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The Coulomb interaction between an s-orbital electron and ionic crystal lattice
O.A. Anikeenok
Kazan State University, Kremlevskaya, 18, Kazan 420008, Russia E-mail: Oleg.Anikeenok@ksu.ru
Received June 1, 2009 Revised July 17, 2009 Accepted July 17, 2009
Volume 11, No. 1. pages 1-6, 2009
http://mrsei.ksu.ru
The Coulomb interaction between an s-orbital electron and ionic crystal lattice
O.A. Anikeenok
Kazan State University, Kremlevskaya, 18, Kazan 420008, Russian Federation E-mail: Oleg.Anikeenok@ksu.ru
Analytical expressions for the Coulomb interaction between an s-orbital electron and the surroundings, infinite ionic crystal lattice are derived. The s-orbital wave functions are presented in the form of a Gaussian expansion. As a test, Madelung constants and interaction energy for NaCl, KMgF3, CaF2 are calculated using a single Gaussian exponent. The calculated values are coincided with known literature data with a high degree of accuracy.
PACS: 31.15.Ar, 61.44.Fw, 61.66.Fn
Keywords: localized orbitals, Madelung energy, orbital energy.
1. Introduction
The calculation of energy of Coulomb interaction of orbital with infinite crystal lattice is one of the most important problems in the ion crystals theory. The value of this interaction is estimated by the electrostatic potential calculation at the lattice site [1-3]. In this paper, the expressions for the calculations of the interaction energy of the 5-orbital electron with infinite crystal lattice are derived. The 5-orbital is taken in the Gaussian expansion form. The expressions are absolutely convergent series, which do not require additional parameter for convergence of series as in [1] and which do not contain the sum taken over direct lattice. This series can be calculated with pinpoint accuracy. One exponent orbital is used for test calculations of NaCl, KMgF3 and CaF2 crystals.
2. General part
Let the ion is situated at the origin of coordinate system and the radial part of its 5-orbital is described by the Gaussian Type Orbital (GTO) function. Then an 5- electron wave function is
Kns =
aiexp (-a 2),
(1)
where n is the main quantum number, s is the angular-momentum quantum number.
The position of the charge q is determined by the vector R = (Rx, Ry, Rz ). The Coulomb interaction of such i electron with the charge q may be written [4]
r - R
Wns) = ■
1 ( 1 ]1 ) = - 2 qS aiak — lj exp \_-a,k r2 x2 ]dx,
V aik J 0
(2)
where aik = ai + ak .
The Coulomb interaction of the ion s-electron at the site r;- with the crystal lattice is the sum of functions (2). Then the electron - lattice interaction, including the ion of the site r;-, is energy Ej. It may be written as follows
( ]
E] = K| -S^| -------------------------rr
V Rm p |r - [Rm - (rj - rp )]|
Vns) =
1 SSqpSaa —]jexp[~aik[Rm-(rj-rp)]2x2]
2 Rm P V aik J 0
1 ( 1 ] 1 (
2Saiak — ljdxS Sqp exp[ - aik[Rm- (rj- rp)]2 x2]
V aik J 0
dx =. ]
(3)
where Rm is the radius vector of unit cell, rj is the radius vector of ion unit cell, q p is the charge of ion with the radius vector rp . According to (2) an interaction energy E0 between the s-electron at the site r;- and the single unit positive charge at the same site r;- (R = 0) may be written as
E0=-2 s.
H k
The interaction energy E^r of s-orbital and crystal lattice is
Ejr = Ej - qjE0 ■-
(4)
where qj is the charge of the site rj ion.
The expressions (3) and (4) are the functions of the lattice variables. Using transformation [5], the expression (4) may be displayed in the term of reciprocal lattice variable
1 exp[i(g. (rj - rp))] , (5)
-J=S eXP[ - [Rm - (rj - rp)]2 x2 ] = ^"“S~Texp
Rm VC g x
where g is the vector of reciprocal lattice, vc is the volume of unit cell. Then the interaction energy Ej may be written as follows
Rm V P
ej —
-X
f 1 \2
atak
X G (g) ~rexP
4a,-,
G; (g) = cos (gr;)F1 (g) + sin(gr; ) F2 (g) ,
Fi (g)=X qpcos (§rp), f2 (g) = X qpsin(grp) •
(7)
(8)
p p The function (7) is named as a structure factor of the crystal.
3. Calculation
Let us take one exponent orbital
a = 2
8a5
Hereinafter all quantities are given in atomic units (au). According to equations (6) and (9)
4n V-1 ✓-> ( ) 1 f
Ej =-—X G (g) ~2exP
c g
Eo =-2
8a
Let us examine NaCl crystal. The unit cell is defined as Na: r =(0,0,0), r3 = (0.5a,0.5a,0.5b). Cl: r2 =( 0.5a,0.5a,0),
where b = V2a, b = 5.63 A or b = 10.63916232186962 .
The energy Ej for NaCl according to (10) is written as follows
= ( 0,0,0.5b ).
(9)
(10)
(11)
(12)
ENa =- — X
bn n n n
GNa (n)
2 2 + 2 2 + 2'eXP
2nv + 2n„ +
2ab
-(2
2 (■( + 2n2y + n1
(13)
where GNa (n) is the structure factor, b is the absolute value of the unite cell vector along the axis z .The structure factor GNa = GNa (n) for NaCl is defined as
GNa = 1 -(-1)nv + ny -(-1)nz +(-1)nv + ny + nz .
The value aN is defined by formula
aor = 2 (ENa qNaE0 )
(14)
Below we use the calculation accuracy up to 25 decimal places. According to equation (14) one can obtain a = 0.01, aNra = 0,8488752444376062993366228 ; a = 0.1, aN = 1.7429785198333593881232629 ;
a = 1,
aNa = 1.7475645946331821906362119 ;
a = 10, aNra = 1.7475645946331821906362120 ;
a = 100 , aNa = 1.7475645946331821906362120 .
Another test is the lattice constant variation. Consider a = 0.1. Let b’ is a new lattice constant and b is the lattice constant we used before. So, we are obtained
b' = 3b , aNa = 1.7475645946331821906361765 ;
b' = 6b , aONa = 1.7475645946331821906362120 .
3
3
The Madelung constant aM for NaCl calculated by direct summation up to 25 decimal places was presented in work [6]
(15)
aM = 1.7475645946331821906362120
The explanation of the coincidence of (15) with our calculations is evident. According to the Gauss theorem if the spherically symmetrical distribution of charge does not overlap with the point charge then this distribution for this point charge looks like the point charge.
Let us examine KMgF3 and CaF2 crystals. The energy Ej for KMgF3 and CaF2 according to (10) is written as follows (for a= 1 )
Ej =-
Gj (n)
an nx ~,n2 n2x + n2y + n]
exp
The structure factors for KMgF3 are defined as
GK = 2 (-1)nv + ny + * -(-1)nv + ny -(-1)nv + nz -(-1)
ny + nz + 1
\nv + ny + n7
(16)
(17)
(18)
GMg = 2-(-1)nv-(-1)ny-(-1) +(-1)
Gf = 2 (-1) -1 -(-1)nv+nz -(-1)ny+nz +(-1)nv+ny . (19)
For the crystal KMgF3 a = 3.973 A or a = 7.507884885397513 . For simplicity we use below the accuracy of calculated values up to 8 decimal places. Then
aKr = 2.69360482, E^ = 0.35877013, aMf = 6.18873401, EMg = 0.82429794,
aFr = - 3.22795440, EFor = - 0.42994191.
The Madelung constants a for KMgF3 according to [7] are aMg = 6.189, aF =-3.228.
The structure factors for CaF2 are defined as
Gca = 2
nv + ny + nz
1 + (-1)nv+ny 1 + (-1)nv+nz -(-1)-----------------------2- 1 + (-1)nv 1 + (-1)ny 1 + (-1)n
gf =
r “1 r “1 r “I nv + ny+nz
1+(-1)nv 1+(-1)ny 1+(-1)nz (-1) 2 -1
(20)
(21)
For CaF2 a = 5.462 A or a = 10.321688206403527 , then
F2 a
aCa = 7.56585221, ECra = 0.73300530, aFr = - 4.07072302, EFor = - 0.39438539.
The Madelung constant aM is defined in handbook [8] and for CaF2 aM = 2.51939. Using Ecr and EFor we can
obtain the same result for aM in the framework of proposed here approach.
So, we can conclude that our calculated values coincide with known values adduced in references. Equation (6) is
absolutely convergent series, which continuous in ai . Therefore all the calculations are mathematically correct.
Conclusions
The expressions for the calculations of the interaction energy of the s-orbital electron with infinite crystal lattice are derived. The equations can be used for the Madelung energy calculations in case of sufficiently localized orbitals. The Madelung energy for the NaCl, KMgF3 and CaF2 crustals are calculated.
References
1. Ewald P.P. Ann. der Physik 64, 253 (1921).
2. Evjen H.M. Phys. Rev. 39, 675 (1932).
3. Sabry A., Ayadi M. Choukn A. Computational Materials Science 18, 345 (2000).
4. Anikeenok O.A. Physics of the Solid State 47, 1100 (2005).
5. Ziman J.M. Principles of the Theory of Solids, 2nd ed, Cambridge University Press, pp.37-42 (1972).
6. Hajj F.Y. J. Chem. Phys. 56, 891(1972).
7. Hubbard J., Rimmer D.E., Hopgood F.R.A. Proc. Phys. Soc. 88, 13 (1966).
8. David R. Lide, ed., CRC Hadbook of Chemistry and Physics, Internet Version 2005 {http://www.hbcpnetbase.com}, CCRC Press, Boca Raton, FL, 2005.
