Quantum Chemistry of Solids / 30-back-matter

A

Matrices of the Symmetrical Supercell

Transformations of 14 Three-dimensional Bravais

Lattices

A.1. The triclinic crystal system

(i) l(P,P) is an arbitrary integer matrix.

A.2. The monoclinic crystal system

n1 n5

0

L = (n1n2 − n4n5)n3;

(A.1)

n1

5

(i) l(P,P)= n4

n2

0

,

0

0

n3

,

n

0

L = 2(n1n2 − n4n5)n3;

(ii) l(A,P)= n4 n2 −n3

n4 n2

n3

n1

n4 n4

n5

L = 2(n1n2 − n4n5)n3;

n1

− n5

(iii) l(P,A)= n5

n2

n2

,

0

n3 n3

,

(iv) l(A,A)= n4 n2 + n3 n2 − n3

L = 4(n1n2

−

n4n5)n3;

n4 n2 − n3 n2 + n3

A.3. The hexagonal crystal system

n1

0

0

0

n1

0

L = n12n2;

(A.2)

(i) l(1)(P,P)=

,

0

0

n2

n1 −n1

0

(ii) l(2)(P,P)= n1

2n1

0 ,

L = 3n12n2;

0

0

n2

A.4. The rhombohedral crystal system

n1 + n2

n2

n2

(i) l(R,R)=

n2

n1 + n2 n2

,

L = n12(n1 + 3n2);

n2

n2

n1 + n2

522 A Matrices of the Symmetrical Supercell Transformations of 14 Three-dimensional Bravais Lattices

A.5. The orthorhombic crystal system

(i)l(P,P) coincides with (A.1);

n1 −n2 0

(ii) l(C,P)= n1		n2			0				,	L = 2n1n2n3;
		0			n
00 n n3						3
(iii) l(F,P)= n1		02	n3				,			L = 2n1n2n3;
n		n		0
n1	n2
− 1				2	n		3
		−	n2		n3					L = 4n1n2n3;
(iv) l(I,P)= n1								,

n1 n2 −n3 n1 n1 0

(v) l(P,C)=

−n2 n2

0

,

L = 2n1n2n3;

0

0

n

n1 n2

0 3

L = (n12 − n22)n3;

(vi) l(C,C)= n2 n1

0

,

n

0

0n10 n13

(vii) l(A,C)=

−n2 n2 −n3

,

L = 4n1n2n3;

−n2nn2 n3n

n

3

,

−

2

2

(viii) l(F,C)=

n1

n1

n3

L = 4n1n2n3;

n

n

n

+ n

0

n1

−n 2

n1

2

− 1 −

2

3

2 2

−n1

n1

n1

L = 2(n1 − n2)n3;

(ix) l(I,C)= n1

n2

n3

,

n2

n1

−n3

−

n2

n2

L = 4n1n2n3;

(x) l(P,F)= n2

,

n3 n3 −n3 −n1 n1 n2

(xi) l(C,F)=

−n2 n2

n1

,

L = 2(n12 − n22)n3;

n3

n3 −n3

n

n

2

+ n

3

2

+ n

3

n

n

3

L = n

n

n

;

1

−

2

−

1

2

3

(xii) l(F,F)=

n1 + n3

n1 − n3

n1, n2, n3

are of

2

−n1 + n3

,

−n1 + n2

n1 − n2

n1 + n2

the same parity;

(xiii) l(I,F)=

n

1

+ n

2

+ n

3

−

n

3

−

n

2

,

−n3

n1 + n2 + n3

−n1

−n2

−n1

n1 + n2 + n3

−

L = 2[(n

1

+ n

2

+ n

)(n

1

n

2

+ n

2

n

3

+ n

n )

n

n

n

];

0

3

3

1

1

2

3

n

n1

(xiv) l(P,I)=

n2

01

n2 ,

n

L = 2n1n2n3;

n

n

n

n3

n3

0

1 − 2

− 2

1

(xv) l(C,I)=

n2

n1 n1 + n2

,

L = 4n1n2n3;

n3

n3

0

A Matrices of the Symmetrical Supercell Transformations of 14 Three-dimensional Bravais Lattices

523

n

+ n

n3

n2

,

(xvi) l(F,I)=

2 n3

3

n1 + n3

n1

L = 4n1n2n3;

n

n1 n

n + n

2n + n

n2

+ n

3

1

+1n

3

2

L = n

n

n

;

1

2

−

− 1

1

2

3

(xvii) l(I,I)=

2

−n2 + n3 n1

+ n3

n1 − n2

,

n1, n2, n3 are of

n2 − n3

n1

− n3

n1 + n2

the same parity;

A.6. The tetragonal crystal system

(i)l(1)(P,P) coincides with (A.2);

(ii) l

(2)

(P,P)=

n1 −n1

0

,

2

n1

n1

0

L = 2n1n2;

0

0

n

2

0

n1 n2

(iii) l(1)(I,P)= n1

n2

,

L = 2n12n2;

n1 n1

0

n2

,

−n1

n1

2

(iv) l(2)(I,P)= n1

−n1

n2

L = 4n1n2;

n1

n − 1 −n2

0

n1 n1

2

−n1

n1

(v) l(1)

(P,I)= n1

0

n1

,

L = 2n12n2;

n2 n2

0

(vi) l(2)(P,I)= n1

0

,

n1 2n1

L = 4n1n2;

n

n2

0

n2

n12+ n2

n1

(vii) l(1)(I,I)=

n2

n1 + n2

n1

L = 4n12n2;

n1

n1

2n1

n

1

+ n

2

−

n

1

+ n

0

L = n2n

;

1

2

1 2

(viii) l(2)(I,I)= 2

−n1 + n2

n1 + n2

0

n1, n2 are of

n1 − n2

n1 − n2

2n1

the same parity;

A.7. The cubic crystal system

0 n

n 0 0

0 n 0

L = n3;

(i) l(P,P)=l(F,F)=l(I,I)=

,

0 0 n

n

n

(ii) l(F,P)=l(P,I)= n 0 n ,

L = 2n3;

n n 0

−

n

n

3n

−n n

L = 4n3;

(iii) l(I,P)=l(P,F)= n

,

−

n

n

−n

n

−

n

2n n n

L = 16n3;

(iv) l(I,F)=

−n 3n

−n

,

−n −n 3n

n

n 2n

L = 4n3.

(v) l(F,I)= n

2n n

,

B

Reciprocal Matrices of the Symmetric Supercell

Transformations of the Three Cubic Bravais

Lattices

	1 0 0			−1	1	1			0 1 1
	0 0 1			1	1	−1			1 1 0
Q1 =	0 1 0		Q2 =	1	−1	1		Q3 =	1 0 1

Q4	=	1 2 1	Q5	=	−1 3	−1
		2 1 1			3 −1	−1
		1 1 2			−1 −1	3

C

Computer Programs for Periodic Calculations in Basis of Localized Orbitals

The information about computer programs applied in molecular quantum chemistry and materials science can be found on di erent Internet sites [856–858]. In particular, on site [856] on-line links are given to the appropriate information. The majority of the existing computer codes for calculations of periodic systems can be divided into two main groups: LO codes using a basis set of localized orbitals (LO) and PW(APW) codes using plane waves (PW) or augmented plane waves (APW) as a basis set. The only exclusion is the hybrid Gaussian LO and PW (GPW) DFT code QIUCKSTEP/CP2K [859]. The sites mentioned include references to both LO and PW codes. The information about periodic codes using LO or PW bases can also be found in [568, 860]. Here, we restrict ourselves only to brief information about computer codes for periodic LO calculations (see Table C.1). The indication of the corresponding homepage and references on the publications are given in the last two columns of Table C.1. Both ab-initio (HF,DFT (LDA,GGA), hybrid HF-DFT, time-dependent DFT (TDFT)) and semi-empirical (SE) codes are included in the list. The codes listed allow all-electron (AE) calculations (also termed full potential-FP) or valence-only ones with the use of pseudopotential (PP) for core electrons or both. For the localized basis we use the acronyms GTO (Gaussian-type orbitals), STO (Slater-type orbitals) and NTO (numerical type orbitals). In particular, f -GTO and f -polarization-GTO mean the possibility to make full GTO calculations for f -elements or the simple extension of Gaussian basis-sets to include f -polarization basis-functions, respectively. In the remaining cases only s, p and d orbitals can be used in the basis set. The geometry optimization can be made over the atomic coordinates only (GO) or including both the lattice parameters optimization (full GO). Some codes allow linear scaling (O(N )) with the basis-set size and molecular-dynamics (MD) calculations. The semiempirical codes can allow the band structure calculations for nonzero wave vectors (BAND) or the cyclic-cluster model (CCM) without the BZ summation can be used. The CRYSTAL suite remains the mostly advanced and developing LCAO periodic program, compared to other codes listed in Table C.1. We refer the reader to the calculations of metal-organic frameworks (MOF), [872] as a case study showing that the CRYSTAL program allows reliable ab-initio prediction of materials properties of complex systems to be made. Note that from January 1st, 2007, CRYSTAL03 will not be supported any more.

528 C Computer Programs for Periodic Calculations in Basis of Localized Orbitals

Table C.1. List of periodic computer programs using localized orbitals basis

No	Program	Description	Web site	Refs
1	CRYSTAL03	HF;DFT (LDA,GGA);	www.crystal.unito.it	[568, 860–863]
		Hybrid HF-DFT; AE;	www.cse.clrc.ac.uk/cmg
		PP; GTO; GO
2	CRYSTAL06	+ f -polarization-GTO, full GO,	www.crystal.unito.it/
		vibrational frequences	home.html
3	CRYSCOR	+ MP2 correlation	www.crystal.unito.it	[201]
4	GAUSSIAN03	HF;DFT (LDA,GGA);	www.gaussian.com	[379]
		PHF; Hybrid HF-DFT;
		AE; PP;f -GTO; GO;O(N)
5	ADF2006	DFT (LDA,GGA);	/www.scm.com/Doc/	[864]
	(BAND)	Hybrid HF-DFT;	Doc2006.01
		TD-DFT; AE; STO;
6	SIESTA	DFT (LDA,GGA);PP;	www.uam.es/	[400]
		NTO; GO; MD ;O(N)	departamentos/ ciencias/
			fismateriac/siesta
7	AIMPRO	DFT(LDA,GGA); PP;	http://newton/ex/	[555]
		LGTO	ac.uk/research
8	SEQUEST	DFT (LDA, GGA);PP;	http://dft.sandia.	[865]
		GTO;O(N);GO	gov/Quest
9	DMOL3	DFT (LDA, GGA);AE;	http://people.web.psi.ch/	[866]
		NTO; GO	delley/dmol3.html
10	FPLO	DFT (LSDA,LSDA+U);	www.fplo.de	[867]
		AE; LO;
11	PLATO	DFT (LDA);	www-sta .lboro.ac.uk	[868]
		NTO;GO	/ masdk/plato.html
12	QUICKSTEP/	DFT(LDA,GGA);	http://cp2k.berlios.de	[859]
	CP2K	PP;GTO/PW
13	CNDO 4.0	SE,BAND,GE	www.wgc.chem.pu.ru	[869]
			/ valera/cndo/manual.html
14	MOPAC2002	SE,BAND,GE	www.cachesoftware.	[312, 870]
			com/mopac
15	MSINDO	SE,CCM,STO,GO,MD	www.theochem.
			uni-hannover.de/	[256]
			Bredow/MSINDO
16	SOLID2000	SE,CCM,QR-INDO,	www.stech.sk/stech	[307]
		STO,GO	soft/htm
17	SYM-SYM	SE,CCM,INDO,GO	www.cmmp.ucl.ac.uk	[871]
			/ lev/recearc4/node8.html

C Computer Programs for Periodic Calculations in Basis of Localized Orbitals

529

The reader can also find useful such codes as XCrysDen( [873], GULP( [875] and BAND-GUI( [345].

XCrysDen program can be used as a graphical user interface (GUI) for the CRYSTAL95/98/03/06 programs. The advantage of the GUI is its rendering possibility, which yields graphical feedback to the user. Every manipulation of the structure is visualized immediately and to enhance the usefulness of XCrySDen even further, an UNDO/REDO option is available, which makes every false move recoverable. Several graphical procedures make the manipulation of the atomic structures easier, while other graphical functions are suitable for checking the geometry of the structure. This program uses the ease of a GUI and the flexibility of manual editing and allows the user to view and edit the input scripts for the CRYSTAL programs on request. Immediately after the manual editing is completed the program visualizes the result. We refer the reader for details to the program description [874].

GULP is a program for performing a variety of types of simulation on materials using boundary conditions of 0D (molecules and clusters), 1D (polymers), 2D (surfaces, slabs and grain boundaries), or 3D (periodic solids). The focus of the code is on analytical solutions, through the use of lattice dynamics, where possible, rather than on molecular dynamics. A variety of force fields can be used within GULP spanning the shell model for ionic materials, molecular mechanics for organic systems, and the embedded-atom model for metals. Analytic derivatives are included up to at least second order for most force fields, and to third order for many.

ADF2006.01 is the first ADF release containing a graphical user interface for the BAND program. BAND input also enables inexperienced ADF-BAND users to easily create BAND jobs. You can use BAND input to define your periodic structure (geometry), and to set details of your BAND job using an easy-to-use graphical user interface. BAND input will generate the complete job script for you. This script takes care of running BAND. You can also use BAND input to run these script files on your local machine in the background.

References

1.R. Dronskowski, Computational Chemistry of Solid State Materials. A Guide for Materials Scientists, Chemists, Physicists and Others (Wiley-VCH, Weinheim, 2005)

2.C.M. Quinn, An Introduction to Quantum Chemistry of Solids (Clarendon Press, Oxford, 1973)

3.A.A. Levin, Solid State Quantum Chemistry (McGraw-Hill, New York, 1977)

4.Ab-initio Calculations of the Properties of Crystalline Materials, ed. C. Pisani, Lecture Notes in Chemistry 67 (Springer Verlag, Berlin, 1996)

5.P. Fulde, Electron Correlations in Molecules and Solids, 3rd edn. Springer Series in Solid State Sciences 100 (Springer, Berlin Heidelberg New York, 1995)

6.W. Harrison, Electronic Structure and Properties of Solids: Physics of the Chemical Bond, 2nd edn. (Dover, New York, 1989)

7.R. Ho mann, Solids and Surfaces: A Chemist View on Bonding in Extended Structures

(Weinheim VCH, 1988)

8.M. Springborg, Methods of Electronic Structure Calculations (Wiley, 2000)

9.E. Kahiras, Atomic and Electronic Structure of Solids (Cambridge University Press, Cambridge, 2003)

10.R. Martin, Electronic Structure. Basic Theory and Practical Methods (Cambridge University Press, Cambridge, 2004)

11.L. Kantorovich, Quantum Theory of the Solid State (Kluwer Academic Publishers, Dordrecht, 2004)

12.K. Ohno, R. Esfarjani, Y. Kawazoe, Computational Materials Science,

Springer Series in Solid State Sciences, 129 (Springer, Berlin Heidelberg New York, 1999)

13.R.A. Evarestov, V.P. Smirnov, Site Symmetry in Crystals: Theory and Applications, 2nd edn. Springer Series in Solid State Sciences, 108 (Springer, Berlin Heidelberg New York, 1997 )

14.R.L. Flurry, Symmetry Groups. Theory and Chemical Applications (Prentice-Hall, Englewood Cli s, NJ, 1980)

15.P.W.M. Jacobs, Group Theory with Applications in Chemical Physics (Cambridge University Press, New York, 2005)

16.Bilbao Crystallographic Server, http://www.cryst.ehu.es/

17. A.P. Cracknell, B.L. Davies, S.C. Miller, W.F. Love, Kronecker Product Tables,

1 General Introduction and Tables of Irreducible Representations of Space Groups

(IFI/Plenum, New York-Washington-London, 1979)

18.M.I. Petrashen, E.D. Trifonov, Applications of Group Theory in Quantum Mechanics

(M.I.T. Press, Massachusetts, 1969)

532References

19.International Tables for Crystallography, ed. T.Hahn, A Space-group symmetry (Reidel, Dordrecht, 1983 )

20.http://cst-www.nrl.navy.mil/lattice/

21.http://cst-www.nrl.navy.mil/lattice/others.html

22.Inorganic Crystal Structure Database (ICSD), http://www.fiz-informationsdienste. de/en/DB/icsd/

23.V.R. Saunders, R. Dovesi, C. Roetti, R. Orlando, C.M.Zicovich-Wilson, N.M.Harrison, CRYSTAL-2003 Users Manual, University of Torino, Torino, 2003; http://www.crystal.unito.it

24.Richard J.D. Tilley, Crystals and Crystal Structures (Wiley, 2006)

25.A. Munoz, J.A. Alonso, M.J. Martinez-Lope, M.T. Casais, J.L. Martinez, M.T. Fernandez-Diaz, Phys. Rev. B 62, 6498 (2000)

26.John Z.H. Zhang, Theory and Application of Quantum Molecular Dynamics (World Scientific Publishing Co., Singapore, 1999)

27.G.L. Bir, G.E. Pikus, Symmetry and Strain-induced E ects in Semiconductors (Wiley, New York, 1972)

28.C. Zicovich-Wilson, R. Dovesi, Int. J. Quantum Chem. 67, 299 (1998)

29.C. Zicovich-Wilson, R. Dovesi, Int. J. Quantum Chem. 67, 311 (1998)

30.C.J. Bradley, A.P. Cracknell, The Mathematical Theory of Symmetry in Solids (Clarendon, Oxford, 1972)

31.O.V. Kovalev, Representations of the Crystallographic Space Groups: Irreducible Representations, Induced Representations and Corepresentations, 2nd edn. ed. by H.T. Stokes, D.M. Hatch (Gordon and Breach, Philadelphia, PA 1993)

32.I. Shur, J.F.Math. 139, 155 (1911)

33.G.Ya. Lyubarskii, The Application of Group Theory in Physics (Pergamon, Oxford, 1960)

34.C.Herring, J. Franklin Inst. 233, 525 (1942)

35.M.W. Schmidt, K.K. Baldridge, J.A. Boatz, S.T. Elbert, M.S. Gordon, J.H. Jensen, S. Koseki, N. Matsunaga, K.A. Nguyen, S. Su, T.L. Windus, M. Dupuis, J.A.J.Jr. Montgomery, J. Comput. Chem. 14, 1347 (1993)

36.C. Edmiston, K. Ruedenberg, Rev. Mod. Phys. 35, 457 (1963)

37.J. Pipek, P.G. Mezey, J. Chem. Phys. 90, 4916 (1989)

38.S.F. Boys, Rev. Mod. Phys. 32, 296 (1960)

39.J. Zak, Phys. Rev. B 23, 2824 (1981)

40.H. Bacry, L. Michel, J. Zak, Lect. Notes Phys. 313, 291 (Springer, Berlin, Heidelberg, 1988)

41.M. Sch¨utz, G. Hetzer, H.-J. Werner, J. Chem. Phys. 111, 5691 (1999)

42.V.P. Smirnov, D.E. Usvyat, Phys. Rev. B. 64, 245108 (2001)

43.G. Wannier, Phys. Rev. 52, 191 (1937)

44.W. Kohn, Phys. Rev. 115, 809 (1959)

45.J. Des Cloizeaux, Phys. Rev. 135, A698 (1964)

46.N. Marzari, D. Vanderbilt, Phys. Rev. B, 56, 12847 (1997)

47.V.P. Smirnov, R.A. Evarestov, D.E. Usvyat, Int. J. Quantum. Chem. 88, 642 (2002)

48.B. Sporkmann, H. Bross, Phys. Rev. B. 49, 10869 (1994)

49.A. Shukla, M. Dolg, P. Fulde, H. Stoll, Phys. Rev. B. 60, 5211 (1999)

50.S. Goedecker, Rev. Mod. Phys. 71 , 1085 (1999)

51.I. Souza, R.M. Martin, N. Marzari, X. Zhao, D. Vanderbilt, Phys. Rev. B. 62 , 15505 (2000)

52.P.Y. Yu, M. Cardona, Fundamentals of Semiconductors. Physics and Materials Properties, 3rd rev. and enlarged edn. (Springer, Berlin, Heidelberg, 2005)

53.C.M. Zicovich-Wilson, R. Dovesi, V.R. Saunders, J. Chem. Phys. 115, 9708 (2001)

54.I. Mayer, G. Rather, S. Suhai, Chem. Phys. Lett. 293, 81 (1998)

References 533

55.R.A. Evarestov, V.P. Smirnov, phys. stat. sol. (b) 180, 411 (1993)

56.J. Des Cloizeaux, Phys. Rev. 129, 554 (1963)

57.W. Kohn, Phys. Rev. B, 7, 4388 (1973)

58.J.C. Slater, G.F. Koster, Phys.Rev. 94, 1498 (1954)

59.V.P. Smirnov, D.E. Usvyat, Phys. Rev. B 59, 9695 (1999)

60.N. Marzari, D. Vanderbilt, in First-Principles Calculations for Ferroelectrics, ed. R.E. Cohen, AIP Conf. Proc. No 436 (AIP, Woodbury, 1998), p.146

61.I. Souza, N. Marzari, D. Vanderbilt, Phys. Rev. B 65, 035109 (2001)

62.http://www.wannier-transport.org

63.C.M. Zicovich-Wilson, A. Bert, C. Roetti, R. Dovesi, V.R. Saunders J. Chem. Phys. 116, 1120 (2002)

64.A. Bert, M. Llunell, R. Dovesi, C.M. Zicovich-Wilson, Phys. Chem. Chem. Phys. 5, 5319 (2003)

65.E. Clementi, C. Roetti, Roothaan–Hartree–Fock Atomic Functions, in Atomic Data and Nuclear Data Tables, 14, 177 (1974)

66.C.C.J. Roothaan, Rev. Mod. Phys. 23, 69 (1951)

67.C.C.J. Roothaan, Rev. Mod. Phys. 32, 179 (1960)

68.J.A. Pople, R.K. Nesbet, J. Chem. Phys. 22, 571 (1954)

69.E. Apra, Lecture Notes in Chem. 67, 101 (1996)

70.R.A. Evarestov, I.I. Tupitsyn, Phys. Solid State 44, 1656 (2002 )

71.P.P. Ewald, Ann. Phys. 64, 253 (1921)

72.A.B. Kunz, Phys. Rev. B 6, 606 (1972)

73.J.J. Ladik, Phys. Rep. 313, 171 (1999)

74.J.-M. Andre, L. Gouverneur, G. Leroy, Int. J. Quantum Chem. 1, 427, 451 (1967)

75.C. Pisani, R. Dovesi, Int. J. Quantum Chem. 17, 501, 517 (1980)

76.R. Dovesi, C. Pisani, C. Roetti, P.Dellarole, Phys. Rev. B 24, 4170 (1981)

77.R. Dovesi, M. Causa, G. Angonoa, Phys. Rev. B 24, 4177 (1981)

78.M. Causa, Lecture Notes in Chem. 67, 91 (1996)

79.C. Pisani, U.Birkenheuer, Comp. Phys. Commun. 96, 221 (1996)

80.http://www.crystal.unito.it

81.http://www.cse.clrc.ac.uk

82.K.N. Kudin, G.E. Scuseria, Phys. Rev. B 61, 16440 (2000)

83.R. Improta, K.N. Kudin, G.E. Scuseria, V. Barone, J. Am. Chem. Soc 124, 113 (2002)

84.N.W. Ashcroft, N.D. Mermin, Solid State Physics (Cole, London, 1976 )

85.M.F. Guest, V.R. Saunders, Mol. Phys. 28, 819 (1974)

86.R.A. Evarestov, V.P. Smirnov, J. Phys.: Condens. Matter 9, 3023 (1997)

87.R.A. Evarestov, V.P. Smirnov, phys. stat. sol. (b) 119, 9 (1983)

88.D. J. Chadi, M.L. Cohen, Phys. Rev. B 8, 5747 (1973)

89.J. Moreno, J.M. Soler, Phys. Rev. B 46, 13891 (1992)

90.H.J. Monkhorst, J.D. Pack, Phys. Rev. B 13, 5188 (1976)

91.R.A. Evarestov, V.P. Smirnov, Phys. Rev. B 70, 233101 (2004)

92.A. Baldereschi, Phys. Rev. B 7, 5212 (1973)

93.D. Vanderbilt, Phys. Rev. B 41, 7892 (1990)

94.M. Causa, R. Dovesi, C. Roetti, Phys. Rev. B 43, 11937 (1991)

95.R.A. Evarestov, V.A. Lovchikov, I.I. Tupitsin, phys. stat. sol. (b) 117, 417 (1983)

96.K.B. Wiberg, Tetrahedron 24, 1083 (1968)

97.V.A. Veryazov, A.V. Leko, R.A. Evarestov Phys. Solid State 41, 1286 (1999)

98.R.A. Evarestov, I.I. Tupitsyn, Russ. J. Phys. Chem. 74, S363 (2000)

99.C. Pisani, E. Apra, M. Causa, Int. J. Quantum Chem. 38, 419 (1990)

100.T. Bredow, R.A. Evarestov, K. Jug, phys. stat. sol. (b) 222, 495 (2000)

101.V. Fock, M. Vesselow, M. Petrashen, Zh. Exsp. Theor. Fiz. 10, 723 (1940)