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50ISSN 2072-5981 doi: 10.26907/mrsej
aänetic Resonance in Solids
Electronic Journal
Volume 21 Special Issue 4 Paper No 19409
1-10 pages
2019
doi: 10.26907/mrsej-19409
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Yoshio Kitaoka (Osaka University,
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Boris Malkin (KFU, Kazan) Alexander Shengelaya (Tbilisi State University, Tbilisi) Jörg Sichelschmidt (Max Planck Institute for Chemical Physics of Solids, Dresden) Haruhiko Suzuki (Kanazawa University, Kanazava) Murat Tagirov (KFU, Kazan) Dmitrii Tayurskii (KFU, Kazan) Valentine Zhikharev (KNRTU,
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* In Kazan University the Electron Paramagnetic Resonance (EPR) was discovered by Zavoisky E.K. in 1944.
Short cite this: Magn. Reson. Solids 21, 19409 (2019)
doi: 10.26907/mrsej-19409
The Lowdin orthogonalization and magnetoelectric coupling
for noncentrosymmetric ions
A.S. Moskvin
Ural Federal University, Lenin str. 51, Ekaterinburg 620083, Russia
E-mail: alexander.moskvin@urfu.ru
(Received May 9, 2019; accepted May 28, 2019; published June 6, 2019)
The Lowdin orthogonalization procedure being the well-known technique, particularly in quantum chemistry, however, gives rise to novel effects missed in earlier studies. Making use of the technique of irreducible tensorial operators, we have developed a regular procedure for the account of the orthogonalization effects. For illustration, we address the emergence of a specific magnetoelectric coupling for noncentrosymmetric 3d or 4f ions.
PACS: 31.10.+z, 75.85.+t.
Keywords: Lowdin orthogonalization, irreducible tensorial operators, magnetoelectric coupling.
Dedicated to Boris Malkin, on the occasion of his 80th birthday
1. Introduction
More than 50 years ago P.-O. Lowdin [1] suggested a regular procedure for the orthogonalization of the atomic functions localized at different sites. The orthogonalization problem was in the focus of the electron structure calculations in 1960-70ies, though later on it was undeservedly forgotten in the shade of the so-called "ab-initio" calculations. However, many interesting points have been missed being overboard the orthogonalization procedure. In this paper, we generalize the Lowdin technique for many-electron atoms and demonstrate that the irreducible tensorial operator method, which is well-known in theory of free atoms, can be successfully applied to "orthogonalized" electrons. As an urgent issue, we address the orthogonalization contribution to the orbital magnetoelectric coupling.
The paper is organized as follows. In Sec. II we demonstrate the conservation of the effective rotation symmetry under the Lowdin orthogonalization technique. In Sec. III and IV we address some novel properties of different operators acting on the orthogonalized basis set. In Sec. V and VI we calculate the electric dipole moment of the noncentrosymmetric quasi-atoms and the orthogonalization contribution to the orbital magnetoelectric coupling.
2. Lowdin orthogonalization procedure
Making use of the one-electron p>/nlm(r) wave functions of a free atom as a basis set for description of the electronic structure in crystals is restricted because of their nonorthogonality for different sites/atoms. One of the more practicable techniques for solution of the problem was suggested by Lowdin [1]. Let us start with a set of standard one-electron functions pK(r) = f/nim(r), where f labels a site, and introduce a Hermitian overlap matrix A = 1 + S as follows: , , Xl ,
AkK = (r)> = SKK> + SkK . (1)
Linear transformation p ^ ^ = pA
^ (r) = E Pk' (r)AK'k > (2)
/
K
yields a set of orthogonalized functions ^K(r):
<^(r)|^' (r)> = SkK .
It is easy to see that matrix elements A obey a matrix equation
AfAA = i , (3)
whose solution can be represented as follows
A = A-1/2B , (4)
where B^B = i, given the existence of the matrix A-1/2 = (i + S)-1/2. Obviously, the choice of the unitary B matrix, hence the certain orthogonalization procedure, is an ambiguous one. The most physically clear and practicable procedure of a so-called symmetric orthogonalization, in the case when B = 1 and A = A-1/2 = ( i + S)-1/2, was suggested by Lowdin [1]. Given a small overlap, when ^K' |SK'K| < q < 1 for all k, the series
(S + S)-1/2 = s - 1S + 3S2 + ... (5)
will converge, and that enables to represent the orthogonalized functions as follows
^(r) = £((1 + S)-1/2)K ipK. (r),
/
K
or
^fnlm(r) = ffnlm(r) — 2 X/ Sf'n'l'm'fnlm ff'n'l'm'(r - Rff') +
f 'n'l'm'
3
+ 8 ^ ^ Sf'n'l'm';f''n''l''m'' S f"n"l"m"; fnlm ff'n'l'm'(r — Rff') + .... f 'n'l'm' f''n''l''m''
(6)
Hereafter the orthogonalized functions ^fnlm can be termed as "quasi-atomic", at variance with nonorthogonalized atomic functions ffnlm.
The two-site overlap integrals obey an analog of the Wigner-Eckart theorem [2]:
J 0i*mi (r)^l2m2 (r - R12)dr = <l1m1|l2m2) =
V(—1)ll-mi( 11 k 12 )sfc(I1I2) Ck*(R12)
^—' \ —rr)i n m^ / H
(7)
-m1 q m2
kq \
where ^ • • • j is a Wigner coefficient [2], k obeys the triangle rule |11 —l2| < k < l1 +l2 given even values of l1 + k + l2. In particular, for l1 = 1, l2 = 2 (pd-overlap), k can have only two values: k = 1,3. In the coordinate system with Oz directed along the radius-vector R12
<10|20) = Spda = —^S^pd) + ^I;S3(pd), (1 ± 112 ± 1) = Spdn = — ^S^pd) — ^3;S3(pd),
where Spda and Spdn are overlap integrals on the a- and n-bonds, respectively.
Thus, taking into account that the bare atomic functions ffnlm(r) form a basis of the irreducible representation of the rotation group, we can rewrite ^fnlm(r) as follows:
^fnlm(r) = ffnlm(r) — -Tl1^ £ £ Sk (f'n'l'; ful) Ck (Rff') X Iff'n'l' (r — Rff')
[ ] k f 'n'l'
l
+
m
+8]l372 £ £ £ [l'']-1/2Ski(f'u'l'; f''u''l'')Sk2(f"u"l"; ful)x (8)
[ ] kik2 f 'n'l' f ''n''l''
Ckl (Rf'f'') x [Ck2 (Rf''f) x fn'l' (r — Rff')]
ll
+ ...
m
for even values of (k + l' + l), (k1 + k2 + l' + l),.... Here we made use of direct products [■ x ■] of irreducible tensorial operators [2], [l] =2l + 1. Unified tensorial form of different terms in the right-hand side unambiguously points to the same transformational properties of the bare atomic function f/nlm(r) and its orthogonalized counterpart ^/nlm(r) (Slater-Koster theorem [3]), however, with regard to rotations of the laboratory system, rather than the physical one. Indeed, the former transformation involves coordinates both of the electron (r) and the lattice (R/), while the latter concerns only the electron coordinates (r). We should emphasize here the preservation of the transformational properties with regard to any rotations rather than certain rotations from the local point group.
Within a linear approximation, the orthogonalization gives rise to a mixing of atomic functions centered at different sites (a nonlocal hybridization), while within a quadratic approximation the on-site mixing of different atomic nlm-functions appears (a local hybridization). The quadratic correction can be written as follows:
A/im(r) = EE ESfc (ff''n'l'; ff''nl) [Ck (R/) x if/nl (r)
k /" n'l'
(9)
where
Sk(ff"n'l'; ff ''nl) =
8[l]1/2
EE (-i)k2 [k]
1/2
kik2 n"l"
k1 0
k2 0
k 0
xSki(fn'l'; f''n''l'')Sk2(f"n"l"; fnl),
k1 k2 k l' l l''
(10)
given even values of k + l' + l,
k1 0
k2 0
k 0
and
k1 l'
k2 l
k l''
are the Clebsch-Gordan
coefficient and 6j-symbol, respectively [2]. Structure of Sk(f f''n'l'; ff''nl), in particular, summation on n''l'' implies significant troubles with its reliable estimates. Usually, this quantity is of the order of Sp2d ~ 0.01 for typical 3d oxides. It is worth noting that the orthogonalization does enhance the role of high-energy unfilled excited states which are characterized by a strong overlap with atomic functions of the filled orbitals of neighboring atoms.
Application of the irreducible tensor technique to the orthogonalized fnlm-functions provides a regular procedure for a revision of different effects which are typical for many-electron atomic systems. Below we address different properties of quasi-atoms which are composed of "orthogonalized" electrons.
3. "Delocalization" of the orbital atomic operators
Delocalization of atomic orbitals in orthogonalized ^/nim functions gives rise to an unconventional effect of a "delocalization" of the orbital atomic operators. Indeed, a local irreducible tensorial operator V^(fnl), working within the conventional nonorthogonalized basis set of the (2l + 1) nlm-states for the f-atom according to the Wigner-Eckart theorem
(nlm|VZ(fnl)|nlm') = (-1)
l—m
l
l
—m a m'
<nl||Ua(fnl)||nl),
(11)
will be described within the basis set of the orthogonalized counterparts ^/nlm by an equivalent operator
m
3
x
a
K(fnl) + 2 [l, a\1/2Sk (ff'nl; ff"nl)
kb
k a b I I I
Vb(fnl) x Ck(Rffn)
£ E [b][k]1/2Sfcl(f'n'l'; fnl)Sk2(f'n'l'; fnl){n'l'\\V0(f'n'l')\\n'l')
[a] k1k2kbf 'n'l'
' ki l' l 1 o k2 l' l 1 [yb(fnl) x Ck(Rff')]0 k a b I 1 ki l' l 1
given even values of k and (a + b), 1 k2 l' l 1 is the 9j-symbol [2], [a, l] = (2a + l)(2l + 1).
Ik a b I
Here vb(fnl) is a b-rank irreducible tensorial operator whose submatrix element equals one. The Exp. (12) describes the delocalization effect for different quantities related to tensorial operators. Note the emergence of tensorial operators lb (fnl) whose rank differs from that of (a) of the bare
k1 k2 k 0 0 0
operator.
4. Effective orbital momentum
Obviously that, at variance with the (pfnlm(r) functions, the ^fnlm(r) functions are not the eigenfunctions for operators l2 and l (the square and z-component of the orbital momentum operator), though they form a basis of the irreducible representation D(l) of the rotation group. Nonetheless, one can introduce an effective orbital momentum, or quasi-momentum l as follows:
ï = -m ( [r x Vr] + E[Rf x Vr,]
(13)
Rf
By analogy with a free atom we can introduce an addition of the orbital quasi-momenta thus forming many-electron configurations and wave functions \LSMlMs) for the 2S+1L terms of the many-electron quasi-free atom, which incorporates all the effects of the orthogonalization of one-electron states for different sites.
The orthogonalization procedure preserves many (though not all) advantages of the free atom theory based on the application of the theory of the rotation group. Thus the Wigner-Eckart theorem for conventional irreducible tensorial operators, such as spherical harmonic Ck(r), orbital momentum l and other orbital operators acting in the conventional r space, does not work on ^fnlm(r)-functions, because matrix elements will depend parametrically on the lattice vectors.
However, in practice, e.g., for Zeeman coupling one should address a true orbital momentum whose relations with effective orbital momentum will have a nontrivial form. Within a \LMl) multiplet, these relations are as follows:
Li = aijLj + aijklLjLkLl + ... , (14)
where we meet unconventional tensorial linear and different nonlinear terms.
The second-rank tensor aij for the systems such as a 3d ion in oxides can be written as follows:
aij = a5ij + A aij , (15)
where a < 1, but Aaj ~ S2 (S is a cation-anion overlap integral), and the difference between a and 1 is of the order of S2. Point symmetry puts distinct limitations on the aj tensor, e.g., for a cubic symmetry, A aj = 0.
a
a
Complex nonlinear relations between true and effective orbital momenta result in a nontrivial form of Zeeman coupling
Vz = »b (L ■ H) = psi L â(2)H + L L L a(4)H + ..] , (16)
where we make use of a symbolic form of a tensorial product of vector operators. Thus the Zeeman coupling acquires novel features due to the orthogonalization procedure: i) a reduction of the effective orbital g-factor as compared with its bare value gl = 1 with its anisotropy for low-symmetry sites; ii) emergence of a nonlinear Zeeman coupling. These features evolve from the orthogonalization procedure as a zero order perturbation effect prior to effects of crystal field and covalency.
5. Electric dipole moment of a "free quasi-atom"
Orthogonalized ^/nlm(r) functions for noncentrosymmetric sites f, at variance with bare f/nlm(r) functions, do not have a definite parity. This immediately gives rise to a nonzero electric dipole polarization of such a "free quasi-atom".
Taking into account the local hybridization effects (10), we can write one-electron matrix element of electric dipole moment as follows
{/almd №/nlm>) = ^=[l]1 EEE^^ 1 ||l'Kl;n'l' Sk (ff''n'l'; ff''nl)x
V3 a.k /'' n'l'
(17)
1 k a I (-W-™ ( I a I l l l' | \ —m a m
a k 1 a q' q
Ck' (R//" ),
where a = 0,2,..., 2l and k = a ± 1 are even and odd numbers, respectively. Here (l||C 1||l') is a submatrix element of a tensorial harmonic [2],
f™ 3 Tnin'i' = Rni(r)r Rn'i'(r)dr Jo
is a dipole radial integral. Thus within the ^fnim basis set with a certain fnl the dipole moment operator can be replaced by an effective (equivalent) operator as follows
dq = E dak(R//' ) Ô°(l) X Ck(R//) , (18)
i—' L J q
a,k,/' q
where va (l) is an orbital a-rank irreducible tensorial operator with the unit submatrix. In Cartesian coordinates the Exp. (18) can be written as follows
di = d(0) + 2 djk {lj, Ik} + terms with a = 4,..., (19)
where {j ,lk } = j I k +1 k j. Of a particular interest are the terms with nonzero values a = 2,4,..., which directly relate the electric polarization to the degenerated or quasi-degenerated orbital state, in particular, to quadrupole (a = 2), octupole (a = 4) electronic momenta of quasi-atoms and give rise to an orbital magnetoelectric coupling. Strictly speaking, the electric polarization induced by the orthogonalization can amount to big values due to both big overlap and radial integrals fnl,n'l' ~ 1 A.
The expression (18) can be easily generalized for many-electron quasi-atomic configurations. For certain terms for the nlN shell of equivalent electrons one should replace the one-
electron operator Va(l) in (18) by the equivalent many-electron orbital operator V^(L) which
x
acts on the \LMl) basis set according to the Wigner-Eckart theorem
(LMl\va (L))\L'Ml } = (-l)L-M
L
L'
-ML a ML
U
(a)
SL ; SL'
where uSL.SL' is the spectroscopic Racah coefficient [4]. For the 2S+1Lj multiplets the V0(l) operator in (18) should be replaced by the equivalent many-electron orbital operator VJ0 (J) which acts on the \SLJMJ) basis set according to the Wigner-Eckart theorem
(SLJMj\Va(J))\SL'j 'MJ '} = (-1)J-Mj
J
J '
U
(a)
-MJ a MJ' ) SLJ;SL'J'
where [4]
U
(a)
SLJ ;SL'J '
= -)S+a+L'+J^(2j + 1) (2 j' + 1) j L' j S ' U(a)
SL;SL' ■
(21)
(22)
A consideration of the linear overlap effects needs the knowledge of two-site dipole matrix elements which can be written as follows:
(Pfnlm\dq \lfif 'n'l'm'} = £ ( —1)
a,a,k,q'
l—m
l
l'
—m a m'
(fnl\\dak \\f 'n'l'}
a
a q
k 1 /
q
(23)
Ck' (Rff ')
Accordingly, the contribution to dipole matrix on the ^fnim basis with a certain set of quantum numbers fnl acquires a tensorial form which is similar to (18). As an obvious practical implication of the Exp. (18), we should point on the calculation of the probabilities of the intra-configurational electro-dipole transitions, e.g., d — d transitions for 3d compounds or f — f transitions for 4f compounds, which are dipole-allowed for noncentrosymmetric quasi-atoms.
Judd-Ofelt theory of effective electric dipole moment for noncentrosymmetric ions
The Judd-Ofelt theory of effective electric dipole moment for noncentrosymmetric ions [5] takes into account the admixing of configurations of opposite parity due to the odd-parity crystal field
vcf = E Akq rk Ck (
(24)
kq
where Akq are crystal field parameters. The effective dipole moment operator can be written out similarly to (18) as follows
i
dq = £ dak va(l) X A
a,k
where
and
dak = -^(2a + l)-(ka),
(ka) = 2(2a + l) E(2l' + l)(-l)l+l' \
l'
(25)
(26)
l a k l l' l
l l l' 000
l' k l \ (4f\r\n'l'}(4f\rk\n'l'}
0 0 0 J "
En'l' — E
nl
(27)
a
a
a
q
6. The overlap contribution to orbital magnetoelectric coupling
For paramagnetic ions with an orbital (quasi)degeneracy, the thermal expectation value ^V£(l) in (18) can strongly depend on the magnetic field, either internal molecular or the external one, thus providing an effective magnetic control of electric polarization. Indeed, in the absence of crystal-field and spin-orbital effects, the thermal expectation value can be represented as follows:
(C(l))T = (V0a(l))T ca (L), (V0a)T = (V0a)0 pa(H,T), (28)
thermal expectation value with (V0a)0 = ( T n T ), and pa(H,T) is a temperature factor,
where (L) is tensorial spherical harmonic with a classical vector L as an argument, <V£) the
L a L -L 0 L
e'g" i (3L2 - L(L + 1))
P0 = 1; P2 = L(2L - 1) * (29)
Within a molecular field approximation Ca (L) = Ca (H). Thus in frames of our simplifications we arrive at a very interesting expression
1
~ ! U „ nki-Q _
a,k,f '
(dq) = £ dak(Rff') <V0°(L))o Pa(H,T) [ca(H) x Ck(R//')] , (30)
that demonstrates a magnetic control of the electric polarization in the most clear way. In Cartesian coordinates the Exp. (18) can be written as follows
di = d(0) + dj(H, T)hj hk + terms with a = 4,..., (31)
where djk(H, T) « p2(H,T), and h = H/H.
For rare-earth ions with 4fn shell and strong spin-orbital coupling, the electric polarization for the SLJ-multiplet can be easily written as follows:
(dq ) = E M)'™ J] {LU
x
i J L k . .
a,k,/' K ) (32)
xdak(R//') <Voa(J))o Pa(T) Ca(H) x Ck(R//)
1
It should be noted that the expressions (30) and (32) define both the field and temperature dependence of the electric dipole moment, however, given a complete neglect of the the crystal field quenching effects. These effects are usually much stronger than Zeeman coupling, in particular, for 3d ions. However, for some rare earth ions with so-called quasi-doublets in ground state the Zeeman and crystal field effects can compete with each other, and our Exps. (18) and (32) predict a rather strong magnetoelectric coupling. Such a situation is realized, e.g., for Tb3+ ion in the multiferroic terbium manganate TbM^Os (see, e.g., Ref. [6]).
For the 3d ions, the crystal field quenches both the orbital momenta and orbital magne-toelectric coupling. However, due to the spin-orbital coupling/mixing, the orbital effects give rise to different effective spin interactions, e.g., the emergence of an orbital contribution to effective spin g-tensor and single-ion spin anisotropy. Furthermore, the orbital operator Va (L) given a = 2, 4, . . . for orbitally nondegenerated ground state can be replaced by an effective spin operator. This is relatively easy to perform for so-called S-type 3d ions, which have an orbitally nondegenerate ground state of the A\ or A2 type for a high-symmetry octahedral, cubic or tetrahedral crystal field. These are ions, e.g., with 3d3 (Cr3+, Mn4+), 3d5 (Fe3+, Mn2+),
q
3d8 (Ni2+, Cu3+) configurations with an octahedral surroundings. Strictly speaking, first we should replace the V^(L) by a linear combination
K1 (L) = £ cVa(L), (33)
a
which form a basis of the irreducuble representation DY (line v) of the point symmetry group, Oh or Td. Here 7 = E or T2 given a = 2, 7 = A\, E, T\ or T2 given a = 4. Next we can replace the orbital operator by a spin equivalent as follows
(V17(L))gs = naiVvaY(S), (34)
where the (...)GS means a mapping on the ground state. The naY parameters depend on the type of the 3d ion: n2Y ~ (Z3d/A)2 < 10-3, here (3d is a spin-orbital constant, A is a mean energy of excited T-terms. Thus for S-type ions we can transform the orbital dipole moment operator (18) into an effective spin operator which acts within the SMS-multiplet of the ground state as follows
V = £ <7)Va7 (S), (35)
ajv
where
4VY) = Z dak(Rff')na7Z C
a,k,f' aq'
a k 1
, Ckq/ (Rff). (36)
—- a q q q
aq' L
As in (33), VV1 (S) = a caaVaa(S), however, with a more simple spin matrix for VOf(S):
(SMslKmsMs,) = (-1)S-M^ -SMs a M:sl) . (37)
The spin irreducible tensorial operators VV1 (S) can be transformed into a Cartesian form, e.g.
E (S)=2 ;(s)=JS-Sw ■... ■ (38)
where (2S + 3)(5) = (2S + 3)(2S + 2)...(2S - 1). In Cartesian coordinates the Exp. (35) can be reduced to a standard form
di = d(0) - 1 Rijk{S j,Sk} + terms with a = 4,..., (39)
widely adopted in the theory of the electric field effects in electron spin resonance (ESR) [7].
The magnetoelectric effect due to local noncentrosymmetry was addressed recently in Ref. [8]. The authors started with the nonorthogonalized basis set of 3d orbitals, next considered an odd-parity crystal field and spin-orbital coupling as perturbations. Finally, they arrived at an expression which is similar to (39). Furthermore, in fact, the microscopic consideration in Ref. [8] reproduces, however, with strong simplification the well-known paper by Kiel and Mims [9] on electric field effect in ESR for Mn2+ ions in CaWO4.
It is interesting that the experimental findings of this and many other papers on the electric field effect in ESR [7] can be used for a direct estimation of the single-ion contribution to the magnetoelectric coupling in different multiferroics. For instance, for well known multiferroic MnWO4 [10] one might use the parameters Rijk measured for Mn2+ ions in CaWO4 [9]: being normalized to unit cell volume of MnWO4 [10] (Vc 138 A3), these are as follows: | R1231 ~ 1.8, |Rii3|~ 1.5, |R3ii|~ 0.3, |R3121 ~ 5.0^Cm-2 (R223 = -R113, R213 = R123). It should be noted
that for Mn2+ ions in SrWO4 these parameters are nearly two times larger [11]. Taking account of two Mn2+ ions in unit cell and a rather large value (< 6) of the spin factor in (39), one may conclude that the single-ion mechanism can be a significant contributor to the ferroelectric polarization observed in MnWO4 (Pb ~ 50 ^Cm-2) [10]. It is interesting that very recent quantitative estimates of the spin-dependent ferroelectric polarization in MnWO4 based on the low-energy model, derived from the first-principles electronic structure calculations [12] showed values which are an order of magnitude less than the experimental ones.
It is worth noting that the single-ion term (39) does not produce the magnetoelectric coupling for quantum spins S = 2 (e.g., Cu2+) due to a kinematic constraint: 0 < a < 2S.
7. Summary
Making use of the Lowdin orthogonalization for one-electron atomic (ffnlm(r) wave functions, we arrive at a basis set of orthogonalized ^fnlm(r) orbitals which formally preserve symmetric properties of the bare (ffnlm(r) orbitals. Instead of many-electron atomic configurations composed of non-orthogonalized nlm-orbitals, we arrive at quasi-atoms composed of the orthogonalized nlm-counterparts. Formal conservation of the rotational symmetry allows to apply the powerful technique of irreducible tensorial operators [2] and Racah algebra [4] to description of the quasi-atoms, in particular, to the overlap contribution in the crystal field parameters [13].
As an illustration, we addressed a single-ion contribution to magnetoelectric coupling which is usually missed in current studies on multiferroics. A regular procedure has been developed for calculation of the overlap contribution to the single-ion orbital magnetoelectric coupling both for 3d- and 4f-ions. In a sense, the overlap contribution resembles the point charge contribution to a crystal field, this correctly describes both the lattice symmetry and the symmetry of electronic states, and provides reasonable semiquantitative estimates. Furthermore, making use of experimental data for the electric field effect in ESR, we have shown that the single-ion magnetoelectric coupling can be a leading mechanism of multiferroicity, e.g., in MnWO4.
Acknowledgments
Supported by Act 211 Government of the Russian Federation, agreement No 02.A03.21.0006 and by the Ministry of Education and Science, projects No 2277 and No 5719.
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