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Band structures by the maximum overlap symmetry molecular orbital method
Journal of Molecular Structure (Theochem), 219 (1993) 47-52 Elsevier Science Publishers B.V., Amsterdam
41
Band structures by the maximum overlap symmetry molecular orbital method Fang Zheng”*b, Chang-Guo Zhan” and Xing-Jiao Lib “Department of Chemistry, Central China Normal University, Wuhan 430070 (People’s Republic of China) bDepartment of Solid State Electronics, Huazhong University of Science and Technology, Wuhan 430074 (People’s Republic of China) (Received 3 February
1992; in final form 27 April 1992)
Abstract It is shown that the maximum overlap symmetry molecular orbital (MOSMO) procedure can be employed to construct the crystal orbitals and study the band structures by using the Bloch function basis set. The concrete MOSMO calculations on the extended Hiickel molecular orbital (EHMO) approximation level using Bloch functions for some organic polymers and graphite show that the calculated band gaps are close to those worked out by using the ordinary linear combination of atomic orbitals (LCAO) method if the same parameterization is adopted. Because the MOSMO procedure is more easily performed than the ordinary LCAO procedure, the MOSMO calculation using Bloch functions may be feasible in large systems.
Introduction
The electrical properties of a crystalline material depend on its electronic structure and ultimately upon the chemical constitution of its unit cell. The calculation of the band structures of various solids comprising repeating units are generally based on the ordinary linear combination of atomic orbital (LCAO) method, in which the atomic orbitals are replaced by Bloch functions [1-4]-the basis vectors for the irreducible representations (IRS) of the translation-symmetry group. This method has been extensively applied to many organic polymers [5-81 and many other perfect or defect solid materials [4,9-l l] for discussing the relation between the calculated energy band and electrical properties. Of all the semiempirical LCAO methods which Correspondence to: C.-G. Zhan, Department of Chemistry, Central China Normal University, Wuhan 430070, People’s Republic of China.
have been used for studying electronic structures of crystalline materials, the extended Hi.ickel molecular orbital (EHMO) method [12] using Bloch functions [6,8] may be regarded as the simplest and can be employed in large systems with unit cells formed of many atoms [4]. However, it is still difficult to use the EHMO method to analyse very large systems. Therefore it is of some interest to find a simple new and reasonable one-electron method for studying these problems. A new maximum overlap symmetry molecular orbital (MOSMO) model, differing from the ordinary LCAO method, has been suggested in a previous paper [13]. The theoretical analysis and the numerical results [13-161 show that the calculated results obtained by performing the MOSMO calculation are as reliable as those of the customary LCAO method on the same approximation level. Furthermore, calculation by the MOSMO procedure requires less computing time than the LCAO method, and therefore the MOSMO procedure may be feasible even in very
0166-1280/93/$06.00 0 1993 Elsevier Science Publishers B.V. All rights reserved.
48
F. Zheng et al./J. Mol. Strut.
large systems [13]. The simplicity of the MOSMO procedure leads us to ask whether it could be used to study the electronic structures of solids. In this paper, the MOSMO method is extended to study the band structures of solid materials. After presentation of the basic equations needed for the construction of the crystal orbitals, the MOSMO method using Bloch functions is examined at the EHMO approximation level, taking some conjugated polymers and graphite as examples. Outline of the basic MOSMO method
The MOSMOs can be constructed by a simple method [13], which may be regarded as an extension of the maximum overlap principle [l&36]. In this method, the atomic orbitals (AOs) in a molecule are partitioned into two sets:
~=eP,M,)...lcp,)) X = (Ix, )Ix2). . *IL>) Each set includes the AOs of indirectly bonded atoms, i.e. all the directly bonded atoms are partitioned into different sets [13]. The linear combinations of these AOs give W” = (Iwp)lo;).
. . lo;))
= @A0
Y’”= w;))llc1;)~~~ ItiP>) = XB” Let q 2 r. The coefficient matrices A” and B” are worked out with the equations iVIM+ A0 = A”Q
(1)
LP = iv+ aO(ibzy-’
(2)
where M = CD+fix, a0 is a submatrix formed from the first r columns of A”, Q a diagonal matrix formed from the eigenvalues of matrix MM+, and the diagonal elements of diagonal matrix My consist of the negative square roots of the r nonzero eigenvalues. With Icop) and I t,bp) the delocalized molecular orbitals are obtained:
IG>=
C&II.@+ &I$:) i I@)
(i= 1,2,...,r) (i=r+
l,.,..,q)
(3)
(Theochem)
279 (1993) 47-52
It has been shown that ]c$), ]I&‘)and ]G) all form the basis vectors for the irreducible representations of the molecular point symmetry group and satisfy the extended maximum overlap criterion [13] - i
(op]Q$~)
= maximum
i=l
(4)
in which the negative sign is used so that the criterion expressed in eqn. (4) is consistent with the previous simple maximum overlap criterion [ 131. All these orbitals are called the maximum overlap symmetry orbitals (MOSOs). Moreover ]c$) and I&‘) are the linear combinations of the corresponding hybrid orbitals [13], also called the maximum overlap symmetry hybrid orbitals (MOSHOs); ]C$‘) are molecular orbitals, called MOSMOs [15,16]. Owing to the interesting properties of ]c()F) and ICI:)‘taking >, them as basis vectors and calculating molecular orbitals, the problem of solving the higher order secular equation is changed into a simpler one of solving Y second-order secular equations. Hence the method for constructing the ]e) MOSMOs is simpler than the ordinary LCAO method. It has also been shown that the MOSMOs obtained are close to the canonical molecular orbitals obtained from the customary LCAO method, especially for the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO), thus illustrating the fact that the MOSMO method is not only a reasonable approximation of the LCAO method, but simpler, and feasible in large 9 molecular systems. Use of the Bloch function basis set
It has been shown [13,17] that the conclusion reached about the symmetry properties of the MOSOs, ]ws), I@) and ]G) is still correct, even if the bases @ and X are replaced by any others, provided @ and X form the bases for q- and rdimensional representations of the point symmetry group, respectively, and fi is a Hermitian operator which commutes with all the transformation operators 6, associated with the symmetry operations i? of the point group. It follows that the
F. Zheng et al./J. Mol. Struct.
‘i’ ‘i’
H
\,4Q”, I
-T
I
H
H
(Theochem)
H
/-
H
H
1
2
H\C=C/H =c
H
/H >-c< >-c\ 2-T Ii
y&H
’
‘Cd’
‘H
H’
‘+Cq
‘c= ‘Ii
49
279 (1993) 47-52
H’
r c
‘“CC
A
3
‘i’ c ‘.\
A 4
(v = I, 2,. . . , r) where N is the number of the unit cells in a crystal, k’the wave vector, and I? the position vector. The overlap integrals between the Bloch functions are given by GP,(~r?lX”(~r’) = ~~,vhv~>
+ 1 [exp (iK_ R)(u,(i - I?)]v,(?)) R’ + exp (- ik’. R)(u,(f)[v,(i
- R))]
H
6 H
H
H
H
+ 1 [exp (iK* ~)(u,(i
- R)ju,(r’))
R’ + exp (- ik’* IZ)(u,(r’>]u,(i - I?))]
= (v,mm>
+ 1 [exp (i,G* IZ)(v,(i - @v,(r’)) R’ + exp(-
Fig. 1. Skeletons of the systems analysed.
MOSMO calculation procedure can be extended, and performed by using various bases that are regarded as @ and X. Therefore, if the atomic orbitals in @ and X are replaced by the corresponding Bloch functions, the ]q) orbitals obtained are a kind of crystal orbital, similar to the crystal orbitals obtained from the ordinary LCAO method. This time, the MOSMO method for molecular systems becomes a simple new method for the construction of the crystal orbitals. Let {u,} and {vy} be the two partitioned sets of orbitals for the atoms of a unit cell, the two sets of Bloch functions {q,, (&?)} and (x, (I@)} are formed as follows: ]q,,(@)) = N-l’*C exp (iIF* R)Ju,(i - R)) + (P’ 1,2,...,q; IX”(@)) = N-“*C exp (iK* R)Jvy (F - I?)) R’
ik’. X)(v,(?)~v,(i-
I?))]
They can be calculated directly from a valence A0 basis set of Slater orbitals using standard Slater exponents. The matrix elements of M constructed from the Bloch functions can be written by using the one-electron hamiltonian, such as that used in the EHMO method [8]:
(cp,(Er’)l&“m)
=
+ C [exp (Se R)(u,(i - R)]A]v,(?)) z + exp (- ik’* IZ)(u,(?)]~v,(?
- R))]
The other elements of the hamiltonian matrix can be expressed in
+ exp ( L ik’. ~)(u,(v’)]fi]u,(i-
~x,GEr’)l&,@D
@))I
=
+ $ [exp(iC* @
R)(v,(r’)]A]v,(i-
Z))]
50
F. Zheng et al./J. Mol. Strut.
(Theochem)
279 (1993) 47-52
A
-15
E-
System 1 +
73 t
System
2
a
E
-ISf-
System
(b)
-5
ELI
I
System 5 -+
System 9 3
3 +
I
System
6+
Fig. 2. The s band structures of six organic polymers.
F. Zhenget al/J. Mol. Struct. (Theochem) 279 (1993) 47-52
51
TABLE 1 The calculated band gaps (ev) for some one-dimensional Method
conjugated
polymers and two-dimensional
graphite
Systema
MOSMO EHMOb
1
2
3
4
5
6
7
8
9
10
0.90 0.96
0.68 0.71
1.18 1.23
0 0
1.55 1.89
0.44 0.47
0 0
0.45 0.45
2.29 2.35
0 0
“See Fig. 1. bRef. 8.
The matrix MM+ and the coefficient matrix respectively be denoted
A” can
MM+ (kt) = R(k) + iZ(k) A”(G) = A”,(k) + i&(k) where R, Ai and Z, A: are real and imaginary parts of complex matrices MM+ and A” respectively. Then the equation MM+ (&I”(@ = A”(k)Q can be changed into
[;;
-:;I[:;; A”,(@ = [ A;(k)
_;;;I A:(@ -A”,(@
I[ 1 Q
0
0
Q
The band structure is determined by solving this eigenvalue equation and performing the subsequent calculation steps in the MOSMO method for various values of k (usually within the first Brillouin zone). The sums of position vectors R’are carried out to first-nearest neighbours. Calculation results and conclusion
To examine the MOSMO calculation procedure using Bloch functions, we consider for all atoms only the valence atomic orbitals and the parameterization used in the EHMO method [12]. A FORTRAN program for calculation with the procedure using Bloch functions has been written for an MV/6000 computer and employed to analyse
some organic polymers and graphite, which often serve as examples for examining new theoretical methods for studying band structures. For convenience of comparison, we have assumed for every system the same orbital exponents, parameter K, and ionization energies of the valence AOs as those used in the EHMO calculation [8] using Bloch functions. The skeletons of these systems are displayed in Fig. 1; the geometries of the calculated polymers have been specified in literature [S]. The calculation results obtained by using the MOSMO procedure show that the band structures (especially for the valence and conduction bands) of these systems are on the whole close to those obtained from the EHMO calculation. As examples, the 7tbands of six of the systems are given in Fig. 2, in which the valence and conduction bands are labelled a and b respectively. The band gaps obtained for the ten analysed systems are listed in Table 1. As seen in Table 1, the band gaps worked out by using the MOSMO and EHMO methods are close to each other, and some important trends in the results of the EHMO calculation [8] are well reproduced by the MOSMO calculations, such as the increase in the band gaps of systems 2, 1 and 3, thus illustrating the reasonableness of the MOSMO calculation scheme using Bloch functions. To summarize, the MOSMO method can be per~formed by using Bloch functions and employed to construct the crystal orbitals and study the band structures. The calculation results obtained are close to those obtained from the ordinary LCAO method if the same parameterization is adopted.
52
F. Zheng et al./J. Mol. Struct. (Theochem) 279 (1993) 47-52
Because the main step during the MOSMO calculation is diagonalizing the matrix MM+-much easier than diagonalizing the hamiltonian matrix of higher order-the MOSMO calculation is simpler and therefore feasible for studying the structureproperty relations in large systems.
14
Acknowledgement
This project was supported by the National Natural Science Foundation of China. References J.-M. Andre, J. Delhalle and J. Ladik (Eds.), Quantum Theory of Polymers, Reidel, Dordrecht, 1978. M. Kertesz, Adv. Quantum Chem., 15 (1982) 161. D.W. Bullett, in H. Ehrenreich, F. Seitz and D. Turnbull (Eds.), Solid State Physics, Vol. 35, Academic Press, New York, 1980. 4 R. Ramirez and M.C. Bohm, Int. J. Quantum Chem., 34 (1988) 47. 5 J.-M. Andre, J. Chem. Phys., 50 (1969) 1536. 6 A. Imamura, J. Chem. Phys., 52 (1970) 3168. 7 M.-H. Whangbo and R. Hoffmann, J. Am. Chem. Sot., 100 (1978) 6093. 8 M.-H. Whangbo, R. Hoffmann, and R.B. Woodward, Proc. R. Sot. London, Ser. A, 366 (1979) 23. 9 A. Zunger, Phys. Rev. B, 17 (1978) 626. 10 E. Orti, P. Lambin, J.-L. Bredas, J.-P. Vigneron, E.G. Derouane, A.A. Lucas and J.-M. Andre, Solid State Commun. 64 (1978) 313. 11 N. Sahoo, S. Markert, T.P. Das and K. Nagamine, Phys. Rev. B, 41 (1990) 220. 12 R. Hoffmann, J. Chem. Phys., 39 (1963) 1398. 13 C.-G. Zhan and F. Zheng, Int. J. Quantum Chem., 39 (1991) 729.
15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
33 34 35 36
C.-G. Zhan, J. Mol. Struct. (Theochem), 236 (1991) 321. C.-G. Zhan, Croat. Chem. Acta, 64 (1991) 469. C.-G. Zhan, Int. J. Quantum Chem., 41 (1992) 773. C.-G. Zhan and F. Zheng, J. Mol. Struct. (Theochem), 226 (1991) 339. L. Pauling, J. Am. Chem. Sot., 53 (193 1) 1367. J.C. Slater, Phys. Rev., 37 (1931) 481. J.N. Murrell,lJ. Chem. Phys., 32 (1960) 767. A. Golebiewski, Trans. Faraday Sot., 57 (1961) 1849. P.G. Lykos and H.N. Schmeising, J. Chem. Phys., 35 (1961) 288. R.J. Bartlett and Y. iihrn, Theor. Chim. Acta, 21 (1971) 215. P. Pelikan and L. Valko, J. Mol. Struct., 28 (1975) 229. R. Boca, P. Pelikan and L. Valko, Chem. Phys., 11 (1976) 229. C.A. Coulson, Valence, Oxford University Press, Oxford, 1952. M. Randic and Z.B. Maksic, Theor. Chim. Acta, 3 (1965) 59; Chem. Rev., 72 (1972) 43. K. Kovacevic and Z.B. Maksic, J. Org. Chem., 39 (1974) 539. Z.B. Maksic and A. Rubcic, J. Am. Chem. Sot., 99 (1977) 4233. Z.B. Maksic, Pure Appl. Chem., 55 (1983) 307. Z.B. Maksic, M. Eckert-Maksic and K. Rupnik, Croat. Chem. Acta, 57 (1984) 1295. K. Jug and Z.B. Maksic, in Z.B. Maksic (Ed.), Theoretical Models of Chemical Bonding, Vol. 3, SpringerVerlag, Berlin, 1991, in press. G. Del Re, J. Mol. Struct. (Theochem), 169 (1988) 487. F. Zheng and C.-G. Zhan, J. Mol. Strnct. (Theochem), 205 (1990) 267. C.-G. Zhan, Q.-L. Wang and F. Zheng, Theor. Chim. Acta, 78 (1990) 129. C.-G. Zhan,,Int. J. Quantum Chem., 40 (1991) 675; J. Mol. Struct. (Theochem), 231 (1991) 33, 39, 47, 57; Chem. Phys. Lett., 179 (1991) 137.
41
Band structures by the maximum overlap symmetry molecular orbital method Fang Zheng”*b, Chang-Guo Zhan” and Xing-Jiao Lib “Department of Chemistry, Central China Normal University, Wuhan 430070 (People’s Republic of China) bDepartment of Solid State Electronics, Huazhong University of Science and Technology, Wuhan 430074 (People’s Republic of China) (Received 3 February
1992; in final form 27 April 1992)
Abstract It is shown that the maximum overlap symmetry molecular orbital (MOSMO) procedure can be employed to construct the crystal orbitals and study the band structures by using the Bloch function basis set. The concrete MOSMO calculations on the extended Hiickel molecular orbital (EHMO) approximation level using Bloch functions for some organic polymers and graphite show that the calculated band gaps are close to those worked out by using the ordinary linear combination of atomic orbitals (LCAO) method if the same parameterization is adopted. Because the MOSMO procedure is more easily performed than the ordinary LCAO procedure, the MOSMO calculation using Bloch functions may be feasible in large systems.
Introduction
The electrical properties of a crystalline material depend on its electronic structure and ultimately upon the chemical constitution of its unit cell. The calculation of the band structures of various solids comprising repeating units are generally based on the ordinary linear combination of atomic orbital (LCAO) method, in which the atomic orbitals are replaced by Bloch functions [1-4]-the basis vectors for the irreducible representations (IRS) of the translation-symmetry group. This method has been extensively applied to many organic polymers [5-81 and many other perfect or defect solid materials [4,9-l l] for discussing the relation between the calculated energy band and electrical properties. Of all the semiempirical LCAO methods which Correspondence to: C.-G. Zhan, Department of Chemistry, Central China Normal University, Wuhan 430070, People’s Republic of China.
have been used for studying electronic structures of crystalline materials, the extended Hi.ickel molecular orbital (EHMO) method [12] using Bloch functions [6,8] may be regarded as the simplest and can be employed in large systems with unit cells formed of many atoms [4]. However, it is still difficult to use the EHMO method to analyse very large systems. Therefore it is of some interest to find a simple new and reasonable one-electron method for studying these problems. A new maximum overlap symmetry molecular orbital (MOSMO) model, differing from the ordinary LCAO method, has been suggested in a previous paper [13]. The theoretical analysis and the numerical results [13-161 show that the calculated results obtained by performing the MOSMO calculation are as reliable as those of the customary LCAO method on the same approximation level. Furthermore, calculation by the MOSMO procedure requires less computing time than the LCAO method, and therefore the MOSMO procedure may be feasible even in very
0166-1280/93/$06.00 0 1993 Elsevier Science Publishers B.V. All rights reserved.
48
F. Zheng et al./J. Mol. Strut.
large systems [13]. The simplicity of the MOSMO procedure leads us to ask whether it could be used to study the electronic structures of solids. In this paper, the MOSMO method is extended to study the band structures of solid materials. After presentation of the basic equations needed for the construction of the crystal orbitals, the MOSMO method using Bloch functions is examined at the EHMO approximation level, taking some conjugated polymers and graphite as examples. Outline of the basic MOSMO method
The MOSMOs can be constructed by a simple method [13], which may be regarded as an extension of the maximum overlap principle [l&36]. In this method, the atomic orbitals (AOs) in a molecule are partitioned into two sets:
~=eP,M,)...lcp,)) X = (Ix, )Ix2). . *IL>) Each set includes the AOs of indirectly bonded atoms, i.e. all the directly bonded atoms are partitioned into different sets [13]. The linear combinations of these AOs give W” = (Iwp)lo;).
. . lo;))
= @A0
Y’”= w;))llc1;)~~~ ItiP>) = XB” Let q 2 r. The coefficient matrices A” and B” are worked out with the equations iVIM+ A0 = A”Q
(1)
LP = iv+ aO(ibzy-’
(2)
where M = CD+fix, a0 is a submatrix formed from the first r columns of A”, Q a diagonal matrix formed from the eigenvalues of matrix MM+, and the diagonal elements of diagonal matrix My consist of the negative square roots of the r nonzero eigenvalues. With Icop) and I t,bp) the delocalized molecular orbitals are obtained:
IG>=
C&II.@+ &I$:) i I@)
(i= 1,2,...,r) (i=r+
l,.,..,q)
(3)
(Theochem)
279 (1993) 47-52
It has been shown that ]c$), ]I&‘)and ]G) all form the basis vectors for the irreducible representations of the molecular point symmetry group and satisfy the extended maximum overlap criterion [13] - i
(op]Q$~)
= maximum
i=l
(4)
in which the negative sign is used so that the criterion expressed in eqn. (4) is consistent with the previous simple maximum overlap criterion [ 131. All these orbitals are called the maximum overlap symmetry orbitals (MOSOs). Moreover ]c$) and I&‘) are the linear combinations of the corresponding hybrid orbitals [13], also called the maximum overlap symmetry hybrid orbitals (MOSHOs); ]C$‘) are molecular orbitals, called MOSMOs [15,16]. Owing to the interesting properties of ]c()F) and ICI:)‘taking >, them as basis vectors and calculating molecular orbitals, the problem of solving the higher order secular equation is changed into a simpler one of solving Y second-order secular equations. Hence the method for constructing the ]e) MOSMOs is simpler than the ordinary LCAO method. It has also been shown that the MOSMOs obtained are close to the canonical molecular orbitals obtained from the customary LCAO method, especially for the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO), thus illustrating the fact that the MOSMO method is not only a reasonable approximation of the LCAO method, but simpler, and feasible in large 9 molecular systems. Use of the Bloch function basis set
It has been shown [13,17] that the conclusion reached about the symmetry properties of the MOSOs, ]ws), I@) and ]G) is still correct, even if the bases @ and X are replaced by any others, provided @ and X form the bases for q- and rdimensional representations of the point symmetry group, respectively, and fi is a Hermitian operator which commutes with all the transformation operators 6, associated with the symmetry operations i? of the point group. It follows that the
F. Zheng et al./J. Mol. Struct.
‘i’ ‘i’
H
\,4Q”, I
-T
I
H
H
(Theochem)
H
/-
H
H
1
2
H\C=C/H =c
H
/H >-c< >-c\ 2-T Ii
y&H
’
‘Cd’
‘H
H’
‘+Cq
‘c= ‘Ii
49
279 (1993) 47-52
H’
r c
‘“CC
A
3
‘i’ c ‘.\
A 4
(v = I, 2,. . . , r) where N is the number of the unit cells in a crystal, k’the wave vector, and I? the position vector. The overlap integrals between the Bloch functions are given by GP,(~r?lX”(~r’) = ~~,vhv~>
+ 1 [exp (iK_ R)(u,(i - I?)]v,(?)) R’ + exp (- ik’. R)(u,(f)[v,(i
- R))]
H
6 H
H
H
H
+ 1 [exp (iK* ~)(u,(i
- R)ju,(r’))
R’ + exp (- ik’* IZ)(u,(r’>]u,(i - I?))]
= (v,mm>
+ 1 [exp (i,G* IZ)(v,(i - @v,(r’)) R’ + exp(-
Fig. 1. Skeletons of the systems analysed.
MOSMO calculation procedure can be extended, and performed by using various bases that are regarded as @ and X. Therefore, if the atomic orbitals in @ and X are replaced by the corresponding Bloch functions, the ]q) orbitals obtained are a kind of crystal orbital, similar to the crystal orbitals obtained from the ordinary LCAO method. This time, the MOSMO method for molecular systems becomes a simple new method for the construction of the crystal orbitals. Let {u,} and {vy} be the two partitioned sets of orbitals for the atoms of a unit cell, the two sets of Bloch functions {q,, (&?)} and (x, (I@)} are formed as follows: ]q,,(@)) = N-l’*C exp (iIF* R)Ju,(i - R)) + (P’ 1,2,...,q; IX”(@)) = N-“*C exp (iK* R)Jvy (F - I?)) R’
ik’. X)(v,(?)~v,(i-
I?))]
They can be calculated directly from a valence A0 basis set of Slater orbitals using standard Slater exponents. The matrix elements of M constructed from the Bloch functions can be written by using the one-electron hamiltonian, such as that used in the EHMO method [8]:
(cp,(Er’)l&“m)
=
+ C [exp (Se R)(u,(i - R)]A]v,(?)) z + exp (- ik’* IZ)(u,(?)]~v,(?
- R))]
The other elements of the hamiltonian matrix can be expressed in
+ exp ( L ik’. ~)(u,(v’)]fi]u,(i-
~x,GEr’)l&,@D
@))I
=
+ $ [exp(iC* @
R)(v,(r’)]A]v,(i-
Z))]
50
F. Zheng et al./J. Mol. Strut.
(Theochem)
279 (1993) 47-52
A
-15
E-
System 1 +
73 t
System
2
a
E
-ISf-
System
(b)
-5
ELI
I
System 5 -+
System 9 3
3 +
I
System
6+
Fig. 2. The s band structures of six organic polymers.
F. Zhenget al/J. Mol. Struct. (Theochem) 279 (1993) 47-52
51
TABLE 1 The calculated band gaps (ev) for some one-dimensional Method
conjugated
polymers and two-dimensional
graphite
Systema
MOSMO EHMOb
1
2
3
4
5
6
7
8
9
10
0.90 0.96
0.68 0.71
1.18 1.23
0 0
1.55 1.89
0.44 0.47
0 0
0.45 0.45
2.29 2.35
0 0
“See Fig. 1. bRef. 8.
The matrix MM+ and the coefficient matrix respectively be denoted
A” can
MM+ (kt) = R(k) + iZ(k) A”(G) = A”,(k) + i&(k) where R, Ai and Z, A: are real and imaginary parts of complex matrices MM+ and A” respectively. Then the equation MM+ (&I”(@ = A”(k)Q can be changed into
[;;
-:;I[:;; A”,(@ = [ A;(k)
_;;;I A:(@ -A”,(@
I[ 1 Q
0
0
Q
The band structure is determined by solving this eigenvalue equation and performing the subsequent calculation steps in the MOSMO method for various values of k (usually within the first Brillouin zone). The sums of position vectors R’are carried out to first-nearest neighbours. Calculation results and conclusion
To examine the MOSMO calculation procedure using Bloch functions, we consider for all atoms only the valence atomic orbitals and the parameterization used in the EHMO method [12]. A FORTRAN program for calculation with the procedure using Bloch functions has been written for an MV/6000 computer and employed to analyse
some organic polymers and graphite, which often serve as examples for examining new theoretical methods for studying band structures. For convenience of comparison, we have assumed for every system the same orbital exponents, parameter K, and ionization energies of the valence AOs as those used in the EHMO calculation [8] using Bloch functions. The skeletons of these systems are displayed in Fig. 1; the geometries of the calculated polymers have been specified in literature [S]. The calculation results obtained by using the MOSMO procedure show that the band structures (especially for the valence and conduction bands) of these systems are on the whole close to those obtained from the EHMO calculation. As examples, the 7tbands of six of the systems are given in Fig. 2, in which the valence and conduction bands are labelled a and b respectively. The band gaps obtained for the ten analysed systems are listed in Table 1. As seen in Table 1, the band gaps worked out by using the MOSMO and EHMO methods are close to each other, and some important trends in the results of the EHMO calculation [8] are well reproduced by the MOSMO calculations, such as the increase in the band gaps of systems 2, 1 and 3, thus illustrating the reasonableness of the MOSMO calculation scheme using Bloch functions. To summarize, the MOSMO method can be per~formed by using Bloch functions and employed to construct the crystal orbitals and study the band structures. The calculation results obtained are close to those obtained from the ordinary LCAO method if the same parameterization is adopted.
52
F. Zheng et al./J. Mol. Struct. (Theochem) 279 (1993) 47-52
Because the main step during the MOSMO calculation is diagonalizing the matrix MM+-much easier than diagonalizing the hamiltonian matrix of higher order-the MOSMO calculation is simpler and therefore feasible for studying the structureproperty relations in large systems.
14
Acknowledgement
This project was supported by the National Natural Science Foundation of China. References J.-M. Andre, J. Delhalle and J. Ladik (Eds.), Quantum Theory of Polymers, Reidel, Dordrecht, 1978. M. Kertesz, Adv. Quantum Chem., 15 (1982) 161. D.W. Bullett, in H. Ehrenreich, F. Seitz and D. Turnbull (Eds.), Solid State Physics, Vol. 35, Academic Press, New York, 1980. 4 R. Ramirez and M.C. Bohm, Int. J. Quantum Chem., 34 (1988) 47. 5 J.-M. Andre, J. Chem. Phys., 50 (1969) 1536. 6 A. Imamura, J. Chem. Phys., 52 (1970) 3168. 7 M.-H. Whangbo and R. Hoffmann, J. Am. Chem. Sot., 100 (1978) 6093. 8 M.-H. Whangbo, R. Hoffmann, and R.B. Woodward, Proc. R. Sot. London, Ser. A, 366 (1979) 23. 9 A. Zunger, Phys. Rev. B, 17 (1978) 626. 10 E. Orti, P. Lambin, J.-L. Bredas, J.-P. Vigneron, E.G. Derouane, A.A. Lucas and J.-M. Andre, Solid State Commun. 64 (1978) 313. 11 N. Sahoo, S. Markert, T.P. Das and K. Nagamine, Phys. Rev. B, 41 (1990) 220. 12 R. Hoffmann, J. Chem. Phys., 39 (1963) 1398. 13 C.-G. Zhan and F. Zheng, Int. J. Quantum Chem., 39 (1991) 729.
15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
33 34 35 36
C.-G. Zhan, J. Mol. Struct. (Theochem), 236 (1991) 321. C.-G. Zhan, Croat. Chem. Acta, 64 (1991) 469. C.-G. Zhan, Int. J. Quantum Chem., 41 (1992) 773. C.-G. Zhan and F. Zheng, J. Mol. Struct. (Theochem), 226 (1991) 339. L. Pauling, J. Am. Chem. Sot., 53 (193 1) 1367. J.C. Slater, Phys. Rev., 37 (1931) 481. J.N. Murrell,lJ. Chem. Phys., 32 (1960) 767. A. Golebiewski, Trans. Faraday Sot., 57 (1961) 1849. P.G. Lykos and H.N. Schmeising, J. Chem. Phys., 35 (1961) 288. R.J. Bartlett and Y. iihrn, Theor. Chim. Acta, 21 (1971) 215. P. Pelikan and L. Valko, J. Mol. Struct., 28 (1975) 229. R. Boca, P. Pelikan and L. Valko, Chem. Phys., 11 (1976) 229. C.A. Coulson, Valence, Oxford University Press, Oxford, 1952. M. Randic and Z.B. Maksic, Theor. Chim. Acta, 3 (1965) 59; Chem. Rev., 72 (1972) 43. K. Kovacevic and Z.B. Maksic, J. Org. Chem., 39 (1974) 539. Z.B. Maksic and A. Rubcic, J. Am. Chem. Sot., 99 (1977) 4233. Z.B. Maksic, Pure Appl. Chem., 55 (1983) 307. Z.B. Maksic, M. Eckert-Maksic and K. Rupnik, Croat. Chem. Acta, 57 (1984) 1295. K. Jug and Z.B. Maksic, in Z.B. Maksic (Ed.), Theoretical Models of Chemical Bonding, Vol. 3, SpringerVerlag, Berlin, 1991, in press. G. Del Re, J. Mol. Struct. (Theochem), 169 (1988) 487. F. Zheng and C.-G. Zhan, J. Mol. Strnct. (Theochem), 205 (1990) 267. C.-G. Zhan, Q.-L. Wang and F. Zheng, Theor. Chim. Acta, 78 (1990) 129. C.-G. Zhan,,Int. J. Quantum Chem., 40 (1991) 675; J. Mol. Struct. (Theochem), 231 (1991) 33, 39, 47, 57; Chem. Phys. Lett., 179 (1991) 137.