Complex Interactions with Single

5.4 Analysis of Complex Interactions

5.4.3 Complex Interactions with Single

Biquadratic exchange: The isotropic linear magnetic exchange can be calculated in a rather straightforward way with single determinant spin unrestricted methods.

Assuming that the BS determinant is a linear combination of the spin states with low- est and highest possible spin moment, the Yamaguchi equation (Eq.4.85) relates the energies of the BS and HS determinants withJin a straightforward way, independent of the number of unpaired electrons on the magnetic sites involved in the coupling.

The biquadratic exchange can however not be addressed from energy differences only, simply because we have only access to one energy difference, obviously too few to determine two parameters.

Instead one can estimate the strength of the biquadratic exchange in an indirect way via the electronic structure parametersU,t andK. To derive the relevant equations we need to compare the expressions of the singlet and triplet states in terms of J andλgiven in Eq.3.75with their fourth-order perturbation estimates using the matrix elements derived in Sect.5.4.1. In the first place, we need a common zero of energy. This is easily achieved by putting the energy of the quintet state to zero. The expressions of singlet, triplet and quintet states in terms of Jandλthen become

E(Q)=0 E(T)=2J

E(S)=3J+3λ (5.46)

The fourth-order perturbation estimates give us expressions in terms of t,U and K, and hence, we can relateλto these electronic structure parameters, which can be calculated with spin-unrestricted single determinant methods. To simplify the perturbation estimates we neglectα,β andγ, and all intersite exchange integrals in the interaction matrices derived in Sect.5.4.1. The interaction matrices for triplet and singlet states then become

|T |N H2 |I2 |I4 |I6 |I8

T| 0

N H2| 0 2K

I2| t13 −t13 U

I4| t13 −t13 0 U

I6| −t₂₄ −t₂₄ 0 0 U^′

I8| −t₂₄ −t₂₄ 0 0 0 U^′

|S |N H3 |I1 |I3 |I5 |I7

S| 0

N H3| 0 4K

I1| −3t13/√

6 t₁₃/√

2 U

I3| −3t13/√

6 t13/√

2 0 U

I5| 3t24/√

6 −t₂₄/√

2 0 0 U^′

I7| 3t24/√

6 −t₂₄/√

2 0 0 0 U^′

The second-order correction to the energy is obtained from the expression E⁽²⁾= Φ_I| ˆH|Φ_αΦ_α| ˆH|Φ_I

E_I−E_α (5.47)

which becomes

E_T⁽²⁾=

i=2,4,6,8

T| ˆH|I iI i| ˆH|T

−U_i = −2t₁₃² U −2t₂₄²

U^′ (5.48)

for the triplet and

E⁽²⁾_S =

i=1,3,5,7

S| ˆH|I iI i| ˆH|S

−U_i = −3t₁₃² U −3t₂₄²

U^′ (5.49)

for the singlet. From these equations we can define J⁽²⁾, the second-order estimate of J, as−^tU¹³² −^tU^242′. The fourth-order contribution is determined with Eq.5.12and counts with much more terms, which are summarized below and illustrated for one of the cases in Fig.5.16.

170 5 Towards a Quantitative Understanding Fig. 5.16 Schematic

representation of one of the fourth-order interactions contributing to the energy of the triplet state. The total contribution of this path equals−t₁₃²t₂₄²/U U^′2K

Φ_I Φ_α Φ_β Φ_γ Φ_I Φ_I Φ_α Φ_β Φ_γ Φ_I

S I1 N H3 I1 S −³₄t₁₃⁴/U²4K T I2 N H2 I2 T −t₁₃⁴/U²2K I3 −³₄t₁₃⁴/U²4K I4 −t₁₃⁴/U²2K I5 −³4t₁₃²t₂₄²/U U^′4K I6 t₁₃²t₂₄²/U U^′2K I7 −³₄t₁₃²t₂₄²/U U^′4K I8 t₁₃²t₂₄²/U U^′2K I3 I3 −³₄t₁₃⁴/U²4K I4 I4 −t₁₃⁴/U²2K

I5 −³4t₁₃²t₂₄²/U U^′4K I6 t₁₃²t₂₄²/U U^′2K I7 −³₄t₁₃²t₂₄²/U U^′4K I8 t₁₃²t₂₄²/U U^′2K I5 I5 −³₄t₂₄⁴/U^′24K I6 I6 −t₂₄⁴/U^′22K

I7 −³4t₂₄⁴/U^′24K I8 t₂₄²/U^′22K I7 I7 −³₄t₂₄⁴/U^′24K I6 I6 −t₂₄⁴/U^′22K The terms with the ionic states interchanged (Φα =I3,Φγ =I1, etc.) should also be added to the perturbational estimate. This is easily done by multiplying all the terms by two, except the ones withΦ_α =Φ_γ. Now the following corrections arise

E_T⁽⁴⁾= 2t₁₃⁴

U²K + 2t₂₄⁴

U^′2K −4t₁₃²t₂₄² U U^′K =2B

K (5.50)

E⁽⁴⁾_S = −3 4

t₁₃⁴ U²K −3

4 t₂₄⁴ U^′2K −3

2 t₁₃²t₂₄²

U U^′K = −3J⁽²⁾²

4K (5.51)

withB =t₁₃²/U−t₂₄²/U^′. Then the total perturbative estimate for the singlet and triplet energies are

E_T =E_T⁽²⁾+E⁽⁴⁾_T = −2t₁₃² U −2t₂₄²

U^′ −2B²

K =2J⁽²⁾−2B²

K (5.52)

ES=E_S⁽²⁾+E⁽⁴⁾_S = −3t₁₃² U −3t₂₄²

U^′ −3J²

4K =3J⁽²⁾−3J⁽²⁾²

4K (5.53)

The comparison with the expression for the singlet and triplet energy eigenvalues of the Heisenberg Hamiltonian with biquadratic terms leads to a fourth-order estimate of Jas

J =J⁽²⁾− B²

K (5.54)

Then from the singlet energy we get E_S =3J+3λ=3

J⁽²⁾− B²

K +3λ=3J⁽²⁾−3J⁽²⁾²

4K (5.55)

from which the perturbative expression forλin terms oft,UandKcan be extracted λ= B²

3K − J⁽²⁾²

4K (5.56)

The next step concerns the calculation of the energy of a collection of spin unrestricted determinants with different occupations and relate their energies to the electronic structure parameters in order to calculate theλparameter and in this way obtain a measure for the biquadratic interaction strength from methods like DFT (Fig.5.17).

Assuming thatU = U^′, the energies of the following five determinants define the parameters that appear in the perturbative expressions ofBandJ, which in turn lead to an estimate ofλ, the biquadratic exchange parameter.

Fig. 5.17 The five determinants that are needed to calculate the electronic structure parameters that defineBandJin the perturbative expression of the biquadratic exchange strength

172 5 Towards a Quantitative Understanding

Fig. 5.18 One of the second-order contributions to the energy ofΦ2. This path contributest₁₃²/−U

M_S=2: Φ₁= |ϕ₁ϕ₂ϕ₃ϕ₄| E₁⁽²⁾=0 (5.57) M_S=0: Φ2= |ϕ1ϕ2ϕ₃ϕ₄| E₂⁽²⁾= −2(t₁₃² +t₂₄²)

U (5.58)

Φ₃= |ϕ₁ϕ₂ϕ₃ϕ₄| E₃⁽²⁾=2K −2(t₁₃² +t₂₄²)

U (5.59)

M_S=1: Φ₄= |ϕ₁ϕ₂ϕ₃ϕ₄| E₄⁽²⁾=K−2t₂₄²

U (5.60)

Φ5= |ϕ1ϕ2ϕ₃ϕ4| E₅⁽²⁾=K−2t₁₃²

U (5.61)

The listed energies are the sum of the zeroth-order energies and second-order cor- rections. The latter are calculated by taking into account all possible interactions of these determinants with the ionic determinants, which are assumed to be degenerate with energyUrelative toΦ1. One example is given in Fig.5.18, the rest is completely analogous. The zeroth-order energiesΦI| ˆH|ΦIonly count the number of on-site exchange interactionsK, all other terms are neglected or the same as in the reference energy E₁⁽⁰⁾.

5.13 Write down the two ionic determinants that interact withΦ4and calculate the interaction matrix elements.

Four-center interactions: In Chap.3, we have seen how the four-center interac- tions can be extracted using an effective Hamiltonian spanned by the six M_S =0 determinants. To address the ring exchange within a spin unrestricted setting, this model space is no longer sufficient, but has to be extended with theM_S =2 and the four M_S = 1 determinants. In any standard implementation of density functional theory, the main area of spin unrestricted methods, one has only access to the diago- nal elements of this 11×11 model Hamiltonian; matrix elements between different determinants are not routinely calculated in most quantum chemistry packages. In addition, it should be realized that the fourMS=1 determinants are all degenerate and as was shown in Eq.3.84, the sixM_S=0 determinants are degenerate in pairs.

Hence, one can count with at most five energies, i.e. four energy differences that can be used to determine four independent parameters. It is therefore intrinsically impos- sible to determine the interaction strength of the three cyclic permutations defined in Fig.3.14, as can in principle be done with wave function based methods through the construction of a numerical effective Hamiltonian. However, in any practical case

there is only one sizeable four-center interaction, namely the one with the Pˆ₁₂₃₄ operator associated to it.

The expectation values of theMS=0 determinants can be found in Eq.3.84, but for theMS =1 andMS=2 determinants we will derive them here. Among the four degenerate MS=1 determinants we will focus on|abcd|, or|αααβ|in a spin-only notation. With the following ingredients for the two-center interactions:

−J₁(Sˆ_ASˆ_B+ ˆS_CSˆ_D)αααβ= −J₁ 1

4αααβ+1

2ααβα−1 4αααβ

−J2(SˆASˆD+ ˆSBSˆC)αααβ= −J2

1

2βααα−1

4αααβ+1

4αααβ (5.62)

−J3(SˆASˆD+ ˆSBSˆC)αααβ= −J3

1

4αααβ+1

2αβαβ−1 4αααβ and for the four-center interaction:

J_r(Sˆ_ASˆ_B)(Sˆ_CSˆ_D)αααβ=J_r(Sˆ_ASˆ_B) 1

2ααβα−1 4αααβ

=J_r 1

8ααβα− 1 16αααβ J_r(Sˆ_ASˆ_D)(Sˆ_BSˆ_C)αααβ=J_r(Sˆ_ASˆ_D)

1

4αααβ (5.63)

=Jr

1

8βααα− 1 16αααβ

−Jr(SˆASˆC)(SˆBSˆD)αααβ= −Jr(SˆASˆC) 1

2αβαα−1 4αααβ

= −J_r 1

8αβαα− 1 16αααβ the matrix element becomes

αααβ| ˆH|αααβ = − 1

16Jr (5.64)

Applying the same procedure to|αααα|leads to αααα| ˆH|αααα = −1

2(J1+J2+J3)+ 1

16J_r (5.65)

5.14 Check the matrix element of theM_S=2 determinant of the spin Hamil- tonian given in Eq.3.83.

174 5 Towards a Quantitative Understanding Taking the expectation value of the M_S =2 determinant as zero of energy, the following relations emerge to determine the four parameters

E(|abcd|)−E(|abcd|)=J1+J2

E(|abcd|)−E(|abcd|)=J1+J3 (5.66) E(|abcd|)−E(|abcd|)=J₂+J₃

E(|abcd|)−E(|abcd|)=1

2(J₁+J₂+J₃)−1 8J_r

Hence, the extraction of the four-spin cyclic exchange parameter within the spin- unrestricted setting of the DFT approach relies on obtaining converged solutions for the determinants with the required spin distributions, which is not always a trivial task.

Problems

5.1 Zeroth-order description. Write down the matrix of the model space that only considers neutral determinants expressed in local orbitals. Diagonalize the matrix and calculate the singlet-triplet energy difference. What is the state of lowest energy?

5.2 Construction of the CAS(2,2)CI matrix in the symmetry adapted CSF basis. The CASCI matrix given in Eq.5.4uses the fourMS=0 determinants as basis. The matrix can be greatly simplified by a basis set change using symmetry adapted CSFs.

a. Write down the four symmetry adapted CSFs that arise from the linear combi- nations of the four M_S = 0 determinants. The expressions of the states after configuration interaction given in Eq.5.5may give a hint on the CSFs.

b. Calculate the energy expectation values of the four CSFs and place them on the diagonal of the matrix.

c. Identify the CSFs as singlet or triplet spin eigenfunctions and label them by gerade/ungeradespatial symmetry, assuming that the system has an inversion center. How many off-diagonal elements have non-zero value?

d. Calculate the remaining matrix elements to complete the CAS(2,2)CI matrix.

5.3 Spin contamination of the BS state. The relaxation of the magnetic orbitals of the BS determinant in the field of the frozen ROKS core orbitals introduces spin contamination. The amount of spin contamination can be determined analytically by rewriting Eq.5.26in terms of spin adapted CSFs instead of the neutral and ionic valence bond structures.

1. Which term in Eq.5.26is an eigenfunction ofSˆ². Give the eigenvalue of this term.

2. The two other terms have to be written in the form of the singlet

|ab| + |ba| and triplet

|ab|−|ba|

CSFs. Use the trigonometric relations sin²φ+cos²φ=1 and

sin²φ −cos²φ=cos(2φ). Hint: Add and subtract ¹₂cos²α|ba| + ¹₂sin²α|ab| and split the first two terms of Eq.5.26in halfs. Then, order the terms to form the given trigonometric relations.

3. CalculateΦB S| ˆS²|ΦB S using the above derived expressionΦB S = (|ab| +

|ba|)/2+(|ab| − |ba|)cos(2α)/2+(|aa| + |bb|)sinαcosα.

5.4 Kinetic exchange by second-order perturbation theory. Make a second- order estimate of the singlet energy taking into account the interaction between S = ^√1₂

|ab| + |ba|

and the ionic states I₁ = ^√1₂

|aa| + |bb|

and I₂ =

√1 2

|aa| − |bb|

. The energy of the triplet,T = ^√1₂

|ab| − |ba|

, equalsE_{r ef}−K_ab and should be taken as reference.

5.5 Biquadratic exchange versus t₁₃/t₂₄. Express the perturbative estimate ofλ (Eq.5.56) in terms oft₁₃ andt₂₄ with a common denominator for the two terms.

Determine for which values oft₁₃andt₂₄the biquadratic exchange vanishes and for which values it can be expected to be maximal.

5.6 Estimating J_r. Accurate calculations on a polynuclear paramagnetic compound with four S = ¹₂ magnetic centers indicate that the only significant interactions are the following bilinear isotropic interactions: J₁₂ = J₃₄ = −25.1 meV, and J₂₃ = J₁₄ = −39.5 meV. Nevertheless, the experimental temperature dependence of the magnetic susceptibility could not be fitted satisfactorily with these values.

Provide a perturbational estimate for the four-center cyclic exchange to improve the fitting (extra data:U=5.3 eV).

References

1. J. Cabrero, C. de Graaf, E. Bordas, R. Caballol, J.P. Malrieu, Chem. Eur. J.9, 2307 (2003) 2. C.J. Calzado, J. Cabrero, J.P. Malrieu, R. Caballol, J. Chem. Phys.116(7), 2728 (2002) 3. C.J. Calzado, J. Cabrero, J.P. Malrieu, R. Caballol, J. Chem. Phys.116(10), 3985 (2002) 4. C.J. Calzado, C. Angeli, D. Taratiel, R. Caballol, J.P. Malrieu, J. Chem. Phys.131, 044327

(2009)

5. E. Coulaud, J.P. Malrieu, N. Guihéry, N. Ferré, J. Chem. Theory Comput.9, 3429 (2013) 6. R. Bastardis, N. Guihéry, C. de Graaf, J. Chem. Phys.129, 104102 (2008)

Chapter 6

Magnetism and Conduction

Abstract After the description of the electron hopping in systems where not all the magnetic centers have the same number of unpaired electrons, a short account is given of the double exchange mechanism in mixed-valence systems. Although this phenomenon can certainly be found in transition metal complexes, it is more common to happen in doped systems in the solid state. Therefore, the second part of this chapter introduces the basics of the quantum chemical approach to magnetic interactions in extended systems. The embedded cluster approach will be contrasted against band structure calculations. Thereafter, some concepts will be introduced that are widely used in the condensed matter physics community. We do not give a full description of all the magnetic phenomena in solid state compounds but rather help the reader with a quantum chemical background to find its way in the rich literature on this topic.