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A Floating Spherical Gaussian Orbital (FSGO) model for polymers: calculation of X-ray diffraction structure factors
Chemical Physics 20 (1977) 367-370 0 North-Holland Publishing Company
A FLOATING SPHERICAL GAUSSIAN ORBITAL (FSGO) MODEL FOR POLYMERS: CALCULATION OF X-RAY DIFFRACTION STRUCTURE FACTORS J.M. ANDRE and J-L. BREDAS Fact&& Universitaires Notre-Dame de la Pair. Laboratoire 61. rue de Bruxelles, B-5000 - Namur, Belgium
de Chimie Thdorique AppliqtrPe.
Received 13 September 1976
The application of the Floating Spherical Gaussian Orbital (FSGO) method for polymers to the calculation of X-ray structure factors is described. The use of off-atomic centered basis functions provides a good description of the bonding density for small values of (sin I~)/Aand gives results in excellent agreement with those obtained by the usual atomic scattering factors for large values of (sin @)/A.
1_ Introduction
3/,(k 4 = Over the last few years, methods for calculating band structures of ab lnitio quality have been suggested [l-4]. One of the most promising approaches is based on the use of Spherical Localized Gaussian Orbitals (FSGOs) as originally proposed by Frost [5] and generalized later by Christoffersen [6,7]. Such an approach differs from the most popular ab initio ones by using gaussians centered on chemical bonds instead of atomic functions. In this paper, we check the validity of the FSGO model by calculating electronic densities and mainly by determining related X-ray scattering factors in polymers and molecular crystals.
F
cpP(k, r) = (l/n)
c i
cxp(ik - ri> xp(r - rj) ,
(2)
where Q defines a translation in the direct space and +(r - rj) is a basis function related to the origin unit cell by translation for a system of N unit cells. The contribution of an energy band to the electronic density at a given position r is, in terms of crystalline orbitals: Jl%,
r) $,(k, r)
(3)
or in terms of basis functions: n,(r) = (l/Wk
In band theory, the one-electron approximation defines crystalline orbitals to represent the wave function of an electron in a periodic potential; with the aid of the LCAO approximation, the crystalline orbitals $,#, r) are expressed as linear combination of Bloch atomic functions $(k, r), where k is a point of tbe first Brillouin zone of the polymer:
(1)
The Bloch orbitals $Jk, r) are expanded in terms of basis functions since:
p,*(r) = T 2. Electronic densities in polymers from an FSGO model
C,,(k) Gp(kr) .
iT, 4 C&,(k) C,&) ., . 3
xexp[ik-(rh-rf)]
xp(r-rj)&&-r,l).
(4)
For analyzing the electronic density, it is useful to introduce the elements of the density matrices: Rpq(k) = G
C;p(k) Cnq(k) 9
(9
and D&j)
=(1/N) T
explik -$,r - 931 R&(@
*
(6)
When restricting our attention to saturated polymers or more generally to those where the spatial distribution of ali n electrons within the unit celi Can be adequately described by n/2 basis functiods, the density matrices elements R(k) are univomliy computed from the elements of the overlap matrices between BIoch Funcrionsr
mentally obseived ones in X-ray diffraction except For a temperature factor. In our approach, the basis functions were FSCOs defmed as x,(r) = Eza,/r43’4 exPi-cu,(~ - .+,21 , c10 where both the location rp and the orbital exponent alp are used as variational parameters. Standard locations and exponents are given in the fiterature [9j. These orbit& provide a simple resolution of the genera&red overlap integrals [ 10 J S’$G)
=lx,(r
- ta) exp(i(;
l
4 x,(r - t$ d3r
= [2o&Qr33i4 [I&r9 frr]3j4 expf-+&-~~-~p~21 by using the inversion procedure: R(k) = s-1(k)
X expfiG - r) exp [-+r
.
The LCAO coefficients for this one~determ~nant PoIymeric wave function and, as a consequence, the dectr~ni~densityare thus completely defined end no ite&ion calculation is necessary. The periodic properties of the electronic densities of moiecular crystals or polymers make them readily suitable to a Fourier development such as: p(r) - 9
FyG) exp(-iG - r) .
(81
The Fourier transforms F(G) are easily obtained by a straightforward integration. In the following, V means the tatal volume of the crystal (V = NV):
- 3 - r,)‘j
d3r .
02) If we develop G = h* + kb* + 2c* in terms of reciprocal tat tice parameters and r = xu +yb + ze in terns of direct lattice parameters, the generahzed overlap integral becomes: S;,$ (G) = [2+[4
314[Zo~,/rrJ~/~ [rrfccu,+ ar,)] 3/2
X expC--toLpo$ofp + ff$I AB2J X exp {-- frr2/(oP f OL 4 )I ($ + k2 i Z*)> X exp D(hC, + kc’ + N$)] Table t
,
FSGO orbitalexponentsand center coordinates
The generalized overlap integrelsSFi(G) atomic orbit& are defined as: ZZ$$G) =s+(r
between
- ro) exp(iC - r> &(r - fj) d3r _ (10)
The generalized overlap integrals which were fhst introduced by Stewart [gf are natbing but the Fourier transform ofan overlap charge distribution.. it is important to_uote that, unlike the over@ integrals $e, the SFq(G)‘s depend on the absolute positions of centers p and q ;ts weU as on their relative positions. It is clear that the coefficients E(G) are the StNCture factors of the polymers related to the experi-
(131
J_M. An&& J.L. Brt?dasjFSGO X-ray diffractionstructurefactors
where AB is the distance between the two orbital centers A and B and Ci=(~~Ai+orqBi)/(orp+cuq).
3. Calculation polyethylene
i=X,y,Z
of X-ray diffraction
-
(14)
factors for
tn this section, the formerly described model is applied to the determination of X-ray diffraction factors for polyethylene using the geometrical structure obtained by neutron diffraction [l l] _ Table 2 Comparison sine T3.9058 0.0975 0.8870
0.8866 0.8696 0.8601 0.8571 0.8495 0.8435 0.8422 0.8290 0.8250 0.8210 o.a1a3 Cl.8116
0.805s 0.8032 O.-l943 O.,SQQ 0.7352 0.7347 0.7120 0.6992 0.6952 0.6848 0.6814 0.676, 0.6749 Cl.6690 0.6569 O&S13 0.645, 0.6324 0.6243 0.6240 Cl.6207 0.616, 0.5920 0.5913 Cl.5610 0.5436 0.5382 0.5239 0.5211 OS*67 0.5125 O.S1co 0.5016 0.4961 0.4843 0.4808 0.4603 0.4795 i---m
between
theoretical
a
1c
L
4 4 4 4 4 4 4 1 4 4 4 4 4
4 0 4 3 3 2 3 3 1 2 2 1 2
: 4 4 4 3
1 0 1
:
3 : 3 3 3 : 3 3 3 3
z 3 3 3 2 2 2 2
2 2 1 0 2 t 2 t 2 0 1
Two polymer chains run through the Pm, orthorhombic unit cell which contains four carbon and eight hydrogen atoms. The atomic coordinates and the FSGO orbital exponents and center coordinates are listed in table 1. The density matrix elements come from a FSGO LCAO CO program for saturated polymers [ 12]_ Since polyethylene is an hydrocarbon, they are calculated in first approximation for an isolated chain and by taking into account the interactions between one CzHd unit group and two C,H, groups at each side. As a result, the R,,(k) elements with orbitals p and q belonging to two different chains are regarded as zero.
and calculated scattering factors a
2
0
4 4 3 3 2 3 3 3 1 2 0 2 I
2 1 Q 0 1 1 4 3 3 2 3 3 4 4 1 2 0 4 : 4
369
0.036 0.051
0.035 0.080 0.030 cl.l%O o.oat 0.020 LO,, O.O,9 0.199 0.150 o.co3 0.098 0.002 0.109 o-125 cl.01.5 0.116 0.114 0.148 0.120 0.360 0.179 0.095 a.3o3 0.107 0.225 0.355 0.065 0.331 0.256 0.046 0.148 0.036 0.084 0.099 0.0'11 0.398 0.252 0.245 0.109 0.665 0.368 0.032 0.059 0.278 0.988 0.044 0.260 0.1cn 0.779 0.228 U-M-
3.4763 D.4681 D.4626 0.4532 0.4484 0.4402 0.4358 0.4357 0.4348 0.430, 0.4298 0.4170 0.4111 0.4OQ6 0.3905 o-x99 0.3831 0.3739 0.3704 0.3636 0.3584 0.3553 0.3328 0.3321 0.3206 0.3tm 0.308, 0.3079 0.2939 0.2759 0.26LS 0.2540 0.2460 0.2450 0.2338 0.2169 0.2168 (1.1831 0.1476 0.1313 ___---
K
L
-2.856 -1.66: -4.099 O.99L -1.84, -2.053 2.439 -4.980 5.255 -5.553 1.841 -3.614 -7.268 4.203 1.444 -,_3,7 6.023 1.452 2.,SO 1.251 -6.297 4.168 -3.8,X 4.29, -S.,,3 3.377 4.518 6.191 5:01s 4.556 32102 -8.451 4.956 -4.988 7.383 3.218 9.463 4.426 15.651 la.053 .__-_-____
-3.379 -1.659 -5.011 a.152 -1.776 -2.Q98 2.654 -6.oa2 s-668 -6.594 1.ca4 -4-169 -9&I, 4.665 1.679 -1.614 ,.,49 1.634 2.943 1.410 -1.528 4,694 -4.130 4.234 -6.239 3.752 5.062 6.243 4.932 4.5'16 3.858 -8.178 4.892 -5.019 6.999 3.327 Q-&,2 4.174 14.,15 16.707
a.923 o.m2 0.912 0.839 0.071 0.045 0.115 1.102 0.613 1.041 0.83, 0.575 1.731 0.662 0.235 0.23, 1.126 0.382 o.*93 c.159 1.231 0.526 o.3t9 0.063 0.466 0.375 0.544 Cl.052 0.064 o.c)zcl 0.156 0.273 0.064 O.C31 0.384 0.1OQ 0.c.z.Q 0.252 0.916 1.346
370
JM. AndrP. J.L. BrS&SGO
X-ray diffraction
structure
factors
Table 2 gives the theoretical scattering factors and compare them with those classically calculated in radiocjstallography as follows:
ionization potentials and photoelectron spectra [15]. Thus, the FSGO procedure can be considered as a very efficient and practical method for the calculation of electronic properties of polymers_
where the fK’s are the atomic form factors as listed in the International tables of X-ray Crystallography [13]. Debye-Wailer (temperature) factors are omitted for both the theoretical and calculated scattering factors.
References
4. Discussion The general agreement between tie theoretical scattering factors and the calculated one appears to be excellent. However, it must be noted that the agreement is better for large values of (sin 0)/A than for small ones. If we look at the amplitudes of the waves scattered by core and valence electrons, we see that, following Stewart’s calculations [ 141, for (sin 0)/A smaller than about 0.45, the valence electrons play an important role in the scattering phenomenon. The quantum approach, which takes into account the chemical binding and is basically different from the traditional crystallographic viewpoint which regards isoIated and spherical atoms, is then more adequate to describe the right electronic distribution and especially the bondiug density. - For (sin 0)/h larger than 0.45, only the core electrons scatter. As a result, the theoretical and calculated scattering factors are then in perfect agreement_
We must add that a Fourier difference analysis shows that the theoretical electronic density is Iarger than the calculated one, in the middle of the C-C bonds and between C and H atoms. As general conclusion, the FSGO model does properly calculate electronic densities and related X-ray structure factors. Other studies have proved that this procedure is equally valuable for computing
[I] J.M. Andre, J. Delhalle, C. Demanet and M.E. GerardLambert, Internat. J_ Quantum Chem, Symposium no. 10 (1976) 99. 121 J. Delhalle. J.M. And&, S. Delhalle, C. Pivont-Malherbe, F. Clarisse, G. Leroy and D. Peeters. Theoret. Chim. Acta, submitted for publication. [3] J. Ladik, in: Electronic Structure of Polymers and Molecular Crystals, eds. J.M. Andre and J. Ladik (Plenum Press, New York, 1975) p-23. [4] N. RGsch and J. Ladik, Chem. Phys. 13 (1976) 285. (51 A.A. Frost, J. Chem. Phys. 47 (1967) 3707. [6J R.E. Christoffersen, J. Amer. Chem. Sot. 93 (1971) 4104. [7i R.E. Christoffersen, Advan. Quantum Chem. 6 (1972) 333. [S] R.F. Stewart, J. Chem. Phys. Sl(1969) 4569. [9] A.A. Frost and R.A. Rouse, J. Amer. Chem. Sot. 90 (1968) 1965. [lo] J.M. Andre, I. Delhalle, J-G. Fripiat and G. Leroy, Intern. J. Quantum Chem. 5 (1971) 67. [ 111 G. Avitabile, R. Napohtano, B. Pirozzi, K.D. Rouse, M.W. Thomas and B.T.M. Willis, Polymer Letters 13 (1975) 351. [12] C..Demanet and J.M. And&, An Automatic Program for FSGO Calculation on Polymers. This program runs on a Digital PDP 1 l/45 system within 32K-words (16 bits) of memory. Typical running time is 10 minutes for the study of polyethylene calculating all interactions between eleven unit cells_ 1131 International Tables of X-ray Crystallography. VoL4
(Kynoch Press, Birmingham, 1974) p. 72; hydrogen
form factors computed by D.T. Cramer and J-B. Mann [Acta Cryst. A24 (1968) 3211, carbon form factors computed by P.A. Doyle and P.S. Turner [Acta Cryst. A24 (1968) 390]_ [14] R.F. Stewart, J. Chem. Phys. 48 (1968) 4882. [15] J.M. And& M.E. G&ud-Lambert and C. Lamotte, Bulletin de la So&t& Chimique de Belgique, submitted for publication.
A FLOATING SPHERICAL GAUSSIAN ORBITAL (FSGO) MODEL FOR POLYMERS: CALCULATION OF X-RAY DIFFRACTION STRUCTURE FACTORS J.M. ANDRE and J-L. BREDAS Fact&& Universitaires Notre-Dame de la Pair. Laboratoire 61. rue de Bruxelles, B-5000 - Namur, Belgium
de Chimie Thdorique AppliqtrPe.
Received 13 September 1976
The application of the Floating Spherical Gaussian Orbital (FSGO) method for polymers to the calculation of X-ray structure factors is described. The use of off-atomic centered basis functions provides a good description of the bonding density for small values of (sin I~)/Aand gives results in excellent agreement with those obtained by the usual atomic scattering factors for large values of (sin @)/A.
1_ Introduction
3/,(k 4 = Over the last few years, methods for calculating band structures of ab lnitio quality have been suggested [l-4]. One of the most promising approaches is based on the use of Spherical Localized Gaussian Orbitals (FSGOs) as originally proposed by Frost [5] and generalized later by Christoffersen [6,7]. Such an approach differs from the most popular ab initio ones by using gaussians centered on chemical bonds instead of atomic functions. In this paper, we check the validity of the FSGO model by calculating electronic densities and mainly by determining related X-ray scattering factors in polymers and molecular crystals.
F
cpP(k, r) = (l/n)
c i
cxp(ik - ri> xp(r - rj) ,
(2)
where Q defines a translation in the direct space and +(r - rj) is a basis function related to the origin unit cell by translation for a system of N unit cells. The contribution of an energy band to the electronic density at a given position r is, in terms of crystalline orbitals: Jl%,
r) $,(k, r)
(3)
or in terms of basis functions: n,(r) = (l/Wk
In band theory, the one-electron approximation defines crystalline orbitals to represent the wave function of an electron in a periodic potential; with the aid of the LCAO approximation, the crystalline orbitals $,#, r) are expressed as linear combination of Bloch atomic functions $(k, r), where k is a point of tbe first Brillouin zone of the polymer:
(1)
The Bloch orbitals $Jk, r) are expanded in terms of basis functions since:
p,*(r) = T 2. Electronic densities in polymers from an FSGO model
C,,(k) Gp(kr) .
iT, 4 C&,(k) C,&) ., . 3
xexp[ik-(rh-rf)]
xp(r-rj)&&-r,l).
(4)
For analyzing the electronic density, it is useful to introduce the elements of the density matrices: Rpq(k) = G
C;p(k) Cnq(k) 9
(9
and D&j)
=(1/N) T
explik -$,r - 931 R&(@
*
(6)
When restricting our attention to saturated polymers or more generally to those where the spatial distribution of ali n electrons within the unit celi Can be adequately described by n/2 basis functiods, the density matrices elements R(k) are univomliy computed from the elements of the overlap matrices between BIoch Funcrionsr
mentally obseived ones in X-ray diffraction except For a temperature factor. In our approach, the basis functions were FSCOs defmed as x,(r) = Eza,/r43’4 exPi-cu,(~ - .+,21 , c10 where both the location rp and the orbital exponent alp are used as variational parameters. Standard locations and exponents are given in the fiterature [9j. These orbit& provide a simple resolution of the genera&red overlap integrals [ 10 J S’$G)
=lx,(r
- ta) exp(i(;
l
4 x,(r - t$ d3r
= [2o&Qr33i4 [I&r9 frr]3j4 expf-+&-~~-~p~21 by using the inversion procedure: R(k) = s-1(k)
X expfiG - r) exp [-+r
.
The LCAO coefficients for this one~determ~nant PoIymeric wave function and, as a consequence, the dectr~ni~densityare thus completely defined end no ite&ion calculation is necessary. The periodic properties of the electronic densities of moiecular crystals or polymers make them readily suitable to a Fourier development such as: p(r) - 9
FyG) exp(-iG - r) .
(81
The Fourier transforms F(G) are easily obtained by a straightforward integration. In the following, V means the tatal volume of the crystal (V = NV):
- 3 - r,)‘j
d3r .
02) If we develop G = h* + kb* + 2c* in terms of reciprocal tat tice parameters and r = xu +yb + ze in terns of direct lattice parameters, the generahzed overlap integral becomes: S;,$ (G) = [2+[4
314[Zo~,/rrJ~/~ [rrfccu,+ ar,)] 3/2
X expC--toLpo$ofp + ff$I AB2J X exp {-- frr2/(oP f OL 4 )I ($ + k2 i Z*)> X exp D(hC, + kc’ + N$)] Table t
,
FSGO orbitalexponentsand center coordinates
The generalized overlap integrelsSFi(G) atomic orbit& are defined as: ZZ$$G) =s+(r
between
- ro) exp(iC - r> &(r - fj) d3r _ (10)
The generalized overlap integrals which were fhst introduced by Stewart [gf are natbing but the Fourier transform ofan overlap charge distribution.. it is important to_uote that, unlike the over@ integrals $e, the SFq(G)‘s depend on the absolute positions of centers p and q ;ts weU as on their relative positions. It is clear that the coefficients E(G) are the StNCture factors of the polymers related to the experi-
(131
J_M. An&& J.L. Brt?dasjFSGO X-ray diffractionstructurefactors
where AB is the distance between the two orbital centers A and B and Ci=(~~Ai+orqBi)/(orp+cuq).
3. Calculation polyethylene
i=X,y,Z
of X-ray diffraction
-
(14)
factors for
tn this section, the formerly described model is applied to the determination of X-ray diffraction factors for polyethylene using the geometrical structure obtained by neutron diffraction [l l] _ Table 2 Comparison sine T3.9058 0.0975 0.8870
0.8866 0.8696 0.8601 0.8571 0.8495 0.8435 0.8422 0.8290 0.8250 0.8210 o.a1a3 Cl.8116
0.805s 0.8032 O.-l943 O.,SQQ 0.7352 0.7347 0.7120 0.6992 0.6952 0.6848 0.6814 0.676, 0.6749 Cl.6690 0.6569 O&S13 0.645, 0.6324 0.6243 0.6240 Cl.6207 0.616, 0.5920 0.5913 Cl.5610 0.5436 0.5382 0.5239 0.5211 OS*67 0.5125 O.S1co 0.5016 0.4961 0.4843 0.4808 0.4603 0.4795 i---m
between
theoretical
a
1c
L
4 4 4 4 4 4 4 1 4 4 4 4 4
4 0 4 3 3 2 3 3 1 2 2 1 2
: 4 4 4 3
1 0 1
:
3 : 3 3 3 : 3 3 3 3
z 3 3 3 2 2 2 2
2 2 1 0 2 t 2 t 2 0 1
Two polymer chains run through the Pm, orthorhombic unit cell which contains four carbon and eight hydrogen atoms. The atomic coordinates and the FSGO orbital exponents and center coordinates are listed in table 1. The density matrix elements come from a FSGO LCAO CO program for saturated polymers [ 12]_ Since polyethylene is an hydrocarbon, they are calculated in first approximation for an isolated chain and by taking into account the interactions between one CzHd unit group and two C,H, groups at each side. As a result, the R,,(k) elements with orbitals p and q belonging to two different chains are regarded as zero.
and calculated scattering factors a
2
0
4 4 3 3 2 3 3 3 1 2 0 2 I
2 1 Q 0 1 1 4 3 3 2 3 3 4 4 1 2 0 4 : 4
369
0.036 0.051
0.035 0.080 0.030 cl.l%O o.oat 0.020 LO,, O.O,9 0.199 0.150 o.co3 0.098 0.002 0.109 o-125 cl.01.5 0.116 0.114 0.148 0.120 0.360 0.179 0.095 a.3o3 0.107 0.225 0.355 0.065 0.331 0.256 0.046 0.148 0.036 0.084 0.099 0.0'11 0.398 0.252 0.245 0.109 0.665 0.368 0.032 0.059 0.278 0.988 0.044 0.260 0.1cn 0.779 0.228 U-M-
3.4763 D.4681 D.4626 0.4532 0.4484 0.4402 0.4358 0.4357 0.4348 0.430, 0.4298 0.4170 0.4111 0.4OQ6 0.3905 o-x99 0.3831 0.3739 0.3704 0.3636 0.3584 0.3553 0.3328 0.3321 0.3206 0.3tm 0.308, 0.3079 0.2939 0.2759 0.26LS 0.2540 0.2460 0.2450 0.2338 0.2169 0.2168 (1.1831 0.1476 0.1313 ___---
K
L
-2.856 -1.66: -4.099 O.99L -1.84, -2.053 2.439 -4.980 5.255 -5.553 1.841 -3.614 -7.268 4.203 1.444 -,_3,7 6.023 1.452 2.,SO 1.251 -6.297 4.168 -3.8,X 4.29, -S.,,3 3.377 4.518 6.191 5:01s 4.556 32102 -8.451 4.956 -4.988 7.383 3.218 9.463 4.426 15.651 la.053 .__-_-____
-3.379 -1.659 -5.011 a.152 -1.776 -2.Q98 2.654 -6.oa2 s-668 -6.594 1.ca4 -4-169 -9&I, 4.665 1.679 -1.614 ,.,49 1.634 2.943 1.410 -1.528 4,694 -4.130 4.234 -6.239 3.752 5.062 6.243 4.932 4.5'16 3.858 -8.178 4.892 -5.019 6.999 3.327 Q-&,2 4.174 14.,15 16.707
a.923 o.m2 0.912 0.839 0.071 0.045 0.115 1.102 0.613 1.041 0.83, 0.575 1.731 0.662 0.235 0.23, 1.126 0.382 o.*93 c.159 1.231 0.526 o.3t9 0.063 0.466 0.375 0.544 Cl.052 0.064 o.c)zcl 0.156 0.273 0.064 O.C31 0.384 0.1OQ 0.c.z.Q 0.252 0.916 1.346
370
JM. AndrP. J.L. BrS&SGO
X-ray diffraction
structure
factors
Table 2 gives the theoretical scattering factors and compare them with those classically calculated in radiocjstallography as follows:
ionization potentials and photoelectron spectra [15]. Thus, the FSGO procedure can be considered as a very efficient and practical method for the calculation of electronic properties of polymers_
where the fK’s are the atomic form factors as listed in the International tables of X-ray Crystallography [13]. Debye-Wailer (temperature) factors are omitted for both the theoretical and calculated scattering factors.
References
4. Discussion The general agreement between tie theoretical scattering factors and the calculated one appears to be excellent. However, it must be noted that the agreement is better for large values of (sin 0)/A than for small ones. If we look at the amplitudes of the waves scattered by core and valence electrons, we see that, following Stewart’s calculations [ 141, for (sin 0)/A smaller than about 0.45, the valence electrons play an important role in the scattering phenomenon. The quantum approach, which takes into account the chemical binding and is basically different from the traditional crystallographic viewpoint which regards isoIated and spherical atoms, is then more adequate to describe the right electronic distribution and especially the bondiug density. - For (sin 0)/h larger than 0.45, only the core electrons scatter. As a result, the theoretical and calculated scattering factors are then in perfect agreement_
We must add that a Fourier difference analysis shows that the theoretical electronic density is Iarger than the calculated one, in the middle of the C-C bonds and between C and H atoms. As general conclusion, the FSGO model does properly calculate electronic densities and related X-ray structure factors. Other studies have proved that this procedure is equally valuable for computing
[I] J.M. Andre, J. Delhalle, C. Demanet and M.E. GerardLambert, Internat. J_ Quantum Chem, Symposium no. 10 (1976) 99. 121 J. Delhalle. J.M. And&, S. Delhalle, C. Pivont-Malherbe, F. Clarisse, G. Leroy and D. Peeters. Theoret. Chim. Acta, submitted for publication. [3] J. Ladik, in: Electronic Structure of Polymers and Molecular Crystals, eds. J.M. Andre and J. Ladik (Plenum Press, New York, 1975) p-23. [4] N. RGsch and J. Ladik, Chem. Phys. 13 (1976) 285. (51 A.A. Frost, J. Chem. Phys. 47 (1967) 3707. [6J R.E. Christoffersen, J. Amer. Chem. Sot. 93 (1971) 4104. [7i R.E. Christoffersen, Advan. Quantum Chem. 6 (1972) 333. [S] R.F. Stewart, J. Chem. Phys. Sl(1969) 4569. [9] A.A. Frost and R.A. Rouse, J. Amer. Chem. Sot. 90 (1968) 1965. [lo] J.M. Andre, I. Delhalle, J-G. Fripiat and G. Leroy, Intern. J. Quantum Chem. 5 (1971) 67. [ 111 G. Avitabile, R. Napohtano, B. Pirozzi, K.D. Rouse, M.W. Thomas and B.T.M. Willis, Polymer Letters 13 (1975) 351. [12] C..Demanet and J.M. And&, An Automatic Program for FSGO Calculation on Polymers. This program runs on a Digital PDP 1 l/45 system within 32K-words (16 bits) of memory. Typical running time is 10 minutes for the study of polyethylene calculating all interactions between eleven unit cells_ 1131 International Tables of X-ray Crystallography. VoL4
(Kynoch Press, Birmingham, 1974) p. 72; hydrogen
form factors computed by D.T. Cramer and J-B. Mann [Acta Cryst. A24 (1968) 3211, carbon form factors computed by P.A. Doyle and P.S. Turner [Acta Cryst. A24 (1968) 390]_ [14] R.F. Stewart, J. Chem. Phys. 48 (1968) 4882. [15] J.M. And& M.E. G&ud-Lambert and C. Lamotte, Bulletin de la So&t& Chimique de Belgique, submitted for publication.