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Simple Lcao band calculation for the ideal polyethylene chain
Volume 2, number 4
CHEMICAL
SIMPLE THE
LCAO IDEAL
PHYSICS LETTERS
BAND
Chemistry
C,%LCULATION
POLYETHYLENE
W. L. McCUBBIN Quantum
August 1968
Group,
FOR
CHAIN
-
** and FL MANNE
Uppsala
University,
Uppsala,
Sweden
Received 20 June 1968 Revised manuscript received 8 July 1968
The extended Iilickel method is used to calculate the hand structure of ideal polyethylene. The results for the valence bands are in reasonable agreement with some earlier work. those for the unoccupied states are not. except in one respect which permits 3 speculation on the origin of an unexplained feature of the optical absorption spectrum.
The ideal polyethylene chain has the unit cell and first Brillouin zone shown in figs. la and lb,
respectively. For a block of Nunit cells with cyclic boundary conditionthe allowed values of the crystal momentum k are 2v?n./Nt where m = = o.*l,k2... ,) A?‘2 (N even). For each value of k in the reduced zone scheme there will be a number of eigenstates which can be expressed as linear combinations of Bloch sums constructed from a minimal atomic basis, i.e. the 2s- and 2porbitals of carbon and the Is-orbital of hydrogen. On this basis, there will be 12 electrons per unit cell and 12 eigenvalues for each value of k for which there is no degeneracy. The wave function for the block of N unit cells can be written
“It A
x
tion constant.
The matrix
elements
between
exp(ik.RZ)lii(Z)
;
x exp[ik.(xj
- xi)]
x exp[ik*(xj
-xi)]
lco
ewdik*RlrSijU)
and
H$
= m,$$,$j
the vector relating the origin of the ith atomic orbital to the center of the cell. K&i is a normalinain part by the Swedish Natural Science Research Council and in part by the Air Force Office of Scientific Research (OSR) through the European Office of Aerospace Research (OAR). United States Air Force under Grant AF EOAR 67-50. ** Present and permanent address: Department of Applied Physics, University of Wales Institute of Ecience and Technology. Cardiff. CFl 3NU, UK.
‘kr
these
are N-l
N-l z.
ui(Z) is the ith atomic function in the Zth unit cell, RZ = It and xi is the projection on the x axis of
230
r
Fig. 1. (a) The unit cell of the polyethylene chain, (b) The first Brillouin zone.
in which
* Sponsored
0
x
tb)
Bloch functions
xki = KZflZeq(ik-xi)
-nit
Ngi elq,(ik.f?Z)Hu(Z)
in which s@(Z) =I u;(0)tcj(Z) dr and
CHEMICAL PHYSICS
Volume 2, number 4
By means of this limited use of the symmetry the original secular equation of order 12Nis thus reduced to a set of N equations of order 12, one for each value of k. The matrix elements Sk- and XI& are in general complex (Hermitian) but ?an in the present case be brought to real (symmetric) form by a phase transformation. In this calculation use is made of the extended Hiickel approximation as used by Hoffmm for hydrocarbons [l]. The matrix elements of the effective Hamiltonian between atomic orbitals are approximated by Hii
=
Hii)/
(i = j).
Here Zi is the atomic valence state ionization energy and F the so-called Woifsberg-Helmholz constant. The calculations employed the same values of Zi as used by Hoffmann [l], viz. ZHls = = -13.6 eV, ZC2s = -21.4 eV and 1~2~ = -11.4 eV, but were carried out for several values of F. Overlap was calculated over Slater orbitals and was included to six and ten unit cells corresponding to N = 12 and N = 20, respectively. Energy eigenvalues were printed out to five decimals and to this accuracy the two sets of results were identical. Therefore, in fig. 2 only the F-dependence of the band structure is illustrated. As regards the choice of F, 1.75 was the value used by Hoffmann [l] whereas F = 2 is compatible with a more rigorous treatment of the off-diagonal matrix elements [2].
-‘Or
50
II-‘”
1968
The valence states of o and B symmetry at I? are ordered in energy as previously predicted by a qualitative argument [3]. In that paper lobe functions centred at each carbon atom were constructed diagramatically from occupied atomic orbitals using elementary symmetry considerations. From these diagrams the composition of molecular states at points of high symmetry in the Brillouin zone were deduced and their relative energy estimated from the magnitude and sign of the Lobe functions. The argument was made semi-quantitative by the use of interaction parameters caIcuIated by Hall for short chain paraffins, thereby permitting estimates of the valence band width and the hole effective mass. The results were given in terms of 6, the phase difference between adjacent carbon atoms, instead of the wave vector k. In comparing fig. l(d) of ref. [3] with the present results, it should be noted that, owing to gIide symmetry, the uppermost state of each band should occur at 0 = 0. It is also clear that the valence states are not very sensitive to the choice of F. Regarding quantitative aspects we note that the B2g state is about 2 eV low [3]. The hole effective mass for F = 1.75 and 2.0 are n$ = 0.46 tn and 0.36 tth respectivelp The agreement with the previous value [3] nlZt = 0.17m is as good as can be expected from approximate calculations. The composition and ordering of the various unoccupied states is different from that suggested an interestin ref. [3]. There is, nevertheless, ing speculation to be made on the basis of the above results. To see this, we refer to previous
-zi ,
Hij = FSij(Hii+
August
LETTERS
bu
50
r7P
CO
co
I
30
30
1
I
(b) Fig.
2. Band structure
Cd)
of ideal polyethylene. (a) and (b) are the valence bands for F = 1.75 and F = 2 rs;pectively: (c) and (d) are the unoccupied bands for the same two values of F-.
231
ultraviolet absorption spectrum (as far as it was known at that time) attention was focussed [5] on the peak - 6.7 eV, part of which we considered could possibly be intrinsic to the paraffin structure. It is clear from Partridge’s experiments [S] that these selection rules ought to have been applied to the 7.6 eV band (had its existence been known),
since this is the first
intense absorption
which certainly relates to the polyethylene chain itself. ‘Ihi3 leaves unanswered the question of the narrow band at 6.7 eV. While a chemical impurity may play some part in this absorption process, one cannot but be struck by the fact that both the earlier qualitative band structure and the one presented here indicate the existence of a narrow band lying some way below a system of broad bands; moreover, in both cases, it can be shown that transitions to this narrow band from the valence band are weakly allowed *. Therefore, al* For the present band structure we also use the results contained in ref. [4].
232
ACKNOWLEDGEMENT Support for one author (R.M.) was provided by Professor P.LO. Liiwdin, who was also instrumental in obtaining the Swedish Institute bursary enjoyed by the other author. For this and for his constructive interest, the authors wish to offer him their warmest thanks.
REFERENCES [l] R. Hoffmann, J. Chem. Phys. 39 (1964) 1397. [2] R. Manne, Theoret. Chim. Acta 6 (1966) 299. 131 - _ W. L. McCubbin and I.D.C. Gurnev._, J.Chem. Phvs. 43 (1965) 463. 141 W. L. McCubbin. Phvs.Stat.Sol. 16 (19661 264. i5j W. L. McCubbinSand>. C. Weeks, J. &pl.‘Phys. 37 (1966) 3644. [6] R. M. Partridge, J. Chem. Phys. 45 (1966) 1685.
CHEMICAL
SIMPLE THE
LCAO IDEAL
PHYSICS LETTERS
BAND
Chemistry
C,%LCULATION
POLYETHYLENE
W. L. McCUBBIN Quantum
August 1968
Group,
FOR
CHAIN
-
** and FL MANNE
Uppsala
University,
Uppsala,
Sweden
Received 20 June 1968 Revised manuscript received 8 July 1968
The extended Iilickel method is used to calculate the hand structure of ideal polyethylene. The results for the valence bands are in reasonable agreement with some earlier work. those for the unoccupied states are not. except in one respect which permits 3 speculation on the origin of an unexplained feature of the optical absorption spectrum.
The ideal polyethylene chain has the unit cell and first Brillouin zone shown in figs. la and lb,
respectively. For a block of Nunit cells with cyclic boundary conditionthe allowed values of the crystal momentum k are 2v?n./Nt where m = = o.*l,k2... ,) A?‘2 (N even). For each value of k in the reduced zone scheme there will be a number of eigenstates which can be expressed as linear combinations of Bloch sums constructed from a minimal atomic basis, i.e. the 2s- and 2porbitals of carbon and the Is-orbital of hydrogen. On this basis, there will be 12 electrons per unit cell and 12 eigenvalues for each value of k for which there is no degeneracy. The wave function for the block of N unit cells can be written
“It A
x
tion constant.
The matrix
elements
between
exp(ik.RZ)lii(Z)
;
x exp[ik.(xj
- xi)]
x exp[ik*(xj
-xi)]
lco
ewdik*RlrSijU)
and
H$
= m,$$,$j
the vector relating the origin of the ith atomic orbital to the center of the cell. K&i is a normalinain part by the Swedish Natural Science Research Council and in part by the Air Force Office of Scientific Research (OSR) through the European Office of Aerospace Research (OAR). United States Air Force under Grant AF EOAR 67-50. ** Present and permanent address: Department of Applied Physics, University of Wales Institute of Ecience and Technology. Cardiff. CFl 3NU, UK.
‘kr
these
are N-l
N-l z.
ui(Z) is the ith atomic function in the Zth unit cell, RZ = It and xi is the projection on the x axis of
230
r
Fig. 1. (a) The unit cell of the polyethylene chain, (b) The first Brillouin zone.
in which
* Sponsored
0
x
tb)
Bloch functions
xki = KZflZeq(ik-xi)
-nit
Ngi elq,(ik.f?Z)Hu(Z)
in which s@(Z) =I u;(0)tcj(Z) dr and
CHEMICAL PHYSICS
Volume 2, number 4
By means of this limited use of the symmetry the original secular equation of order 12Nis thus reduced to a set of N equations of order 12, one for each value of k. The matrix elements Sk- and XI& are in general complex (Hermitian) but ?an in the present case be brought to real (symmetric) form by a phase transformation. In this calculation use is made of the extended Hiickel approximation as used by Hoffmm for hydrocarbons [l]. The matrix elements of the effective Hamiltonian between atomic orbitals are approximated by Hii
=
Hii)/
(i = j).
Here Zi is the atomic valence state ionization energy and F the so-called Woifsberg-Helmholz constant. The calculations employed the same values of Zi as used by Hoffmann [l], viz. ZHls = = -13.6 eV, ZC2s = -21.4 eV and 1~2~ = -11.4 eV, but were carried out for several values of F. Overlap was calculated over Slater orbitals and was included to six and ten unit cells corresponding to N = 12 and N = 20, respectively. Energy eigenvalues were printed out to five decimals and to this accuracy the two sets of results were identical. Therefore, in fig. 2 only the F-dependence of the band structure is illustrated. As regards the choice of F, 1.75 was the value used by Hoffmann [l] whereas F = 2 is compatible with a more rigorous treatment of the off-diagonal matrix elements [2].
-‘Or
50
II-‘”
1968
The valence states of o and B symmetry at I? are ordered in energy as previously predicted by a qualitative argument [3]. In that paper lobe functions centred at each carbon atom were constructed diagramatically from occupied atomic orbitals using elementary symmetry considerations. From these diagrams the composition of molecular states at points of high symmetry in the Brillouin zone were deduced and their relative energy estimated from the magnitude and sign of the Lobe functions. The argument was made semi-quantitative by the use of interaction parameters caIcuIated by Hall for short chain paraffins, thereby permitting estimates of the valence band width and the hole effective mass. The results were given in terms of 6, the phase difference between adjacent carbon atoms, instead of the wave vector k. In comparing fig. l(d) of ref. [3] with the present results, it should be noted that, owing to gIide symmetry, the uppermost state of each band should occur at 0 = 0. It is also clear that the valence states are not very sensitive to the choice of F. Regarding quantitative aspects we note that the B2g state is about 2 eV low [3]. The hole effective mass for F = 1.75 and 2.0 are n$ = 0.46 tn and 0.36 tth respectivelp The agreement with the previous value [3] nlZt = 0.17m is as good as can be expected from approximate calculations. The composition and ordering of the various unoccupied states is different from that suggested an interestin ref. [3]. There is, nevertheless, ing speculation to be made on the basis of the above results. To see this, we refer to previous
-zi ,
Hij = FSij(Hii+
August
LETTERS
bu
50
r7P
CO
co
I
30
30
1
I
(b) Fig.
2. Band structure
Cd)
of ideal polyethylene. (a) and (b) are the valence bands for F = 1.75 and F = 2 rs;pectively: (c) and (d) are the unoccupied bands for the same two values of F-.
231
ultraviolet absorption spectrum (as far as it was known at that time) attention was focussed [5] on the peak - 6.7 eV, part of which we considered could possibly be intrinsic to the paraffin structure. It is clear from Partridge’s experiments [S] that these selection rules ought to have been applied to the 7.6 eV band (had its existence been known),
since this is the first
intense absorption
which certainly relates to the polyethylene chain itself. ‘Ihi3 leaves unanswered the question of the narrow band at 6.7 eV. While a chemical impurity may play some part in this absorption process, one cannot but be struck by the fact that both the earlier qualitative band structure and the one presented here indicate the existence of a narrow band lying some way below a system of broad bands; moreover, in both cases, it can be shown that transitions to this narrow band from the valence band are weakly allowed *. Therefore, al* For the present band structure we also use the results contained in ref. [4].
232
ACKNOWLEDGEMENT Support for one author (R.M.) was provided by Professor P.LO. Liiwdin, who was also instrumental in obtaining the Swedish Institute bursary enjoyed by the other author. For this and for his constructive interest, the authors wish to offer him their warmest thanks.
REFERENCES [l] R. Hoffmann, J. Chem. Phys. 39 (1964) 1397. [2] R. Manne, Theoret. Chim. Acta 6 (1966) 299. 131 - _ W. L. McCubbin and I.D.C. Gurnev._, J.Chem. Phvs. 43 (1965) 463. 141 W. L. McCubbin. Phvs.Stat.Sol. 16 (19661 264. i5j W. L. McCubbinSand>. C. Weeks, J. &pl.‘Phys. 37 (1966) 3644. [6] R. M. Partridge, J. Chem. Phys. 45 (1966) 1685.