1. What happens if we do not use the SPINLOCK directive?
2. Why do we apply the SPINLOCK directive along 30 cycles?
3. How can we visualize spin density distribution?

Answers:

1. If the SPINLOCK directive is omitted, there is no guarantee a priori that the SCF cycle may proceed to the FM solution spontaneously, when starting from a guess Fock matrix of isolated atoms or ions. In this case, if the SCF cycle is not driven by SPINLOCK to the proper FM phase spin configuration, it drifts towards an unstable metallic nonmagnetic state (i.e. a non polarized closed shell or diamagnetic phase).
2. In general, the SPINLOCK directive needs to be active until the SCF process has found a numerically stable path along which the system converges to the required configuration regularly. Actually, in this case, the FM solution can be obtained by applying the SPINLOCK constraint only once, at the beginning of the SCF cycle.
3. Spin density maps are certainly the best representations of spin density distribution. However, useful and synthetic information is gained from Mulliken population analysis (MULPOPAN directive in the SCF input section of CRYSTAL), that allows to see how spin density is distributed to atoms or, in more detail, to atomic orbitals. In this case, Mulliken population analysis shows very clearly that all the 5 unpaired electrons are almost completely localized on the Mn ion and populate d atomic orbitals. This picture is in substantial agreement with the very simple model of an isolated ion in a crystal field, that is commonly referred to in chemical language and indicates that the d atomic orbitals are hardly part of the valence structure of the crystal.

Exercise 2: Verify that, when the SPINLOCK directive in exercise 1 is omitted, a diamagnetic phase is obtained with metallic properties.

Exercise 5: Repeat exercise 4 adding the ATOMSPIN directive in the SCF input section as specified above.
 

Exercise 6: Run CRYSTAL for the FM KMnF₃ using the (double) super cell specified above, save fortran unit 9 and copy to unit 20, then repeat exercise 4 adding the GUESSP and SPINEDIT directives in the SCF input section as specified above.

1. Which of the two methods, a) and b), is to be preferred?

Answer:

1. In many cases the two techniques perform equally well and the choice is arbitrary. However, in those cases where the isolated atom initial guess is too far from the final solution, convergence is not easily achieved following a). Although the AFM phase is not simply like a spin alternant FM phase, the use of the FM solution through the GUESSP and SPINEDIT directives is often a valid alternative.

The FM-AFM energy difference (E)

Exercise 7: Use the results of exercises 1 and 4 to calculate the FM-AFM energy difference for KMnF₃.

Which is the more stable phase?

Is the method accurate enough for the calculation of E?

Since magnetic phase transitions involve quite small E, of the order of 10^-4 hartree, special care must be paid in the determination of the total energy for the two phases. In order to check whether they are accurate enough to give meaningful energy differences, the following tests can be done:

1. Does the use of super cells affect E determination?
   We compare the total energy of the FM phase obtained in exercise 1 (unit cell, 1 Mn ion) with that obtained with the super cell defined by vectors 1 (2 Mn ions per cell, 10 unpaired electrons): -2047.643082 and -4095.286163 hartree, respectively. The super cell energy is almost exactly twice the unit cell energy, the error is smaller than 1 ^.10^-6 hartree, i.e. two orders of magnitude less than E.

   Note: One of the crucial points of computational solid state chemistry and physics in this decade is the implementation of algorithms able to provide high numerical accuracy. If you have a periodic code at home, check one aspect of its numerical accuracy by performing the following test: evaluate the total energy of a system with a unit cell and with a super cell. The total energy should obviously scale linearly with the volume or the number of atoms.

2. Experimental or calculated equilibrium geometry?
   The small value of E certainly leads to exclude a priori that any significant geometry variation may take place during the FM-AFM transition, so that both phases can be described as having essentially the same geometry. On the other hand, as E decreases with the Mn-Mn distance very rapidly, small errors in the determination of equilibrium geometry (typically of the order of 2% within the HF approximation) may affect E with large errors. For these reasons we are discouraged from optimizing geometry, and working at the experimental geometry is certainly to be preferred.
3. Is the basis set used in the exercises adequate?
   Data in table 1 show how E varies when improving the basis set.
   

   Table 1: Total energy difference between the ferro- and antiferromagnetic phases (E) of KMnF₃ as a function of the basis set.

   case Basis set DE a as used in exercise 1 0.293 b case a + d on F (=0.7) 0.293 c case a with 5-1d on Mn instead of 4-1d 0.292 d case c + sp (=0.25) on Mn 0.334 e case d + d (=0.4) on K 0.325

   
   Case e can be considered close to the HF limit. E changes within 15% in the variational basis set range considered. Basis set incompleteness may certainly not be invoked as an explanation of the much larger discrepancies between the calculated and experimental E.

The calculation of the magnetic exchange coupling constant J is particularly important for comparison with experiment. Ising's simple model of a spin lattice allows to relate J to E [3, 4], provided the unpaired electrons are fairly well localized and the magnets interact as Ising magnets. In fact, Ising hamiltonian [5] is a particular case of Heisenberg spin hamiltonian where it is assumed that spin-spin interaction takes place through only one magnetic component (along the z direction):

(2)

where the spin operators are applied to ions at the R and R' lattice nodes. Although, in general, this is not the case, the application of the Ising model is much easier than the use of Heisenberg hamiltonian and is more compatible with our monodeterminantal solution, which corresponds to non pure spin state for general open-shell systems. In addition, it has been shown [6] that eq. 2 is obtained from the Heisenberg hamiltonian when the mean field approximation is applied.
How is eq. 2 further simplified?

• since our model is periodic, if we neglect spin density at all atoms other than Mn, we may limit the first summation in the equation to the Mn ions of one unit cell, that are symmetric (or spin-antisymmetric).
• considering that the super exchange interaction is short ranged, as a first approximation, coupling with only next nearest neighbour Mn ions needs to be included.

E = - N Z Sz2 J

(3)

i.e. E is proportional to the number (N) of symmetric Mn atoms in the (super)cell, the number (Z) of the next nearest (Mn) neighbours to each of the N Mn with opposite spin, the square of the Mn spin component along the n direction. The proportionality constant is J.

Exercise 8: Use the result of exercise 7 to calculate J for the FM-AFM transition. ( Note: experimental J are usually reported in Kelvin; the conversion factor to express J in K from E in atomic units is 3.1577^.10⁵).
 

Breaking symmetry: KCoF3 [2]

Exercise 9: Run CRYSTAL with file kcof3.d12 as an input and discuss Mulliken population analysis data with particular reference to the d atomic orbitals of Co.
 

The result of exercise 9 shows that spin density is large only at the d atomic orbitals of the transition metal, just as happens with KMnF₃. Therefore, the simple model of a Co ion in a crystal field can be helpful in the interpretation of the transition metal electronic configuration in this case, too. According to crystal-field theory, the d-levels of each Co ion, at the centre of a regular octahedron formed by 6 F ions (point group: O_h, or m-3m), split into two sets of degenerate levels, which group theory classifies as t_2g (triply degenerate) and e_g (doubly degenerate), respectively. The d_xy , d_xz , d_yz atomic orbitals are a basis for the t_2g irreducible representation, d_2z²-x²-y² and d_x²-y² for e_g and elementary electrostatics considerations suggest that t_2g be lower in energy.

Is the result of exercise 9 correct?
Hund's rules say how the d levels of an atom must be populated: since the atomic configuration of Co includes 7 electrons in the d orbitals, maximum spin multiplicity is gained with two doubly- and three singly-occupied d levels. Actually, Mulliken atomic orbital population data in CRYSTAL output from exercise 9 do not comply with Hund's rules. In fact, due to symmetry constraints only the two e_g levels are singly-occupied and degeneracy imposes that nearly 1.7 |e| are accommodated in each of the t_2g orbitals. A more stable configuration can be obtained only by partly removing the t_2g degeneracy and allowing Jahn-Teller distortion. This means that crystalline KCoF₃ must have lower symmetry than cubic. For example, if the CoF₆ octahedron is distorted along one of its principal directions, say z, symmetry is lowered to D_4h (4mmm) and the t_2g representation is no longer irreducible, so that it can be decomposed into two terms: e_g and b_2g. d_xz and d_yz are the basis for the doubly degenerate e_g and d_xy for the monodimensional representation. This corresponds to the transformation from cubic symmetry to tetragonal.

How do we tell CRYSTAL to reduce symmetry?
One possibility is consulting the International Tables of Crystallography to find the new space group in the list of the non-isomorphic subgroups of Pm-3m. The group we are interested in is that where all symmetry relationship between z and the orthogonal coordinates x and y are missing, i.e. P4mmm. This is the only piece of information we need to prepare the new geometry input for CRYSTAL.
The other possibility is to cheat CRYSTAL. The ELASTIC keyword in the geometry input section (see CRYSTAL User's Manual) allows distortion of the lattice for the calculation of elastic constants. In this case, input of the following cards:

ELASTIC -2 0. 0. 0. 0. 0. 0. 0. 0. 0.000000001

Keyword to perform an elastic deformation of the lattice deformation in terms of e strain tensor   eij matrix elements

has the effect of automatically reducing symmetry (to P4mmm, in this particular example) without actually changing the lattice geometry. In fact the matrix given in input is summed to identity and multiplied by the starting lattice vectors from the left, so that, in this case, nor the c lattice parameter is changed, being multiplied by 1.000000001.

Exercise 10: Repeat exercise 9 breaking lattice symmetry (from cubic to tetragonal). Compare total energy and Mulliken atomic orbital populations to those previously obtained.

Is there a way to converge CRYSTAL more rapidly to the desired solution?
To speed up convergence in the SCF cycle the EIGSHIFT directive in the SCF input section can be conveniently used as follows:

EIGSHIFT 1 1 6 5 0. .3

Keyword to alterate the orbital occupation before SCF number of elements to be shifted (diagonal only) atom label, nr. of shells in the selected atom basis set, nr. of AO in the selected shell, a-spin shift, b-spin shift

This directive allows to modify diagonal Fock matrix elements at the beginning of the SCF cycle. If this is properly done, the system is encouraged to converge to the solution corresponding to the hypothesized electronic configuration. In this particular case, we identify the first atom in the output list as Co, its sixth shell as that of the d atomic orbitals and, among these, the fifth component of the d shell, i.e. the d_xy atomic orbital. We add 0.3 hartree to the diagonal element corresponding to the d_xy atomic orbital in the -spin matrix and leave the a-spin matrix unmodified.
 

Exercise 11: Repeat exercise 10 using the EIGSHIFT directive to speed up convergence.

1. Why has KCoF₃ been modelled as a cubic crystal?
2. What action is taken with the EIGSHIFT directive?
3. Is the EIGSHIFT directive only used to speed up convergence?

Answers:

1. The removal of degeneracy in the t_2g orbitals of Co (Jahn-Teller distortion) due to symmetry loss corresponds to a real deformation of the KCoF₃ crystal, that becomes tetragonal with lattice parameters a ¹ c. However, since the difference between them is small, a = c have been used in the exercise, as for a cubic crystal. Symmetry breaking is sufficient to allow for Jahn-Teller distortion, even if it does not correspond to a real change in geometry.
2. The -d_xy diagonal element in the initial Fock matrix is altered in such a way that the corresponding orbital is emptied and becomes part of the virtual manifold. The expected result is that the SCF cycle proceeds to a solution where the -d_xy energy band is raised above Fermi energy.
3. In this case, the use of the EIGSHIFT is not necessary to drive the system to a particular state, which is simply obtained by breaking symmetry; however, there are many cases where the energy surface is characterized by a more complex structure of local minima and EIGSHIFT may become important in driving the system to one particular spin configuration. For example, if a double super cell is used for the calculation of the FM wave function of the KCoF₃ crystal, convergence is not gained as easily as in exercise 10, because there is now more variational freedom, and use of the directive is necessary to get the result.

Exercise 12: Calculate E (the FM to AFM transition energy) of KCoF₃, following the same procedure as for KMnF₃.

The asymmetry introduced by Jahn-Teller distortion complicates the calculation of the magnetic coupling constants in this case. Since the nearest neighbouring Co ions of each Co along the z direction now differ from those on the xy-plane, two different coupling constants, J^xy and J^z, will correspondingly appear in eq. 2, when limiting the summation to six nearest neighbours. Eq. 3 can be applied to KCoF₃ only if we consider an average J, otherwise a more appropriate equation is needed:

E = - N Sz2 (Zxy Jxy + Zz Jz)

(4)

where Z^xy and Z^z denote the number of the next nearest neighbours Co with opposite spin of a Co ion in and out of the xy-plane, respectively, i.e. Z^xy = 4 and Z^z = 2. The hypothesis of additivity suggests that the other relation that is necessary to calculate J^xy and J^z can be found by considering a different AFM phase. A good candidate is that phase with the spin configuration shown in the right picture of fig. 1, we refer to as AFM'. What shape has the super cell in this new lattice? A suitable choice is the following linear combination of the unit cell lattice vectors (a_i):

b1 = a1

b2 = a2

b3 = 2 a3

Exercise 13: Run CRYSTAL for AFM' KCoF₃ and calculate the FM to AFM' energy difference E.
 

From the application of eq. 2 to AFM' a new relation follows:

E = - N Sz2 Zz Jz

(5)

Exercise 14: Calculate J^xy and J^z from eqs. 4 and 5.
 

Bibliography

1.

R. Dovesi, V.R. Saunders, C. Roetti, M. Causà, N.M. Harrison, R. Orlando, C.M. Zicovich-Wilson, CRYSTAL98 User's Manual, Università di Torino, Torino, 1999.

2.

R. Dovesi, F. Freyria Fava, C. Roetti and V.R. Saunders, Faraday Discuss. 106, 173 (1997).

3.

L.J. de Jongh and R. Block, Physica B 79, 568 (1975).

4.

L.J. de Jongh and R.A. Miedema, Adv. Phys. 23, 1 (1974).

5.

N.W. Ashcroft, N.D. Mermin, Solid State Physics, Holt Saunders International Editions, New York, 1976, Chapter 33.

6.

Yen-Sheng Su, T.A. Kaplan, S.D. Mahanti and J.F. Harrison, Phys. Rev. B59, 10521 (1999).

Further reading

(by I. de P.R. Moreira)

Basis treateses on the general theory of magnetism: from introductory to more general aspects of molecular and periodic magnetic systems:

[1]

"Solid State Physics" by N.W. Ashcroft, N.D. Mermin, Saunders College Publishing, New York, 1976, Chap.s 31-33 (clear exposition of basic concepts).

[2]

J.H. van Vleck, Rev. Mod. Phys. 50, 181 (1978). (Brief history of magnetism up to his actual status)

[3]

"The theory of magnetism" by D.C. Mattis, Springer Series in Solid State Physics, Vols 17 (Part I: statics and dynamics) and 55 (Part II: thermodynamics and statistical mechanics), Springer-Verlag, Berlin Heidelberg, 1981-1985. (Part I is maybe the clearer exposition of the macroscopic theory of magnetism)

[4]

"Quantum theory of magnetism" by R.M. White, Springer Series in Solid State Physics, Vol. 32, Springer-Verlag, Heidelberg 1983.

[5]

"Theory of magnetism" by K. Yosida, Springer Series in Solid State Physics, Vol. 122, Springer-Verlag, Berlin Heidelberg, 1996. (Excellent exposition of the fundamental theory for solids)

[6]

"Molecular Magnetism" by O. Kahn, VCH Publishers, New York, 1993. (Clear and recent review of magnetism in molecules and low dimensional systems. Gives an accurate analysis of super exchange from the quantum chemistry point of view)

[7]

"Magneto chemistry" by R.L. Carlin, Springer-Verlag, Berlin, Heidelberg, 1986

[8]

L.J. de Jongh and A.R. Miedema, Advances in Physics, Vol. 32, Taylor and Francis, London, 1974. (review of the magnetic properties of 1-, 2-, and 3-dimensional magnetic systems. Also very interesting magnetic data (type of coupling, dimensionality or magnetic coupling constants) a large number of solid systems is given)

[9]

"Spin systems" by W.J. Caspers, Word Scientific Publishing, Singapore, 1989. (Small treatise on the spin models and hamiltonians used to describe the magnetic systems)

Papers with applications of the CRYSTAL code to study several magnetic systems:

[10]

M.D. Towler, N.L. Allan, N.M. Harrison, V.R. Saunders, W.C. Mackrodt, and E. Aprà, "Ab initio study of MnO and NiO", Phys. Rev. B 50, 5041-5054 (1994).

[11]

J.M. Ricart, R. Dovesi, C. Roetti and V.R. Saunders, "The electronic and magnetic structure of KNiF₃ perovskite", Phys. Rev. B 52, 2381-2389 (1995); (E) ibid 55, 15942 (1997).

[12]

R. Dovesi, J.M. Ricart, V.R. Saunders and R. Orlando, "Superexange interaction in K₂NiF₄. An ab initio Hartree-Fock study", J. Phys.: Cond. Matter 7, 7997-8007 (1995)

[13]

M.D. Towler, R. Dovesi and V.R. Saunders, "Magnetic interactions and the co-operative Jahn-Teller effect in KCuF₃", Phys. Rev. B 52, 10150-10159 (1995).

[14]

M. Catti, G. Sandrone, G. Valerio and R. Dovesi, "Electronic, magnetic and crystal structure of Cr₂O₃ by theoretical methods", J. Phys. Chem. Solids 57, 1735-1741 (1996)

[15]

R. Dovesi, F. Freyria Fava, C. Roetti and V.R. Saunders, "Structural, electronic and magnetic properties of KMF₃ (M = Mn, Fe, Co, Ni)", Faraday Discuss. 106, 173-187 (1997).

[16]

A. Chartier, P. D'Arco, R. Dovesi and V.R. Saunders, "An ab initio Hartree-Fock investigation of the structural, electronic and magnetic properties of Mn₃O₄-Hausmannite", Phys. Rev. B 60, 14042-14048 (1999)

[17]

P. Reinhardt, I. de P.R. Moreira, R. Dovesi, C. de Graaf and F. Illas, "Detailed ab-initio analysis of the magnetic coupling in CuF₂", Chem. Phys. ett. 319, 625-630 (2000)

Acknowledgements

Thanks to R. Dovesi and I. de P.R. Moreira for reading the notes and for fruitful suggestions