Hubbard-Like Hamiltonians for Interacting Electrons in S, P and D Orbitals

Hubbard-like Hamiltonians for interacting electrons in s, p and d orbitals

M. E. A. Coury∗ Imperial College London, London SW7 2AZ, United Kingdom and CCFE, Culham Science Centre, Abingdon, Oxfordshire OX14 3DB, United Kingdom

S. L. Dudarev CCFE, Culham Science Centre, Abingdon, Oxfordshire OX14 3DB, United Kingdom

W. M. C. Foulkes Imperial College London, London SW7 2AZ, United Kingdom

A. P. Horsﬁeld Imperial College London, London SW7 2AZ, United Kingdom

Pui-Wai Ma CCFE, Culham Science Centre, Abingdon, Oxfordshire OX14 3DB, United Kingdom

J. S. Spencer Imperial College London, London SW7 2AZ, United Kingdom

Hubbard-like Hamiltonians are widely used to describe on-site Coulomb interactions in magnetic and strongly-correlated solids, but there is much confusion in the literature about the form these Hamiltonians should take for shells of p and d orbitals. This paper derives the most general s, p and d orbital Hubbard-like Hamiltonians consistent with the relevant symmetries, and presents them in ways convenient for practical calculations. We use the full conﬁguration interaction method to study p and d orbital dimers and compare results obtained using the correct Hamiltonian and the collinear and vector Stoner Hamiltonians. The Stoner Hamiltonians can fail to describe properly the nature of the ground state, the time evolution of excited states, and the electronic heat capacity.

PACS numbers: 31.15.aq, 31.15.xh, 31.10.+z

I. INTRODUCTION the Hamiltonians is problematic. Indeed, according to Dagotto2, “the discussions in the current literature re- garding this issue are somewhat confusing”. The starting point for any electronic structure calculation The Hubbard Hamiltonian9 is both the simplest and best is the Hamiltonian that describes the dynamics of the known Hamiltonian of the form we are considering, and electrons. Correlated electron problems are often formu- is valid for electrons in s orbitals. Hubbard was not the lated using Hamiltonians that involve only a small num- first to arrive at this form; two years earlier, in 1961, ber of localized atomic-like orbitals and use an effective Anderson wrote down a very similar Hamiltonian10 for Coulomb interaction between electrons that is assumed electrons in s orbitals with on-site Coulomb interactions. to be short ranged, and thus confined to within atomic Furthermore, in the appendix of the same paper, the sites. However, there is widespread variation in the liter- Hamiltonian was extended to two orbitals, introducing ature over the form chosen for these Hamiltonians, and the on-site exchange interaction, J. Unfortunately, this they often do not retain the correct symmetry proper- extra term is not rotationally invariant in spin space. ties. Here we derive the most general Hamiltonians of In 1966, Roth11 also considered two orbitals and two pa- this type that are rotationally invariant and describe elec- rameters, producing a Hamiltonian that is rotationally trons populating s, p and d orbitals. invariant in spin space. In 1969 Caroli et al.12 corrected

arXiv:1507.04613v3 [cond-mat.str-el] 12 Nov 2015 Despite often missing terms, multi-orbital Hamiltonians Anderson’s Hamiltonian for two orbitals, again making it with on-site Coulomb interactions have been used suc- rotationally invariant in spin space. However, the Hamil- cessfully to describe strong correlation effects in sys- tonians of Roth and Caroli et al. do not satisfy rota- tems with narrow bands near the Fermi energy. Ap- tional invariance in orbital space. This was corrected by plications include studies of metal-insulator transitions1, Dworin and Narath in 197013, who produced a Hamilto- colossal magnetoresistant manganites2, and d band su- nian that is rotationally invariant in both spin and orbital perconductors such as copper oxides3 and iron pnictides4. space. They generalized the multi-orbital Hamiltonian to The Hubbard Hamiltonian has also been used to sim- include all 5 d orbitals, although they only included two ulate graphene5,6, where both theory and experiment parameters: the on-site Hartree integral, U, and the on- have found magnetic ordering in nanoscale structures7,8. site exchange integral, J. However, the lack of consistency in the forms used for The next contribution to the multi-orbital Hamiltonian 2 was by Lyon-Caen and Cyrot14, who in 1975 considered two-electron inter-site Coulomb interactions means that a two-orbital Hamiltonian for the eg d orbitals. They screening effects are missing. Thus, even though the pa- introduced U0 U = 2J, where U0 = Vαα,αα is the self- rameters that define the s, p and d Hamiltonians appear − interaction term, U = Vαβ,αβ is the on-site Hartree in- as bare on-site Coulomb integrals, screened Coulomb integral, and J = Vαβ,βα is the on-site exchange integral, tegrals should be used when modelling real systems; this important for linking these parameters. Here V is the significantly reduces the values of the parameters. Coulomb interaction between electrons, and α and β are Although it is possible to find the on-site Coulomb inte- indices for atomic orbitals. Lyon-Caen and Cyrot also grals using tables of Slater-Condon parameters26,27 or the † † included the pair excitation, JcÎ1,↑cÎ1,↓cÎ2,↓cÎ2,↑, where closed forms for the integrals of products of three spheri- cˆ† andc ˆ are fermionic creation and annihilation opera- cal harmonics found by Gaunt28 and Racah29, the point tors. The new term moves a pair of electrons from a of this paper is to remove that layer of obscurity and completely occupied orbital, 2, on site I, to a completely present model Hamiltonians written in a succinct form unoccupied orbital, 1, on site I. Such double excitations with clear physical meaning. The Slater-Condon param- are common in the literature15–18. eters and Gaunt integrals were not used in the deriva- In 1978, Castellani et al.19 wrote down the three-orbital tions of the on-site Coulomb interaction Hamiltonians 20 Hamiltonian for the t2g d orbitals in a very clear and presented here; however, they did provide an indepen- concise way: its form is exactly what we find here for p dent check to confirm that the tensorial forms derived orbital symmetry. They too make note of the equation here are correct. The link between the spherical har- U0 U = 2J and include the double excitation terms. monics Coulomb integrals, the Slater-Condon parameters − 30 The next major contribution was by Ole´sand Stollhoff21, and the Racah parameters is explained clearly in . The who introduced a d orbital Hamiltonian with three inde- transformation to cubic harmonics is straightforward; an pendent parameters. The third parameter, ∆J, repre- example of which has been included for the p-orbitals in sents the difference between the t2g and the eg d orbital Section II C. exchange interactions; they estimated ∆J to be 0.15J. We note that Rudzikas31 has derived rotationally invari- In our discussion of d orbitals we adopt their notation. ant Hamiltonians for p and d orbital atoms. However, his Unfortunately, their Hamiltonian is not rotationally in- derivation is different from ours and his d orbital Hamil- variant in orbital space. tonian is partially expressed in terms of tensors; the form In 1989, Nolting et al.22 referenced the Hamiltonian pro- we present in this paper is written in terms of physically posed by Ole´sand Stollhoff but discarded the double ex- meaningful operators. citation terms and ignored the ∆J term. Their Hamilto- The p and d symmetry Hamiltonians are described in nian will be called the vector Stoner Hamiltonian in this Section II (see Appendix B for full derivations) and com- paper. Since then, most papers have used a Hamiltonian pared with the vector Stoner Hamiltonian. In Section similar to that of Nolting et al., with only two parame- III we show the ground state results for both our Hamil- ters; see for example references 23–25. The vector Stoner tonian and the vector Stoner Hamiltonian for p and d Hamiltonian is often simplified by replacing the rotation- orbital dimers. We find that, although the results are of- 2 ally invariant moment-squared operator, mˆ = mˆ mˆ , by ten similar, there are important qualitative and quanti- 2 · mˆ z, yielding the collinear Stoner Hamiltonian. tative differences. In Section IV we show the differences The multi-orbital Hamiltonians derived here consist that emerge when evolving a starting state for a p or- of one-particle hopping (inter-site) integrals and two- bital dimer under our Hamiltonian and the vector Stoner electron on-site Coulomb interactions. The on-site Hamiltonian. We also calculate a thermodynamic prop- Coulomb interactions for s, p and d orbital symmetries erty, the electronic heat capacity of a p orbital dimer, are presented in a clear tensor notation, as well as in and obtain significantly different results when using our terms of physically meaningful, rotationally invariant (in Hamiltonian and the vector Stoner Hamiltonian. The both orbital and spin space), operators that include all eigenstates of all Hamiltonians are found using the full of the terms from the previous papers as well as addi- configuration interaction (FCI) method as implemented tional terms of the same order of magnitude that have in the HANDE code32. not been previously considered. The d orbital Hamil- tonian presented in section II D corrects the d orbital Hamiltonian proposed by Ole`sand Stollhoff21 by restor- ing full rotational invariance in orbital space. For an II. THEORY s orbital Hamiltonian, the on-site Coulomb interactions may be described using one independent parameter. Two independent parameters are required for p orbital sym- A. General form of the Hamiltonian metry, and three for d orbital symmetry. Furthermore, the method presented can be extended to f and g orbital We start with the general many-body Hamiltonian of symmetry and generalized to atoms with valence orbitals electrons in a solid, expressed in second-quantized form of multiple different angular momenta. using a basis of localized one-electron orbitals and ig- The use of a restricted basis set and the neglect of the noring terms that do not include electronic degrees of 3 freedom33: B. The one band Hubbard model: s orbital ˆ X X † symmetry H = tIαJβcÎα,σcˆJβ,σ IαJβ σ The Hubbard Hamiltonian is applicable only to one band X X X † † + VIαJβ,KχLγ cÎα,σcˆJβ,ξcˆLγ,ξcˆKχ,σ. models where the one orbital per atom has s symmetry. IJKL αβχγ σξ This gives a simple form for the on-site Coulomb inter- (1) action tensor, U0 = Vαα,αα; as there is only one type of † orbital, there is only one matrix element. We can use this Herec Îα,σ andc Îα,σ are creation and annihilation op- result to simplify the Coulomb interaction part of Eq. (2) erators, respectively, for an electron in orbital α on site and express it in terms of the electron number operator or I with spin σ. Upper case Roman letters such as I, J, the magnetic moment vector operator, defined as follows: K and L refer to atomic sites, the lower case Greek letters α, β, χ and γ represent localized orbitals, and σ and X † nˆ = cˆ cˆ , (5) ζ indicate spin (either up or down). The one-electron α,ζ α,ζ αζ contributions to the Hamiltonian are encapsulated in the X † tIαJβ term that includes the electronic kinetic energy and mˆ = cˆα,ζ σζζ0 cˆα,ζ0 , (6) electron-nuclear interaction. The electron-electron inter- αζζ0 action terms are represented by the Coulomb matrix el- x y z σζζ0 = (σζζ0 , σζζ0 , σζζ0 ), ements VIαJβ,KχLγ . We now make two approximations: we only retain where σ are the Pauli spin matrices and the sum over α electron-electron Coulomb interactions on-site and we re- has only one term as there is only one spatial orbital for strict the basis to a minimal set of localized angular mo- the s case. Hence mentum orbitals. Thus we have just one s orbital per site for the s-symmetry Hamiltonian, three p orbitals per site 1 X X † † 1 2 V cˆ cˆ cˆ cˆ = U0 :n ˆ :, (7) 2 αβ,χγ α,σ β,ξ γ,ξ χ,σ 2 for the p-symmetry Hamiltonian, and five d orbitals per αβχγ σξ site for the d-symmetry Hamiltonian. The Hamiltonian can then be written as follows: or, equivalently, ˆ X X † H tIαJβcÎα,σcˆJβ,σ ≈ 1 X X † † 1 2 IαJβ σ V cˆ cˆ cˆ cˆ = U0 : mˆ :, (8) 2 αβ,χγ α,σ β,ξ γ,ξ χ,σ −6 1 X X X αβχγ σξ + V I cˆ† cˆ† cˆ cˆ , (2) 2 αβ,χγ Iα,σ Iβ,ξ Iγ,ξ Iχ,σ I αβχγ σξ where we have use the normal ordering operator, ::, to remove self interactions. The action of this operator is to where V I = V . To improve readability, from αβ,χγ IαIβ,IχIγ rearrange the creation and annihilation operators such this point onwards we drop the site index, I, from the that all the creation operators are on the left, without Coulomb integrals and creation and annihilation opera- adding the anticommutator terms that would be required tors as we shall always be referring to on-site Coulomb to leave the product of operators unaltered; if the rear- interactions. rangement requires an odd number of flips, the normal The on-site Coulomb interaction, V , is a rotation- αβ,χγ ordering also introduces a sign change. For example ally invariant tensor if the atomic-like orbitals are angular momentum eigenstates. To demonstrate this we note X :n ˆ2 : = :c ˆ† cˆ cˆ† cˆ :, that applying the rotation operator Rˆ to an atomic-like α,ζ α,ζ β,σ β,σ αβσζ orbital φα of angular momentum ` gives X † † X = cˆ cˆ cˆ cˆ . (9) Rˆ φ = φ d` (R), (3) α,ζ β,σ β,σ α,ζ | αi | βi βα αβσζ β where d` (R) is the (2`+1) (2`+1) matrix that corre- If the normal ordering operator is not used in Eqs. (7) βα × and (8) then additional one electron terms have to be Rˆ sponds to the rotation in the irreducible representation subtracted to remove the self interaction. of angular momentum `. By making the change of vari- The mean-field form of this Hamiltonian has been in- x Rrˆ x0 Rrˆ 0 able = and = , it is straightforward to show cluded in Appendix A 1. that ∗ ∗ 0 0 Z φασ(r)φ 0 (r )φχσ(r)φ 0 (r ) V = drdr0 βσ γσ , αβ,χγ r r0 C. The multi-orbital model Hamiltonian: p orbital | − | ` ∗ ` ∗ ` ` symmetry = dα0α(R) dβ0β(R) Vα0β0,χ0γ0 dχ0χ(R)dγ0γ (R), (4) Consider the case where the local orbitals, α, β, χ and γ which demonstrates that Vαβ,χγ is indeed a rotationally appearing in Eq. (4) are real cubic harmonic p orbitals invariant tensor. with angular dependence x/r, y/r, and z/r. In this case 4

` the rotation matrices, d 0 (R), are simply 3 3 Carte- where the vector angular momentum operator for p or- α α × sian rotation matrices. This makes Vαβ,χγ a rotationally bitals on site I is deﬁned to be invariant fourth-rank Cartesian tensor, the general form of which is34, X Lˆ = i ( , , )ˆc† cˆ , (17) V = Uδ δ + Jδ δ + J 0δ δ , (10) 1βα 2βα 3βα α,σ β,σ αβ,χγ αχ βγ αγ βχ αβ χγ αβσ 0 where U = Vαβ,αβ, J = Vαβ,βα and J = Vαα,ββ, with α = β. By examination of the integral in Eq. (4), we note6 that the Coulomb tensor has an additional symme- and µβα is the three-dimensional Levi-Civita symbol. try when the orbitals are real (as are the cubic harmonic p An equivalent expression is orbitals), namely Vαβ,χγ = Vχβ,αγ and Vαβ,χγ = Vαγ,χβ, and hence J must be equal to J 0. Thus we ﬁnd 1 X X † † Vαβ,χγ cˆα,σcˆβ,ζ cˆγ,ζ cˆχ,σ Vαβ,χγ = Uδαχδβγ + J (δαγ δβχ + δαβδχγ ) . (11) 2 αβχγ σζ

This recovers the well known equation U0 = U + 2J, 1 1 1 14,19 = U J :n ˆ2 : J : mˆ 2 : where U0 = Vαα,αα , and shows that the most gen- 2 − 2 −2 eral cubic p orbital interaction Hamiltonian is defined by 35 ! exactly two independent parameters . In Appendix B 1 X 2 we give a full derivation of Eq. (11) using representation + J : (ˆnαβ) : , (18) theory. αβ In passing we observe that symmetric fourth-rank isotropic tensors can be found in other areas of physics, such as the stiffness tensor in isotropic elasticity theory36 where the final term corresponds to on-site electron hopping37, Ciklm = λδikδlm + µ(δilδkm + δimδkl), (12) where λ is the Lamécoefficient and µ is the shear mod- X † ulus. nˆαβ = cˆα,σcˆβ,σ. (19) Transforming Eq. (11) into a basis of complex spherical σ harmonic p orbitals with angular dependence of the form Y`m(θ, φ) is straightforward. We write Vmm00,m0m000 = P ∗ ∗ Eq. (16) embodies Hund’s rules for an atom. By mak- lmαl 00 lm0χlm000γ Vαβ,χγ , where lmα is defined by αβχγ m β ing the substitution mˆ = 2Sˆ, we see that the spin is sph X maximized first (prefactor 2J) and then the angular p = l p , (13) m mα α momentum is maximized (prefactor− 1 J). α − 2 and has the following values The mean-field form of the above p orbital Hamiltonian is given in Appendix A 2.  √1 √i 0  2 − 2 [lmα] =  0 0 1  . (14) √1 √i 0 − 2 − 2 The result of applying this transformation is D. The multi-orbital model Hamiltonian: d orbital Vmm00,m0m000 =Uδmm0 δm00m000 + Jδmm000 δm00m0 symmetry m+m0 + ( 1) Jδ−mm00 δ−m0m000 , (15) − where m, m0, m00 and m000 are spherical harmonics indices The on-site Coulomb interaction for cubic harmonic d orbitals can be expressed as going from 1 to +1, U = V10,10, and J = V10,01. Using Eq.− (11) as a convenient starting point, it is straightforward to rewrite the on-site Coulomb Hamil- tonian in terms of rotationally invariant operators:31 1 Vαβ,χγ = Uδαχδβγ 2 1 X X † † Vαβ,χγ cˆα,σcˆβ,ξcˆγ,ξcˆχ,σ 5 2 + J + ∆J (δαγ δβχ + δαβδγχ) αβχγ σξ 2 ! 1 2 2 2 X = (U J) :n ˆ : J : mˆ : J : L : , (16) 48∆J ξαabξβbdξχdtξγta , (20) − 2 − − − abdt 5 where ξ is a five-component vector of traceless symmetric Since each term in the sum acts on the coordinates of 3 3 transformation matrices defined as follows one electron only, the operators Lˆµ and Lˆν appearing in ×  1  √ 0 0 that term measure the angular momentum of one elec- − 2 3 tron only, not the whole system. More detail on the Qˆ2 [ξ ] =  0 √1 0  , 1ab  − 2 3  operator is included in Appendix C. Eq. (22) is similar √1 31 0 0 3 in form to that found in Rudzikas , the difference being  0 0 1  that our Hamiltonian is expressed in terms of physically 2 meaningful operators. [ξ2ab] =  0 0 0  , 1 0 0 The mean-field form of this Hamiltonian is given in Ap- 2 pendix A 3.  0 0 0  1 [ξ3ab] =  0 0 2  , 1 0 2 0 E. Comparison with the Stoner Hamiltonian  1  0 2 0 1 [ξ4ab] =  2 0 0  , The Stoner Hamiltonian for p and d orbital atoms, as 0 0 0 generally defined in the literature, has the following form 25  1  for the on-site Coulomb interaction: 2 0 0 1 [ξ5ab] = 0 0 . (21)  2  1 1 2 1 2 − VˆStoner = U J :n ˆ : J :m ˆ :, (24) 0 0 0 2 − 2 −4 z We use the convention that index numbers (1, 2, 3, 4, 5) correspond to the d orbitals (3z2 r2, zx, yz, xy, x2 y2). where J, in this many-body context, is the Stoner pa- See Appendix B 2 for a derivation− of Eq. (21) using− rep- rameter, I. However, in the mean field, the Stoner pa- resentation theory. We have chosen U to be the Hartree rameter should not be taken to be equal to J due to the self-interaction error; see reference38. This form clearly term between the t2g orbitals, J to be the average of the breaks rotational symmetry in spin space, which can be exchange integral between the eg and t2g orbitals, and 2 2 ∆J to be the difference between the exchange integrals restored by substituting :m ˆ z : with : mˆ : for eg and t2g. That is, U = V(zx)(yz),(zx)(yz), J = 1 1 1 2 1 2 V(zx)(yz),(yz)(zx) + V(3z2−r2)(x2−y2),(x2−y2)(3z2−r2) , Vˆ ˆ 2 = U J :n ˆ : J : mˆ : . (25) 2 m Stoner 2 − 2 −4 and ∆J = V(3z2−r2)(x2−y2),(x2−y2)(3z2−r2) V . This choice of parameters is the− (zx)(yz),(yz)(zx) In this paper we will compare our Hamiltonians, Eqs. (18) same as was used by Ole´s and Stollhoff21; Eq. (20) and (22), with Eq. (25), which we shall call the vector generalizes their result and restores rotational invariance or mˆ 2 Stoner Hamiltonian. By starting with the vector in orbital space. Note that, in the mean field, the on-site Stoner Hamiltonian and working backwards, it is possible block of the density matrix in a cubic solid is diagonal; to find the corresponding tensorial form of the Coulomb this will cause the mean-field version of Eq. 22) to reduce interaction: to that of Ole´sand Stollhoff21. However the presence of interfaces, vacancies or interstitials will break the mˆ 2 Stoner V = Uδαχδβγ + Jδαγ δβχ. (26) cubic symmetry make the on-site blocks of the density αβ,χγ matrix non-diagonal. Therefore the use of the complete This is very similar to the tensorial form for the p-case Hamiltonian is recommended in mean-field calculations Coulomb interaction, Eq. (11), except that it is missing for systems that do not have cubic symmetry. a term, Jδαβδγχ. The lack of this term means that the Rewriting Eq. (20) in terms of rotationally invariant op- vector Stoner Hamiltonian does not respect the symme- erators gives try between pairs αχ and βγ evident from the form of mˆ 2 Stoner mˆ 2 Stoner 1 1 2 1 2 the integral in Eq. (4), i.e., Vχβ,αγ = Vαβ,χγ . As Vˆtot = U J + 5∆J :n ˆ : (J 6∆J): mˆ : 6 2 − 2 −2 − we shall see later, this omission changes the computed results. X 2 2 2 + (J 6∆J) : (ˆnαβ) : + ∆J : Qˆ : , (22) − 3 αβ III. GROUND STATE RESULTS where Qˆ2 is the on-site quadrupole operator squared. The quadrupole operator for a single electron is defined A. Simulation setup as

1 1 2 Qˆµν = LˆµLˆν + Lˆν Lˆµ δµν Lˆ . (23) Here we investigate the eigenstates of dimers using both 2 − 3 our Hamiltonians, Eqs. (18) and (22), and the vector The quadrupole operator for a system of many electrons Stoner Hamiltonian, Eq. (25). We employ a restricted is obtained by summing the operators for each electron. basis with a single set of s, p and d orbitals for the s, p 6 and d Hamiltonians respectively. We perform FCI cal- 32 culations using the HANDE computer program to ﬁnd 2.5 exact eigenstates of the model Hamiltonians. The Lanc- 1.0 zos algorithm was used to calculate the lowest 40 wavefunctions for our Hamiltonian, more than suﬃcient to 2.0 0.5

ﬁnd the correct ground state wavefunction, while the full 3 1 0.0 : spectrum was calculated for the vector Stoner Hamilto- 1.5 m ˆ | 1 nian. t 0.5 .

− m / | ˆ

The single particle contribution to the Hamiltonian for J 1.0 2 all models (the hopping matrix) is defined for a dimer 1.0 − : aligned along the z-axis: in this case it is only non-zero 1.5 − between orbitals of the same type, whether on the same 0.5 2.0 site or on different sites. The sigma hopping integral, − tσ , was set to 1 to define the energy scale. The other 2.5 | | − hopping matrix elements follow the canonical relations 2 4 6 8 10 39 suggested by Andersen , and we use the values of Paxton U/ t and Finnis40: ppσ : ppπ = 2 : 1 for p orbitals, and | | ddσ : ddπ : ddδ = 6 : 4 : 1 for d− orbitals. The directionally averaged− − magnetic correlation between the two sites has been calculated for the ground state. If positive it shows that the atoms are ferromagnetically correlated (spins parallel on both sites) and if it is neg- ative it shows that the atoms are antiferromagnetically correlated (spins antiparallel between sites). The correlation between components of spin projected onto a direction described by angles θ and φ, in spherical coordinates, is defined as follows

1 2 Cθφ = :m ˆ mˆ : , (27) h θφ θφ i where the :: is the normal ordering operator, used to remove self-interaction, and

I X θφ † mˆ θφ = σξξ0 cÎα,ξcÎα,ξ0 , (28) αξξ0 FIG. 1: The magnetic correlation between two p-shell θφ z x y σξξ0 = σξξ0 cos θ + σξξ0 sin θ cos φ + σξξ0 sin θ sin φ. atoms, each with 2 electrons, for a large range of parameters U/ t and J/ t , where t is the sigma bond The magnetic correlation averaged over solid angle is hopping. The different| | regions| | of the graph are labeled 1 by the symmetry of the ground state. The top graph is Cavg = : mˆ 1.mˆ 2 : . (29) generated from our Hamiltonian and the bottom from 3h i the vector Stoner Hamiltonian. The bottom graph has a 3 − region with symmetry Σg extending a long way up the B. Ground state: p orbital dimer J axis, which is not present in the ground state of our Hamiltonian. The bottom graph also includes a region 1 with two degenerate states with symmetries ∆g and The ground states of our Hamiltonian and the vector 1 + Stoner Hamiltonian have been found to be rather similar Σg ; this degeneracy is broken when our Hamiltonian is for 2, 6 and 10 electrons split over the p orbital dimer, used. but qualitatively different for 4 and 8 electrons. The magnetic correlation between the two atoms for 4 electrons is shown in Fig. 1. The symmetries of the wavefunctions41 total spin. The differences between the ground states 2S+1 ± are indicated on the graph by the notation Λu/g: the of our Hamiltonian and the vector Stoner Hamiltonian is only used for Lz = 0 (i.e. Σ states) and corresponds shown in Fig. 1 are as follows: the ground-state wave- ±to the sign change after a reflection in a plane parallel function of the vector Stoner Hamiltonian has a region 3 − to the axis of the dimer; the u/g term refers to ungerade with Σg symmetry extending far up the J axis, which 1 (odd) and gerade (even) and corresponds to a reflection is not present for our Hamiltonian; the region with ∆g through the midpoint of the dimer; Λ is the symbol cor- symmetry is doubly degenerate for our Hamiltonian, as responding to the total value for Lz (e.g. Lz = 1 is Π, total Lz = 2, whereas for the Stoner Hamiltonian it is ± Lz = 2 is ∆, Lz = 3 is Φ, Lz = 4 is Γ); and S is the triply degenerate, as it is also degenerate with the state 7

1 + of symmetry Σg (this state appears at a higher energy for our Hamiltonian). 2.5 4.5

. . 3 0 C. Ground state: d orbital dimer 2 0 1.5 3 1 :

1.5 0.0 m The ground states of our Hamiltonian and vector Stoner ˆ | 1 t .

Hamiltonian have been found to be rather similar for 2, 4, m / |

1.5 ˆ

J − 6, 14, and 18 electrons split over the d orbital dimer when 2 1.0 3.0 : ∆J is small. The simulations for 10 electrons were not − carried out as they are too computationally expensive. 4.5 − Qualitative differences were found for 8 and 12 electrons; 0.5 6.0 for an example see Fig. 2, which shows the magnetic cor- − relation between two atoms in the ground state of the 7.5 − d-shell dimer with 6 electrons per atom. From Fig. 2 we 2 4 6 8 10 U/ t see that the results for our Hamiltonian with ∆J/ t = 0.0 | | (top graph) and with a small value of ∆J/ t = 0|.1| (middle graph) are qualitatively different from| those| for the 2.5 vector Stoner Hamiltonian (bottom graph). The vector 4.5 1 1 + Stoner Hamiltonian makes the Γg and Σg states de- 3.0 generate, which is not the case for our Hamiltonian. The 2.0 largest differences are between the ∆J/ t = 0.1 graph 1.5 3 1 and the vector Stoner Hamiltonian: new| | regions with :

1.5 0.0 m 1 + 3 5 − ˆ symmetry Σ , Φg and Σ appear, and the region with | 1 g g t .

3 − m / | symmetry Σ almost disappears. This shows that the 1.5 ˆ g J − 2 inclusion of the quadrupole term in Eq. (22) can make a 1.0 3.0 : qualitative difference to the ground state. − 4.5 − 0.5 6.0 IV. EXCITED STATE RESULTS − 7.5 − 2 4 6 8 10 Differences between our Hamiltonian and the vector U/ t Stoner Hamiltonian are also observed for excited states. | | Here we present three examples, one demonstrating ex- plicitly the effect of the pair hopping, one showing more 2.5 4.5 general differences in the excited states through a cal- 3.0 culation of the electronic heat capacity as a function of 2.0 temperature, and one showing a difference in the spin 1.5 3 1 dynamics of a collinear and a non-collinear Hamiltonian. :

1.5 0.0 m ˆ

The full spectrum of eigenstates was calculated for all of | 1 t . m

these examples. / |

1.5 ˆ

J − 2

1.0 3.0 : − A. Pair hopping 4.5 0.5 − 6.0 P † † − The pair hopping term, J αβσσ0 cˆα,σcˆα,σ0 cˆβ,σ0 cˆβ,σ = 7.5 P 2 − J αβ : (ˆnαβ) :, is found in both our p and d or- 2 4 6 8 10 bital Hamiltonians, but is absent from the vector Stoner U/ t Hamiltonian. To demonstrate the eﬀect of the pair hop- | | ping term we initialize the wavefunction in a state with FIG. 2: The magnetic correlation between two d-shell two electrons in the x orbital on site 1 of a p-atom dimer. atoms, each with 6 electrons, for a large range of Figure 3 shows the evolution of the wavefunction in time parameters U/ t and J/ t , where t is the sigma bond using our Hamiltonian and the vector Stoner Hamilto- hopping. The diﬀerent| | regions| | of the graph are labeled nian, with U = 5.0 t and J = 0.7 t . For our Hamilto- | | | | by the symmetry of the ground state. The top graph is nian the pair of electrons in orbital x on atom 1 hops to generated using our d-shell Hamiltonian with the y and z orbitals on atom 1 (mediated by the pair hop- ∆J/ t = 0.0; the middle graph is generated using our ping term) as well as between the two atoms (mediated Hamiltonian| | with a small value of ∆J/ t = 0.1; and the bottom graph is generated using the| vector| Stoner Hamiltonian. 8

1x 2.0 1y 0.8 p orbital Hamiltonian 1z 0.7 Stoner Hamiltonian 2x 1.5 2y 0.6

2z )

a 0.5 N i B α k I 0.4 ( ˆ 1.0 n / h V . C 0 3

0.5 0.2

0.1

0.0 0.0 3 2 1 0 1 2 0 2 4 6 8 10 12 14 16 10− 10− 10− 10 10 10 time(h¯/ t ) kBT/ t | | | | 1x FIG. 4: The electronic heat capacity per atom of a 2.0 1y dimer with four electrons split over two p-shell atoms 1z with U = 5.0 t and J = 0.7 t . The solid line is 2x generated by solving| | our Hamiltonian| | exactly and the 1.5 2y dashed line by solving the vector Stoner Hamiltonian 2z exactly. i α I

ˆ 1.0 n h B. Heat capacity

0.5 The heat capacity is calculated from

CV 1 2 2 = 2 E (T ) E(T ) , (30) 0.0 kBNa Na(kBT ) h i − h i 0 2 4 6 8 10 12 14 16 time(h¯/ t ) | | where P n εi FIG. 3: The time evolution under our Hamiltonian εi exp( ) En(T ) = i − kB T , (31) (top) and the vector Stoner Hamiltonian (bottom) of a h i P exp( εi ) starting state with two electrons in 1x, the x orbital on i − kB T atom 1, of a p-atom dimer with U = 5.0 t and T is the temperature, kB is Boltzmann’s constant, εi is a J = 0.7 t . We see that the two electrons in 1|x| are able | | many-electron energy eigenvalue, and Na is the number to hop into 1y and 1z when the wavefunction is evolved of atoms. The result for both Hamiltonians as a function using our Hamiltonian but not when it is evolved using of temperature for four electrons split over two p-shell the vector Stoner Hamiltonian. atoms with U = 5.0 t and J = 0.7 t is shown in Fig- ure 4. There is a qualitative| | difference| | between the results for our Hamiltonian (the solid line) and the vector Stoner Hamiltonian (dashed line). The energies of the eigenstates of our Hamiltonian are more spread out than those of the Stoner Hamiltonian. This is due to the inclusion of the pair hopping term, which causes states with on-site paired electrons to rise in energy (by J/ t ). ∼ | | by the single electron hopping term). The result is that The unphysical reduction of the heat capacity to zero as there is a finite probability of finding the pair of electrons the temperature tends to infinity is a consequence of the in any of the x, y and z orbitals on either atom. However, use of a restricted basis set. for the vector Stoner Hamiltonian, the two electrons in the x orbital on site 1 are unable to hop into the y and z orbitals on atom 1; they are only able to hop to the x C. Spin dynamics orbital on atom 2. This means tha there is no possibility of observing the pair of electrons in the y or z orbitals Here we show how the collinear Stoner Hamiltonian can on either atom. give rise to unphysical spin dynamics. We consider an 9

Sz = +1 triplet with both electrons on one of the two V. CONCLUSION p orbital atoms, Ψ = T+1 . Written as linear com- | i | i binations of two-electron Slater determinants, the three We have established the correct form of the multi-orbital states in the triplet are model Hamiltonian with on-site Coulomb interactions for 1 atoms with valence shells of s, p and d orbitals. The T+1 = ( 1x 1y + 2x 2y ) , | i √2 −| ↑ ↑i | ↑ ↑i methodology used may be extended to atoms with f and 1 g shells, and to atoms with valence orbitals of several T−1 = ( 1x 1y + 2x 2y ) , different angular momenta. The results presented show | i √ −| ↓ ↓i | ↓ ↓i 2 that there are important differences between our p- and 1 T0 = ( 1x 1y + 1x 1y ) d-shell Hamiltonians and the vector Stoner Hamiltonian. | i − √2 | ↓ ↑i | ↑ ↓i The vector Stoner Hamiltonian misses both the pair hop- 1 ping term, which is present in our p and d orbital Hamil- + ( 2x 2y + 2x 2y ) . (32) √2 | ↓ ↑i | ↑ ↓i tonians, and the quadrupole term, which is present in our d orbital Hamiltonian. The pair hopping term pushes These states are simultaneously eigenstates of the states with pairs of electrons up in energy, whereas the collinear Stoner Hamiltonian, the vector Stoner Hamil- magnetism term pulls states with local magnetic mo- tonian, and our p-case Hamiltonian. They are degener- ments down in energy. The pair hopping term has the ate, with eigenvalue U J, for both our p-case Hamil- largest effect on the ground state for p orbitals with 2 tonian and for the vector− Stoner Hamiltonian. For the and 4 electrons per atom and for d orbitals with 4 and collinear Stoner Hamiltonian the states T and T +1 −1 6 electrons per atom, for J/ t < 2. This is because the have eigenvalue U J whereas state T | hasi eigenvalue| i 0 number of possible determinants| | with paired electrons U. This means that− the degeneracy of| thisi triplet is bro- on site is large for these filling factors and the low ly- ken in the collinear Stoner Hamiltonian. We now rotate ing states can be separated based on this. At values of Ψ in spin space so that the spins are aligned with the J/ t > 2 the magnetism becomes the dominant effect x| -axis,i i.e. S = +1, x upon| | the selection of the ground state and the difference rot 1 1 between the ground state of our Hamiltonian and that Ψ = ( T+1 + T−1 ) + T0 . (33) | i 2 | i | i √2| i of the vector Stoner Hamiltonian becomes small. How- ever, the differences are rather more pronounced for the Ψrot is still an eigenstate, with eigenvalue U J, of both excited states. This is evidenced by the hopping of pairs the| vectori Stoner Hamiltonian and our Hamiltonian,− but of electrons between orbitals on the same site, which is it is no longer an eigenstate of the collinear Stoner Hamil- allowed by our Hamiltonian but not by the vector Stoner tonian. The wavefunction Ψrot evolves in time as | i Hamiltonian, and in differences in the electronic heat ca- tot − iHτˆ tot pacity as a function of temperature. We also find clear Ψ (τ) = e h¯ Ψ (0) , (34) | i | i evidence that the collinear Stoner Hamiltonian is inap-  − iJτ !  − i(U−J)τ 1 e h¯ propriate for use in describing spin dynamics as it breaks  e h¯ ( T+1 + T−1 ) + T0 ,  2 | i | i √ | i rotational symmetry in spin space. We would expect sim-  2  | {z } ilar problems when using collinear time dependent DFT = collinear Stoner Hamiltonian simulations to model spin dynamics. − i(U−J)τ 1 1  h¯  e ( T+1 + T−1 ) + T0 ,  2 | i | i √2| i  | {z }  mˆ 2 Stoner and our Hamiltonian ACKNOWLEDGEMENTS where τ is time. If we take the expectation value of the magnetic moment on each site using the collinear Stoner The authors would like to thank Toby Wiseman for help Hamiltonian, we find with the representation theory and Dimitri Vvedensky, David M. Edwards, Cyrille Barreteau and Daniel Span- rot rot rot rot Jτ Ψ (τ) mˆ 1x Ψ (τ) = Ψ (τ) mˆ 2x Ψ (τ) = cos ,jaard for stimulating discussions. M. E. A. Coury was h | | i h | | i ¯h supported through a studentship in the Centre for Doc- rot rot rot rot Ψ (τ) mˆ 1y Ψ (τ) = Ψ (τ) mˆ 2y Ψ (τ) = 0, toral Training on Theory and Simulation of Materials at h | | i h | | i Imperial College funded by EPSRC under grant num- Ψrot(τ) mˆ Ψrot(τ) = Ψrot(τ) mˆ Ψrot(τ) = 0. 1z 2z ber EP/G036888/1. This work was part-funded by the h | | i h | | i (35) EuroFusion Consortium, and has received funding from In contrast, using the vector Stoner Hamiltonian or our Euratom research and training programme 20142018 un- p-shell Hamiltonian, the expectation value of the mag- der Grant Agreement No. 633053, and funding from the netic moment is independent of time, equal to (1, 0, 0) RCUK Energy Programme (Grant No. EP/I501045). in vector format. This demonstrates that calculations The views and opinions expressed herein do not neces- of spin dynamics using the collinear Stoner Hamiltonian sarily reflect those of the European Commission. The can give rise to unphysical oscillations of the magnetic authors acknowledge support from the Thomas Young moments. Centre under grant TYC-101 10

Appendix A: Mean-field Hamiltonians for which the total energy is Coulomb 1 J 2 J 2 1. The one band Hubbard model: s orbital EMF = U nˆ mˆ symmetry 2 − 2 h i − 4 h i X J + nˆ nˆ + nˆ 2 42 2 h αβih βαi h αβi Application of the mean-field approximation to the on- αβ site Coulomb interaction part of the Hubbard Hamilto- X U nian with s orbital symmetry, Eq. (7), yields the follow- cˆ† cˆ cˆ† cˆ − 2 h ασ βζ ih βζ ασi ing, αβσζ J † † ! + cˆασcˆβζ cˆαζ cˆβσ , (A6) X † † 2 h ih i VˆMF =U0 nˆ nˆ cˆ cˆ cˆ cˆ , (A1) h i − h σ ζ i ζ σ σζ where the double counting correction has been included. for which the total energy is 3. The multi-orbital model Hamiltonian: d-orbital   symmetry Coulomb 1 2 X † † EMF = U0  nˆ cˆσcˆζ cˆζ cˆσ  , (A2) 2 h i − h ih i 42 σζ Application of the mean-field approximation to the model Hamiltonian with d-orbital symmetry, Eq. (22), where the double counting correction has been included. yields the following, Equivalently, applying the mean-field approximation to 1 J Eq. (8) yields the following, VˆMF = U J + 2∆J nˆ nˆ mˆ .mˆ − 2 h i − 2 h i ! 1 X ˆ X † † + (J 6∆J) nˆαβ nˆβα + nˆαβ nˆαβ VMF = U0 2 nˆ nˆ mˆ .mˆ cˆσcˆζ cˆζ cˆσ , (A3) − h i h i 3 h i − h i − h i αβ σζ 2 X ˆ ˆ + ∆J Qµν Qνµ for which the total energy is 3 µν h i !   X † † † † U cˆασcˆβζ cˆβζ cˆασ + J cˆασcˆβζ cˆαζ cˆβσ Coulomb 1 2 2 X † † − h i h i EMF = U0 2 nˆ mˆ cˆσcˆ cˆ cˆσ  , αβσζ 6 h i − h i − h ζ ih ζ i σζ X X + 48∆J ξ ξ ξ ξ cˆ† cˆ cˆ† cˆ , (A4) αst βtu χuv γvsh ασ γζ i βζ χσ αβγχσζ stuv (A7) where again the double counting correction has been included. for which the total energy is

1 1 J ECoulomb = U J + 2∆J nˆ 2 mˆ 2 MF 2 − 2 h i − 4 h i 2. The multi-orbital model Hamiltonian: p orbital X J 6∆J symmetry + − nˆ nˆ + nˆ 2 2 h αβih βαi h αβi αβ Application of the mean-ﬁeld approximation42 to the 1 X + ∆J Qˆ Qˆ model Hamiltonian with p orbital symmetry, Eq. (18), µν νµ 3 µν h ih i yields the following, X U cˆ† cˆ cˆ† cˆ − 2 h ασ βζ ih βζ ασi J J αβσζ VˆMF = U nˆ nˆ mˆ .mˆ − 2 h i − 2 h i J † † X + cˆασcˆβζ cˆαζ cˆβσ + J nˆ nˆ + nˆ nˆ 2 h ih i h αβi βα h αβi αβ X X αβ + 24∆J ξαstξβtuξχuvξγvs ! αβγχσζ stuv X † † † † U cˆασcˆβζ cˆβζ cˆασ + J cˆασcˆβζ cˆαζ cˆβσ , † † − h i h i cˆασcˆγζ cˆβζ cˆχσ , (A8) αβσζ × h ih i (A5) where the double counting correction has been included. 11

Appendix B: Representation theory γ; where X 1. p orbital symmetry s0 = Vαβ,χγ δαχδβγ , χγ X X As a precursor to our treatment of the d case in Appendix s2 = Vαβ,χγ δαγ δβχ = Vαβ,χγ δαβδχγ . (B4) 43 B 2, we use representation theory to derive Eq. (11) for χγ χβ cubic harmonic p orbitals. First we define the following irreducible objects: “0” is a scalar, “1” is a three- This is equivalent to Eq. (11), where s0 = U and s2 = J. dimensional vector, “2” is a 3 3 symmetric traceless second-rank tensor, “3” is a third-rank× tensor consisting of three “2”s, and “4” is a fourth-rank tensor consisting of 2. d orbital symmetry three “3”s. The Coulomb interaction Vαβ,γδ transforms as a tensor product of four “1”s, 1 1 1 1, but we ⊗ ⊗ ⊗ To find the isotropic five-dimensional fourth-rank tensor know also that it must be isotropic and that from the that describes the on-site Coulomb interactions for d or- above irreducible objects only the “0” is isotropic. bital symmetry, we find it convenient to map from a five- Objects such as 1 1, represented as Aij, may be ex- dimensional fourth-rank tensor to a three-dimensional ⊗ panded just as in the addition of angular momentum: eighth-rank tensor. We do this by replacing the five- dimensional vector of d orbitals by the three-dimensional 1 1 = 0 1 2, (B1) ⊗ ⊕ ⊕ traceless symmetric B matrix of the d orbitals: P  2 2 1 2 2  where “0” is a scalar, s = ij Aijδij, “1” is a vector, 3 x 3 y + z xy xz B = − xy 2 y2 1 x2 + z2 yz . v = P A , and “2” is a traceless symmetric ten-  3 3  i jk jk ijk xz− yz 2 z2 1 x2 + y2 1 1 P 3 − 3 sor, Mij = (Aij + Aji) δij( kl Aklδkl). The reader (B5) 2 − 3 may be more familiar with combinations of angular mo- We refer to the elements of B as Bab, and use the sub- mentum. In the language of angular momentum, the scripts a, b, c, d, s, t, u and v as the indices for irre- transformation properties of a tensor product of two ob- ducibles of rank 2 or higher. The transformation between jects of angular momentum with l = 1 are described by the irreducible Bab and the cubic harmonic d orbitals is a 9 9 matrix, which is block diagonalisable into a 1 1 matrix× (which describes the transformations of an l =× 0 X 1 φασ(r) = Nd ξαabBab Rd(r)S(σ), (B6) object), a 3 3 matrix (which describes the transforma- r2 ab tions of an ×l = 1 object), and a 5 5 matrix (which × describes the transformations of an l = 2 object). In the where ξ is a traceless, symmetric transformation matrix, angular momentum representation the matrix elements q defined in Eq. (21), N = 1 15 is a normalisation factor, are complex. Here we are using cubic harmonics and the d 2 π matrix elements are real. Similarly 1 1 1 1 may be Rd is the radial function for a cubic d orbital, and S is the expanded out as: ⊗ ⊗ ⊗ spin function. The orbital indices α, β, χ and γ run over the five independent d orbitals, whereas the irreducible 1 1 1 1 = (1 1) (1 1), indices a, b, c, d, s, t, u and v run over the three Cartesian ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ directions. The mapping between the five-dimensional = (0 1 2) (0 1 2), ⊕ ⊕ ⊗ ⊕ ⊕ fourth-rank tensor and the three-dimensional eighth-rank = (0 0) (0 1) (0 2) (1 0) (1 1) ⊗ ⊕ ⊗ ⊕ ⊗ ⊕ ⊗ ⊕ ⊗ tensor is as follows (1 2) (2 0) (2 1) (2 2). (B2) Z ∗ ∗ ⊕ ⊗ ⊕ ⊗ ⊕ ⊗ ⊕ ⊗ φασ(r1)φβσ0 (r2)φχσ(r1)φγσ0 (r2) Vαβ,χγ = dr1dr2 , The only isotropic, “0”, objects arise from 0 0, 1 1 and r1 r2 ⊗ ⊗ Z | − | 2 2. A general isotropic three-dimensional fourth-rank 4 2 2 2 2 ⊗ = dr1dr2N Rd(r1)/r Rd(r2)/r tensor, Tijkl, therefore has three independent scalars that d | 1| | 2| are found from symmetric and antisymmetric contrac- P P ∗ ∗ ∗ ∗ abcd stuv ξαabBabξβcdBcdξχstBstξγuvBuv tions of Tijkl. We are interested in the form of the p or- , r r bital on-site Coulomb interactions, Vαβ,χγ , which have an × 1 2 X X | − | additional symmetry, such that they remain unchanged = ξαabξβcdξχstξγuvVab,cd,st,uv, (B7) under exchange of α and χ and under exchange of β and abcd stuv γ. We therefore only require the symmetric scalars that arise from the 0 0 and 2 2 contractions to describe where this equation defines the isotropic three- ⊗ ⊗ Vαβ,χγ : dimensional eighth-rank tensor for the on-site Coulomb integrals, Vab,cd,st,uv, and we have dropped the complex Vαβ,χγ = s0δαχδβγ + s2 (δαγ δβχ + δαβδχγ ) , (B3) conjugates on the ξ matrices as they are real. The B matrix is a three-dimensional traceless symmetric “2” which is symmetric under exchange of α and χ or β and and thus contains 5 independent terms. As we show 12 below, it is possible to represent the isotropic three- necker delta products, each multiplied by one of the three dimensional eighth-rank tensor as a list of quadruple Kro- independent coefficients: necker deltas, with 5 independent parameters in general 2 and 3 independent parameters for the on-site Coulomb Vab,cd,st,uv = c0∆ab,cd,st,uv interactions. The isotropic three-dimensional eighth- 1 3 + c1 ∆ab,cd,st,uv + ∆ab,cd,st,uv rank tensor is then converted back into an isotropic five- 5 dimensional fourth-rank tensor. For an independent ver- + c2∆ab,cd,st,uv. (B9) ification of the form of the isotropic three-dimensional We find V by transforming this expression back to eighth-rank tensor see Ref. 44. αβ,χγ the five-dimensional fourth-rank representation, as dic- The power of representation theory is that one can follow tated by Eq. (B7). For completeness we include all six of an analogous procedure for shells of higher angular mo- the transformed contractions: mentum, using a “3” to represent the seven f orbitals, a X X “4” to represent the nine g orbitals, and so on. One ∆1 ξ ξ ξ ξ = can also construct interaction Hamiltonians for atoms ab,cd,st,uv αab βcd χst γuv abcd stuv with important valence orbitals of several different an- X gular momenta. 4ξαabξβbaξχstξγts = δαβδχγ , We now return to the case of a d shell and explain the abst X X 2 procedure used to find the general isotropic Coulomb ∆ab,cd,st,uvξαabξβcdξχstξγuv = Hamiltonian in more detail. We start by represent- abcd stuv ing the isotropic three-dimensional eighth-rank tensor, X 4ξαabξχbaξβcdξγdc = δαχδβγ , Vab,cd,st,uv, as a 2 2 2 2. Proceeding as we did for the p case, we write⊗ ⊗ ⊗ abcd X X 3 ∆ab,cd,st,uvξαabξβcdξχstξγuv = 2 2 2 2 = (2 2) (2 2), ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ abcd stuv 2 2 = 0 1 2 3 4, X ⊗ ⊕ ⊕ ⊕ ⊕ 4ξαabξγbaξβcdξχdc = δαγ δβχ, = 2 2 2 2 = (0 1 2 3 4) abcd ⇒ ⊗ ⊗ ⊗ ⊕ ⊕ ⊕ ⊕ (0 1 2 3 4). (B8) X X 4 ⊗ ⊕ ⊕ ⊕ ⊕ ∆ab,cd,st,uvξαabξβcdξχstξγuv = abcd stuv The terms in Eq. (B8) that can generate a rank 0 (a X scalar) are 0 0, 1 1, 2 2, 3 3 and 4 4. This im- 16ξαabξβbdξγdvξχva, plies that a⊗ general⊗ isotropic⊗ three-dimensional⊗ ⊗ eighth- abdv rank tensor is defined by five independent parameters. X X 5 ∆ab,cd,st,uvξαabξβcdξχstξγuv = We know that, for Vαβ,χγ , there exists a symmetry be- abcd stuv tween the pairs αχ and βγ. It follows that Vab,cd,st,uv has X a symmetry between the pairs (ab)(st) and (cd)(uv). We 16ξαabξβbdξχdtξγta, therefore require only the even contractions, 0 0, 2 2 abdt X X and 4 4, reducing the five parameters to three⊗ 45; this⊗ ∆6 ξ ξ ξ ξ = ⊗ ab,cd,st,uv αab βcd χst γuv is the same number proposed in Chapter 4 of Ref. 35. abcd stuv To get the scalars from Vab,cd,st,uv we have to contract X 16ξαabξχbtξβtdξγda. (B10) it. There are six possible contractions of Vab,cd,st,uv, out- abdt lined in Table I, of which only five are independent. Note that one does not contract within the indices ab, cd, st or We find that the last three of the transformations of P uv because Bab is traceless, hence ab δabBab = 0. By the contractions, although rotationally invariant in or- examining the diagrammatic contractions from Table I, bital space, are not expressible in terms of Kronecker and maintaining the symmetry between the pairs (ab)(st) deltas of α, β, χ and γ. This is to be expected as and (cd)(uv), we conclude that the relevant contractions there exist terms in V that are non-zero and have αβ,χγ 2 1 3 5 more than two orbital indices, e.g. V(3z2−r2)(xy),(yz)(xz) are ∆ab,cd,st,uv, ∆ab,cd,st,uv + ∆ab,cd,st,uv , ∆ab,cd,st,uv or V(3z2−r2)(xy),(xy)(x2−y2). 4 6 and ∆ab,cd,st,uv + ∆ab,cd,st,uv . This is evident because We can now write down Vαβ,χγ concisely, switching two symmetry-related pairs [that is, switching 5 (ab) and (st) or (cd) and (su)] in diagram 1 yields dia- Vαβ,χγ = Uδαχδβγ + J + ∆J (δαγ δβχ + δαβδχγ ) gram 3. Similarly, switching two symmetry-related pairs 2 in diagram 4 yields diagram 6. We see that there are four X 48∆J ξαabξβbdξχdtξγta, (B11) − contractions but we have only three parameters. This abdt implies that out of these four contractions there are only three linearly independent contractions. where we have defined U to be the Hartree term We can now write down a general interaction Hamilto- between the t2g orbitals on-site, U = V(zx)(yz),(zx)(yz), nian Vab,cd,st,uv as a linear combination of three Kro- J to be the average of the exchange integral 13 between the eg and t2g orbitals on-site, J = By finding the expectation value and substituting into 1 2 V(zx)(yz),(yz)(zx) + V(3z2−r2)(x2−y2),(x2−y2)(3z2−r2) , Eq. (C5) we obtain and ∆J to be the difference between the X X exchange integrals for eg and t2g, ∆J = ˆ † Lµ = 4i µmkξαkjξβjmcˆασcˆβσ. (C9) V(3z2−r2)(x2−y2),(x2−y2)(3z2−r2) V(zx)(yz),(yz)(zx). jkm αβσ We could equally well have chosen− to use the sum of the fourth and sixth contractions instead of the fifth contraction, although with different coefficients. 2. Quadrupole operator

Appendix C: Angular momentum and quadrupole We deﬁne the quadrupole operator for a single electron operators as

1 1 2 1. Angular momentum Qˆµν = LˆµLˆν + Lˆν Lˆµ δµν Lˆ . (C10) 2 − 3 The angular momentum operators are most naturally ex- Here we interpret Lˆµ and Lˆν in Eq. (C10) as compo- pressed in spherical polar coordinates, even when using nents of the angular momentum of a single electron. them to operate on cubic harmonics. Their forms are: The quadrupole operator for a system of N electrons is then a sum over contributions from each electron: ˆ ∂ ∂ ˆ PN ˆ ˆ Lx = i sin φ + cot θ cos φ , (C1) Qµν = i=1 Qµν (i). This makes Qµν a one-electron ∂θ ∂φ operator, respresented in second quantization as a lin- ∂ ∂ ear combination of strings of one creation operator and Lˆy = i cos φ + cot θ sin φ , (C2) − ∂θ ∂φ one annihilation operator. Squaring and tracing the one- ˆ ∂ electron tensor operator Qµν yields a two-electron oper- Lˆz = i , (C3) ˆ2 P ˆ ˆ − ∂φ ator Q = µν Qµν Qνµ. To find the form of this operator, we start by applying the where we have seth ¯ to 1 for simplicity. When applying single electron version of Qˆµν to the B matrix, making the above operators to the on-site cubic harmonic p or- use of Eq. (C7). Starting with LˆµLˆν bitals (we have dropped the site index for clarity) it is straightforward to show that X LˆµLˆν Bjk = iLˆµ (νjs Bsk + νks Bjs ) , | i | i | i ˆ X s Lµpj = i µjkpk. (C4) k 2 X = (i) νjs (µss0 Bs0k + µks0 Bss0 ) | i | i ˆ ss0 A general one particle operator O may be expressed in terms of creation and annihilation operators as follows: + νks (µjs0 Bs0s + µss0 Bjs0 ) , | i | i ˆ X ˆ † O = φασ O φβσ0 cˆασcˆβσ0 . (C5) h | | i = ( 1) δjµ Bνk + δkµ Bjν 2δνµ Bjk αβσσ0 − | i | i − | i X Substituting Eq. (C4) into Eq. (C5) yields + (νjsµks0 + νksµjs0 ) Bss0 , (C11) | i ss0 ˆ X † Lµ = i µβαcˆασcˆβσ, (C6) αβσ Lˆν Lˆµ Bjk = ( 1) δjν Bµk + δkν Bjµ 2δµν Bjk | i − | i | i − | i where µ is a Cartesian direction and α and β are cubic X + (µjsνks0 + µksνjs0 ) Bss0 , (C12) harmonic p orbitals (x, y or z). The d case is slightly | i more complicated, but is greatly simplified by the use of ss0 Eq. (B6). Applying the angular momentum operators to where the last equation was found by exchanging µ and the B matrix reveals that ν in the previous equation. Following the same process, X the final term in Eq. (C10) becomes Lˆµ Bjk = i (µjs Bsk + µks Bjs ) , (C7) | i | i | i s X Lˆν Lˆν Bjk = 6 Bjk , (C13) with ν | i | i

X Rd(r) Lˆµ φασ = iNd S(σ) ξαjk (µjs Bsk + µks Bjs ) . This is not a surprising result as the B matrix contains | i r2 | i | i jks linear combinations of spherical harmonics l, lz with l = 2 | i (C8) 2, and Lˆ l, lz = l(l + 1) l, lz . Combining Eqs. (C11), | i | i 14

(C12) and (C13) we ﬁnd 3. The quadrupole operator squared and the on-site Coulomb integrals 1 Qˆµν Bjk = (δµj Bνk + δkµ Bjν | i − 2 | i | i

+δjν Bµk + δkν Bjµ ) The quadrupole operator is of interest to us because one | i | i of the terms in the d-shell on-site Coulomb interaction X 2 + (νjsµks0 + νksµjs0 ) Bss0 . (C14) can be represented in terms of Qˆ . This term is: | i ss0

Equation (C5) yields the corresponding operator for a system of many electrons: X X † † 48∆J ξ ξ ξ ξ cˆ cˆ 0 cˆ 0 cˆ , − αst βtu χuv γvs ασ βσ γσ χσ X X αβχγσσ0 stuv Qˆµν = 2 ξ ξ + ξ ξ (C18) − ανk βkµ αµk βkν αβσ k which is proportional to the ﬁnal term in Eq. (C17). The X † other terms in Eq. (C17) also exist elsewhere in the d- + 4 ξαmnξβjkνjmµkn cˆασcˆβσ. (C15) shell on-site interaction Hamiltonian, so introducing an mnjk explicit Qˆ2 term is straightforward: We now deﬁne the normal ordered quadrupole squared as

2 X X X † † : Qˆ := : Qˆµν Qˆνµ : . (C16) 48∆J ξαstξβtuξχuvξγvscˆασcˆβσ0 cˆγσ0 cˆχσ − 0 µν αβχγσσ stuv 2 2 X † † = ∆J : Qˆ : +2∆J cˆ cˆ 0 cˆ 0 cˆ By substituting Eq. (C15) into Eq. (C16) and dropping 3 ασ βσ βσ ασ the site index for clarity, one obtains the following simple αβσσ0 X † † † † formula for the quadrupole squared operator: 6∆J (ˆc cˆ 0 cˆ 0 cˆ +c ˆ cˆ 0 cˆ 0 cˆ ) − ασ βσ ασ βσ ασ ασ βσ βσ αβσσ0 X : Qˆ2 : = 3δ δ + 9 δ δ + δ δ 2 − αχ βγ αγ βχ αβ χγ = ∆J : Qˆ2 : +2∆J :n ˆ2 : +3∆J(:n ˆ2 : + : mˆ 2 :) αβγχσσ0 3 2 X † † X 72 ξ ξ ξ ξ cˆ cˆ 0 cˆ 0 cˆ . (C17) 6∆J : nˆαβ : . (C19) αst βtu χuv γvs ασ βσ γσ χσ − − stuv αβ

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ab cd ab cd 1 1. ∆ab,cd,st,uv = δad2.δbcδsvδtu + δacδbdδsvδtu + δadδbcδsuδtv + δacδbdδsuδtv ab cd ab cd abst uvcd abst uvcd 1. 2. 1. 2. ab cd abst uvcd st uv st uv 2 st uv 1. 2. ∆ab,cd,st,uv = δatδbsδcvδdu + δasδbtδcvδdu + δatδbsδcuδdv + δasδbtδcuδdv ab cd ab cd abst cduv ababst cduvcd ab cd 1. 2. 1. 2.3. 4. st uv abst uvcd ab cd st st uv abst uvuvcd ∆3 = δ δ δabstδ + δ δ δuvcdδ + δ δ δ δ + δ δ δ δ 3. ab,cd,st,uv av4.bu ct ds au bv ct ds av bu cs dt au bv cs dt ab cd 3. abst uvcd 4. st uv st uv st uv 3. 4. ab cd ab cd ∆4 = δ δ δ δ + δ δ δ δ + δ δ δ δ + δ δ δ δ + st uv st uv ab,cd,st,uv at bd cv su ad bt cv su at bc dv su ac bt dv su ab cd ab cd δatδbdδcuδsv + δadδbtδcuδsv + δatδbcδduδsv + δacδbtδduδsv+ 3. 4. δ δ δ δ + δ δ δ δ + δ δ δ δ + δ δ δ δ + 3. 4. ab cd as bd cvabtu ad bs cvcdtu as bc dv tu ac bs dv tu st uv st uv δasδbdδcuδtv + δadδbsδcuδtv + δasδbcδduδtv + δacδbsδduδtv st uv 5. st uv 6. ab cd ab cd st ab uvcd 5 abst uvcd 5. ∆ab,cd,st,uv = δav6.δbdδctδsu + δadδbvδctδsu + δavδbcδdtδsu + δacδbvδdtδsu+ 5. δau6.δbdδctδsv + δadδbuδctδsv + δauδbcδdtδsv + δacδbuδdtδsv+ ab cd abst uvcd δavδbdδcsstδtu + δadδbvδcsuvδtu + δavδbcδdsδtu + δacδbvδdsδtu+ st uv δauδbdδcsstδtv + δadδbuδuvcsδtv + δauδbcδdsδtv + δacδbuδdsδtv 5. 6. ab cd ab cd abst cduv abst cduv 6 5. 6. ∆ab,cd,st,uv = δavδbtδcuδds + δatδbvδcuδds + δauδbtδcvδds + δatδbuδcvδds+ 5. 6. δavδbsδcuδdt + δasδbvδcuδdt + δauδbsδcvδdt + δasδbuδcvδdt+ st uv st uv δavδbtδcsδdu + δatδbvδcsδdu + δavδbsδctδdu + δasδbvδctδdu+ st uv st uv δauδbtδcsδdv + δatδbuδcsδdv + δauδbsδctδdv + δasδbuδctδdv

TABLE I: Table to show the node contractions for a d shell. Every contraction is represented both diagrammatically and mathematically.