHo|v>. |V> 15 a ground state. The above expression can further become 2 v.. | -2 Z X 11 21 + + - . + + ij 2 o Uij ; -e [nlonlocjociociocja niénlocjociociocjo] ’ = -z 2 Ivii zzlz 1i 2 U.. ’ -e ['1 * ni+nj++ni+nj+‘si+sj-'Si-Sj+] 1j,£ Ivij £1|2 1 1 1 = — 4L - _- .. .. 2 _ [ 1 + (7+Siz) (2 sz) + ('2' Siz) 13 2. Uij 2’ e: 1 (37552) - 51+Sj-‘51-sj+] 2 |v.. I = 11,12 [7 + zsizs.z+s.+s._+s._ s.+] ij 1 U.. — 8 J 1 J 1 J 11.2 lvi' kill 1 = Z 2 3’ [— + 2 §.. §.] . (513) i. 2 U 2 1 J J .. 8 13,1 nd Another term in the 2 order perturbation theory is called the kinetic exchange term. It comes from the "single hopping", and has the form 78 z (619) Y EW where Iv>,|u> are ground states, and Iy> is an excited state. But = = ; + IV) = Ciockolu> ... = If k is on a b-site, this matrix element is 0(A3) for the TSDA basis from eq. (10) of Section V. If k is on an a-site, D2 is 0(A2), therefore, we can drop D2. Hence the Operator corresponding to (619) is 1 'P(H1+C1+B1+D1)Q W Q(H1+B1+C1+D1)P. ((320) 0 Clearly Q(H1+B1+C1+D1)P + + = X'h.. X c. c. + 2'2 .. .. .-+v.... ._ . . Q[ij 1] o 10 J0 ij 0 (v13:31 n10 IJJJnJg)c10CJO . ij + = . + + £3 E Evi£,jzciocjon2] P Q1? gTijoCio Cjop' (G21) where 1.. = h.. + v.. .. n.- + v.. ,,n.- 13 13 11,31 10' 313,33 30 13 + i vi£,j£ “2 79 and Zi’j means sum over 2 with 2% i,j. Taking the matrix element of (620) between ground states, we were able to do the intermediate state sum over the y which contribute in order A4. Because the energy denominator is a constant for all such y, we obtain 2' X 1 + + ij 00' U3; (622) 2'21 = - + ij 0 U};(VITijonion-130)Tjio-TijociociocjocjoTjiol11> Here i,j run over 3 -sites only, and + Uij - c .. ij “ij hij * i V12,j2 Using the expression of 1.. and making a little bit of 130 algebraic manipulation, the part of the effective Hamil- tonian coming from this term is found to be 1 - - - 2 - Z 1 2 hi n - h n - - h.. ij —U;;— o [( 3+vji jo) nio( ji +vji jo) I 13| nionjo - (hij +v: i) CiOCiOCjOC jO(hji’r vji)] lfii.|2 1 - - a: = - 2' ___l__ - 2' ___ z[h n + h..v..n. n.- U <1 ij vji ion jo 31.31 10 30 U.. .. 13 1] 80 2 ’ * 2 + + + Ivjil nionjo - Ihij+vjil (nionjo+ciociocjocjo) 2 _ + (hijvji fijiv31 lvjil ) nionjo] (G23) It |7- 2|t..|2 _-%2__1_L_+21__}J__§1§J 13 13 UiJ’ where ij .. = h.. + 2 . . + .. .. + v. . t11 13 i V1£.J£ v13.31 i 1k.Jk- "1111111111111111mms 1 7967