Calculation of the matrix elements of the long-range Coulomb interaction in low-symmetry crystals Текст научной статьи по специальности «Физика»

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Short cite this: Magn. Reson. Solids 20, 18105 (2018)

Calculation of the matrix elements of the long-range Coulomb interaction in low-symmetry crystals

O.A. Anikeenok

Kazan Federal University, Kremlevskaya 18, Kazan 420008, Russia

E-mail: oleg.anikeenok@kpfu.ru

(Received November 27, 2017; revised May 26, 2018; accepted May 28, 2018; published June 4, 2018)

The expressions for calculating the matrix elements of the Coulomb interaction between an electron and a low-symmetry infinite crystal lattice have been obtained. One-center matrix elements are considered. The Gaussian type of orbitals (GTO) is used in calculations. All expressions are absolutely and rapidly converging series in the space of reciprocal lattice vectors.

PACS: 61.50.Ah, 61.72.S

Keywords: localized orbitals, infinite crystal lattice, Madelung energy, orbital energy

1. Introduction.

Currently, the value of the long-range Coulomb interaction (LRCI), i.e. Coulomb interaction between an electron and an infinite crystal lattice in the case of low symmetry is estimated using the Madelung constant, in other words, by calculating the electrostatic potential at the lattice site, i.e., a point [1-4]. The expressions for calculating the LRCI matrix elements in the case of orthorhombic lattices are given in [5]. Using the results [5], the expressions were obtained in [6] for calculating the LRCI matrix elements on p and d-orbitals and the crystal NaV2O5 was considered. For this crystal, in particular, the Madelung energies and diagonal matrix elements on p and d orbitals were calculated. If for vanadium ions the values of these quantities coincide well enough, they differ markedly for oxygen ions. For example, the following estimates were obtained for one of the oxygen ions: EM =-1.18856 a.u., E (pz) = - 0.96122 a.u. The difference between these quantities is on the order of 6 eV. The energy of px, py orbitals is E(px) = -1.02686 a.u., E(py) = -1.18351a.u, respectively. Thus, the splitting ofthe diagonal matrix elements is on the order of 2-6 eV.

The LRCI matrix elements enter, for example, the expressions for the ab initio calculations of the amplitudes of electron transfer between ions [7]. It can be seen that the LRCI estimates for the transition amplitudes in a given crystal by the methods [1-4] can lead to the incorrect interpretation of the experimental data. In [8], the expressions were obtained for calculating the LRCI matrix elements on f-orbitals. The expressions obtained were used in estimating the crystal field parameters for the impurity centers considered in [8]. The improvement in agreement with experiment was obtained in comparison with standard methods. In this paper we obtain expressions for the calculation of such matrix elements in the case of low symmetry.

2. General part

Let the radial part Rnl (r) of the ionic orbital ynlm (r) on which the electron is located, have the Gaussian type of orbitals (GTO) form

Rnl (r) = !a/e-*r: (1)

i

Let the vectors a1, a2, a3 are the vectors of the cell of the triclinic lattice. We denote by Rl = l1a1 + l2a2 + l3a3 the vector determining the position of the unit cell, a r and rp are vectors of ions in the unit cell. We consider the isolated ion determined by the vectors R 0 + r, R 0 = 0. The charge qp is in the site Rl + rp. Then the matrix element of the Coulomb interaction of an electron with a charge

Calculation of the matrix elements of the long-range Coulomb interaction in low-symmetry crystals qp defined on the wave functions of the isolated ion has the form (in a.u.) [5, 8]

JH (r -r )(-qp lr - (R'+rp )\1) h (r - r )dV=JH (r) (-qP lr - (R'+rp - r )l1) h (r) dV

Let us denote R = Rl + rp - r-. We define the functions F'p (n1n2n3) as

2xFp,j (n1n2n3) = -qpZatbk Jxnynz"3 |r-R|-1 exp[-(a,. + )r2]dxdydz.

(3)

The matrix elements (2) on the orbitals of the isolated ion can be expressed in terms of the F'p (n1n2n3) functions. For example, the matrix element for the y/pz (r) = (3/4^)12 Za,.z exp (-a,.r2) orbital has the form

J H (r) (-qp |r - R|-1) Hp z (r)dV=-F (002).

We write the unit vectors of the triclinic lattice with respect to the Cartesian coordinate system in the form

a1 = ae1, a 2 = be 2, a3 = ce3, (4)

e1 = i, e 2 = cosai + sinaj, e3 = sin0cospi + sin0sinpj + cos0k. (5)

Then for the Rl, rp , r- vectors we obtain the expressions

R l = l1a1 + l2a2 + l3a3 =

= (l1a + l2b cos a + l3c sin 0 cos p) i + (l2b sin a + l3c sin 0 sin p) j + l3c cos 0 k,

rp = Xpa1 + ypa2 + Zpa3 =

= (xpa + ypb cosa + zpc sin0cosp)i + (ypb sina + zpc sin 0 sin p) j + zpc cos 0 k,

rj = Xa1 + y-a 2 + Zja3 =

= (xja + yb cosa + zc sin0cos p) i + (yjb sina + zc sin0sinp) j + zc cos0k.

The components of the |r - (Rl + rp - rj)] vector in the Cartesian coordinate system according to (6)-(8) are written as

x - |(l1 + xp - xj)a + (l2 + yp - yj)bcosa + (l3 + zp - zj)csin0cospj = x - A1, (9)

y -[(4 + yp - y-)bsina + (l3 + zp - z-)csin0sinp] = y -4, (10)

z-(l3 + zp -zj)ccos0 = z-A3. (11)

Further we present the function F'p j (n1n2n3) in the form convenient for calculations. To do this, we use the transformation

(6)

(7)

(8)

1 2 w

—R = -J=J dv exp [-(r - R)

2 v2

(12)

After the transformation (12) and integration with respect to x, y, z in (3), we obtain

K, j ("1"2 "3) = - qP Z abbk J

dv

x exp

,k 0

/ 2 \3/2

(k + v )

n

[",/2] "s! Z

ms =0

(Av=)"

4^m s - 2ms) ( + v2) "

(13)

2 I akV

a k + V

where ak = a + fSk, \ns /2] is the integer part of the number in brackets ns /2.

Using the transformation

v 2 = aitU

dv

1 -i

( + v 2)

du

a

we obtain

F

ab. r

, (n1n2n3) = -qp X — j dun

i,k aik 0 s=1

X (1 - u2 f (2 )n

exp

X A

(14)

(15)

(4a. )m ms !(ns - 2ms)!

Let us ks = ns - 2ms. After multiplying the three braces in (15), we select the products

A. A.2 A.3 exp\_-alku2 (Af + A2 + A)] , (16)

which are present in each term of the resulting sum. We introduce the r = (x,y,z) vector setting

(xp - Xj) a = x, (yp - yj) b = y, (zp - zj) c = z. Then Aj are written as

A1 = (l1a + x) + (l2b + y) cosa + (l3c + z) sin^cos^, (17)

A2 =((2b + y )sina + ((3c + z )sin^sin^, A3 =((3c + z )cos#. (18)

Since the positions of the ions in the unit cell are arbitrary, we assume that the r = (x,y,z) vector is defined at all points of the unit cell. We introduce the function D (r, k1, k2, k3) defined in the unit cell

D (r, .1, k2, k3 )= X A A22 A3. exp [-a.u2 (A + A22 + A32)] . (19)

l ,l2 ,l3

The function D (r, k1, k2, k3) is a periodic function of r, in the space with elementary translations a, b, c of the orthorhombic system. The same as in [5], integration over the unit cell can be transformed into integration over the whole space when finding the Fourier transform of the function D (r, k1, k2, k3). Then we have

D (r, .1, .2, .3) = X D ( g, .1, .2, .3) exp [i (gr )], (20)

g

D (g, .1, .2,.3) = V- j xf1 x22 x.3 exp {-a. [xf + x22 + x32 ]u2 - i (gx + gyy + gzz)dxdydz, (21)

x1 = x + y cosa + z sin^cos^, x2 = y sina + z sin^sin^, x3 = z cos#,

2k n

2nn„

2k n

(22)

(23)

a b c

where vc = abc is the volume of the unit cell of the orthorhombic system, g is the reciprocal lattice vector, nx, ny, nz are integer numbers, i is the imaginary unit. We introduce new integration variables x1, x2, x3. Then (21) will be written as

D (g, .1, .2, .3 ) = — j x1*1 xf2 x3 exp {-a. [ x^ + x2 + x32 ] u2 - i ( x1 + g2 x2 + g3 x3 )|dx1dx2dx3, (24)

Vc V

where vc = abc sin a cos d is the volume of the unit cell of the triclinic symmetry crystal under consideration, and

g1 =.

g 2 ="

^os agx + gy

sin a

g3 =

sin^sin (^-a)gx - sin#sin^gy + sin agz

sin a cos d

(25)

c V

Calculation of the matrix elements of the long-range Coulomb interaction in low-symmetry crystals Performing the integration in (24), we obtain

.3/2

D (g, ki, k2, k3 ) = "3 3/2 eXP

f

3 3/2

vu3al

4aku

23

П

V ^ ik" J

k.-2h.

k I

k. ! ^ _

^ h.=0 hsI(k. -2h.) I(ikU2)

We introduce the function F(n2n31 r), using the expressions (15), (20), (26).

-3/2 и 1 A 3

(ab, гйИтп

F(„«I r) = -Ц-YBidП

i ,k aik 0 u .=1

nslY (1 -u2r

m.. =0

(4aik J

m„ I

Y

2

h =0 hs ( - 2ms - 2hs )!

V a J

(27)

exp

4aku

ig r

V ik1

where Z. =- ig. /2a,k.

We substitute x = (xp - xj)a, y = (yp - ) b, z = (zp - zj) c into the expression obtained (27) and perform the summation over the rp = (xpa, ypb, zpc) vectors of the unit cell. These transformations are

cumbersome, but fairly simple and analogous to the transformations in [5, 6] for the orthorhombic lattice. As a result, we obtain

Fj (П1П2П3 ) = Y F (П1П2П3 |rp - Г j )

n

.3/2

Zabk г du

J 7T

XY f (^ g1 )f (^ g 2 )f (nз, g3 )eXp

i ,k "ik 0 ;2 V

4a,u

V "ik" J

Y qp exp \'g (rp - ]

(28)

V p

where g (rp - rj ) = gx (xp - xj ) a + gy (yp - yj ) b + gz (zp - zj ) c,

[v? (1 - u2)

f (n., g. )= Y z—^—: Y

=0 (4ak )). ms I h,=0 h. ( - 2m. - 2h.)1

n, - 2m. - 2h. f 2 [n/2] vn-2t

= Y Z

V 4ak J

( л V

t I(n - 2t )I

V 4ak J

(29)

For example f (0,gs) = 1, f (1,gs) = Zs, f (2, ~g,) = 1Z2 + J-, f (3,g,) =1Z3

2I . 4ak

3I . 4ak

f (4, g. )=1 Z.4 Z2

1

4I 8a." 32aik

It can be seen that the functions f (ns, gs) do not depend on the integration variable u. We call the sum overp in parentheses in (28) as the structure factor Gj (g) and present it in the form

Gj (g ) = G«(g) + iGf (g ) = X qp exp [ig (fp - r.)] , G«( g ) = cos (gr,) F (g) + sin (gr,) F2 (g), (30)

p

Gi2)(g) =cos(grj)F2(g)-sin(grj)F1 (g), F1 (g)=Xqpcos(grp), f2(g)=Xqpsin(grp). (31)

p p

Substituting (30), (31) into (28), and integrating over u, we obtain the final expression for the function Fj (n1n2n3):

2n

3/2

ab.

f

Fj (П1П2 П3 ) =--n1In2In3IY_i3i Y f (nl, g1 )f (П2, g 2 )f (n3, g3 )exp

i,k aik

g 2 1 ( g)

V 4ak J

(32)

g

It can be seen that the structural factor G, (g) is the invariant of deformations, under which the difference of the relative coordinates is constant.

We denote the LRCI Hamiltonian as HLR. Then the matrix element HLR on orbitals W (r - rj), № (r - rj) is written as

¡W (r - rj )HlrW,(r - rj )dv = (r - rj ))-Z 'qp |r-(R, + rp )) jw(r - r, ). (33)

The stroke denotes that in the case % = the term corresponding to the interaction of the charge qj with the electron on the orbital w (r - r) is absent in the sum.

As an example, we give the expression for the diagonal matrix element of the Hamiltonian HLR in the case of the y/px (r) = (3/4")1/2 Ia*exp(-air2) orbital [6]:

Wl* (r - rjKr ¥px (r - rj)dv

3"/2 ^ aflk

i ,k aik

"l

|JL -11 GM

2a..,

-exp

( > 1 (aS1/2

c g V ik /

v a,

(34)

The diagonal matrix elements on the \yp (r), wp2 (r) orbitals can be obtained from (34) by replacing gi by g2, g3, respectively.

3. Double-oblique crystal

As the first step, we perform test calculations on s-orbitals. According to [5], the energy of the s-orbital E (s) determined by (32) can be written as

"2 ^ aa,

Ej (s) = VI

3/2 k aik

- Gj (gK

2

exp

\ 4a.k ,

ik

+ 2qt I

a

(35)

We denote the expression in square brackets in formula (35) as Ej1 (s) and put ak = 2s.

Ej 1)(s ) = - -1

4" Gj (g)

i

exp

v 8s,

+ 2qt

2s

(36)

The expression (36) is the energy of the one-exponential s-orbital with exponent s,

№1s = (1 / 4")1/2 a exp ( - sr2), a = 2 (8s3 /

We assume the angle d = 0 and a is an arbitrary angle. We determine the ion charge and the basis vectors of the unit cell of the crystal in relative units as q1 = 1, r1 (0, 0, 0), q2 = -1, r2 (a /2, b /2, c /2).

The energy Ej1 (s) for such lattice is written as

(1)(s)=. sina1 -(-1)n

Ei (s) = -

abc" n^Q

d.

-exp

" d.

2s sin2 a

+ 2q I —— I,

(37)

where d = + --— cosa + sin2 a.

1 a2 b2 ab c2

Calculation of the matrix elements of the long-range Coulomb interaction in low-symmetry crystals For a = b and a = 120° we obtain the expression

2n2d

)=-# I ^

where

/2 = «2 + n2y + ^ +

2nc n^o

3a2 n2

_z_

4c2 '

-exp

3sa

-7.qt

2s n

Let a = b = c = 7.838587 be constant lattices. Then according to (38) we obtain s = 2: E^s) = 0.291432860377413,

s = 8: Ef(s) = 0.291432860377413.

It can be seen that the energy Ej1 (s) for the exponent s > 2 is the same (with the given accuracy).

The explanation of this fact is obvious. According to Gauss's theorem, if the spherically symmetric charge distribution does not overlap with a point charge, then this charge distribution can be regarded as a point charge.

Let a = b = c = 7.838587 be lattice constants, and a = 120°, p = 60° and d = 60° the angles of the unit cell. The expressions (25) for the g vector are

g = gx, g2 (( + 2g, ^ g3 =-V3g*-V3gy + 2 gz. Substituting these lattice constants and the expressions (39) into (36), we obtain

I n2d3

e;"(S )=- 4nn 11- (-1)

nr + n., + n7

an

d

-exp^

6as

■2q,

2s n

(39)

(40)

where d3 = 13(nf + n^) + 22nxny + 12nz (nz

ny).

s = 4: E^p (s) = 0.68670778474898,

s = 20: Ey (s) = 0.68670778474898. It is seen that the energy E^1 (s) is the same (with the given accuracy) for the values s > 4.

4. BaTiO3 crystal

We consider the application of the obtained expression (36) to the hexagonal BaTiO3 structure. According to [9], the lattice constants are a = b = 5.7238A, c = 13.9649 A. The basis vectors of the unit cell ions in relative units have the form

Ba:: (0, 0, 0.25), Ba4 (2/3, 1/3, 0.59671), Ti1: (0, 0, 0), Ti4: (0, 0, 0.5), O1: (0.3302, 0.1651, 0.0802), O4: (0.51849, 0.03699, 0.25), O7: (0.3302, 0.1651, 0.4198), O10: (0.1651, 0.3302, 0.5802), O13: (0.03699, 0.51849, 0.75), O16: (0.1651, 0.3302, 0.9198),

Ba2: (1/3, 2/3, 0.09671), Ba5: (0, 0, 0.75), Ti2: (2/3, 1/3, 0.15367), Ti5: (1/3, 2/3, 0.65367), O2: (0.8349, 0.1651, 0.0802), O5: (0.96301, 0.48151, 0.25), O8: (0.84903, 0.1651, 0.4198), O11: (0.6698, 0.83490, 0.58020), O14: (0.48151, 0.51849, 0.75), O17: (0.6698, 0.8349, 0.9198),

Ba3: (1/3, 2/3, 0.40329), Ba6: (2/3, 1/3, 0.90329), Ti3: (2/3, 1/3, 0.34633), Ti6: (1/3, 2/3, 0.84633), O3: (0.8349, 0.6698, 0.0802), O6: (0.51849, 0.48151, 0.25), O9: (0.8349, 0.6698, 0.4198), O12: (0.1651, 0.8349, 0.5802), O15: (0.48151, 0.96301, 0.75), O18: (0.1651, 0.8349, 0.91980).

For BaTiO3 F2 (g) = 0, and F1 (g) has the form

F (g)= 4{ 1 + (-1)nz + 4(-1)nx+ny+nz cos[K(nx /3-ny /3-nz /2)]cos[K(0.19266nz)]j +

+ 4{(-1)nz cos(k nz/2) + 2(-1)nx+n>cos[*(nx /3-ny/3 + nz/2)]cos[K(0.30658nz))J -4(-1)n+nz { cos[k(0.03698(nx -ny)-0.5nz)] + 2cos[k(o.4815(nx -ny)-0.5nz) ] (41) xcos[o.44452K(nx + ny) ] + 2cos[K(0.3396nz) ]{ cos k(o.6698(nx -ny) - 0.5nz) ] + 2cos[0.5047K(nx + ny) ] cos ?r(o.1651(nx -ny)-0.5nz)] J J. According to (36) and (38), we have for BaTiO3

E « (s) =

cos (gr )Fj (g) f 2n2d

exp

2nc d2 1 I 3a2 s

(42)

The ions in BaTiO3 have two non-equivalent positions. We present further the values of energies

E(p (s ) (in a.u.) for ions in BaTiO3 (s> 4):

Ti4+(1): cos (grj ) = 1, Ti4+(2): cos (grJ ) = cos

Ej1 ( s )=1.4582510001,

2n

( 2n n

-+ 0.15367nz

Ej1 ( s ) = 1.6877963539,

Ba2+(1): cos ( grj ) = cos [2a- (0.25nz )],

Ej1 ( s ) = 0.7947950353,

Ba2+(2): cos (gr ) = cos

2n

2n

-+ 0.09671n

Ej1 ( s ) = 0.6515284065,

(43)

(44)

(45)

(46)

O2-(1): cos(grj ) = cos[2k(0.5149nx + 0.03699ny + 0.25nz)], Ej1 (s) = -0.9016606636, (47)

O2-(2): cos(gr) = cos[2k(0.3302nx + 0.1651ny + 0.0802nz)], E(() (s) = -0.6683044552. (48)

It can be seen that the energy Ej1 (s) has Table 1. The values of the energies of electron in

i r- ^ /i tt, i the lattice site, i.e., in a point is obtained

the same value for s> 4. The values are ^ m

given with the accuracy indicating the in the work [3] and the energies Ej (s) of

absolute convergence of the series with the present work (in a.u.). respect to the reciprocal lattice vectors. Table 1 shows the energies E^p (s) of the

present work and the energies obtained in [3]. A small difference in the values of the

energies E^1 (s) is due to the fact that the

lattice constants in [3] were taken from [10], in which a = 5.735 A, c = 14.05 A.

The work [3] The present work

Ba(1) 0.81144 0.7947950353

Ba(2) 0.70417 0.6515284065

Ti(1) 1.57409 1.4582510001

Ti(2) 1.68474 1.6877963539

O(1) - 0.88351 - 0.9016606636

O(2) - 0.86721 - 0.6683044552

In this paper, we also calculated the diagonal matrix elements (34) on the

fpx(r)=J4kXa,xexp(a2) ¥Py(r)=A0KXa,yexp(a2) wri(r)=\SKXarzexp(a2)

Calculation of the matrix elements of the long-range Coulomb interaction in low-symmetry crystals

orbitals of oxygen O2-(1). The Hartree-Fock function of /-orbitals were taken from [11]. The following values were obtained.

jy*x (r - r, )HLR ¥px (r - r, )dV = - 0.77608 a.u., (49)

jW'py (r - r, )Hlr ¥py (r - r, )dV = -1.10451a.u., (50)

jypi (r -r,)Hlr Wpi (r -r,)dV = - 0.857581 a.u. (51)

The values (49)-(51) have the splitting on the order of 2-8 eV and differ markedly from the value Ej1 (s) in (47) obtained for a sufficiently localized s-orbital.

5. Summary

It is shown that there is the possibility of calculating LRCI matrix elements on orbitals of the arbitrary symmetry in the case of the low symmetry of the crystal. It can be seen from the expressions (32), (34) and (35) that the structure of the functions F, (n1n2n3) of this paper is the same as the structure of the

functions (20) for the orthorhombic system [5]. The difference lies in the redefinition of the structural factor and the reciprocal lattice vectors. Thus, all expressions for the matrix elements on s, p, d, and /-orbitals obtained in [5, 6, 8] for orthorhombic systems can be used for low-symmetric systems with allowance for the above redefinition.

This approach can also be used to derive two-centered LRCI matrix elements [5]. Acknowledgments

This work was supported by the subsidy of the Ministry of Education and Science of the Russian Federation allocated to Kazan Federal University in the frame of the state assignment in the area of scientific activities for performing research project (Grant No. 3.2166.2017).

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