Estructura cognitiva
3.4 La SEP frente al acto lector: Programa Nacional de Lectura y Escritura y Programas de estudio
This leads us to b eliev e that, with su itable adjustment o f the
screening parameter, resu lts comparable to those calculated with SCF wave functions can be obtained using a sin gle S later function with a
considerable saving in labour«
I t is the purpose o f this chapter to in vestigate the e ffe c t s o f molecular vibrations on the calculated values o f the m obility ratios w h ile, at the same time, attempting to assess the use o f sin gle S la ter,
functions in energy band structure calcu lation s« 3.2 Construction o f symmetry adapted wave functions
Naphthalene cry s ta lliz e s in the m onoclinic system with space group C|h and has two molecules per unit c e l l . The fa cto r group o f the space group contains the follow in g operations t
( i ) inversion at any s it e
( i i ) r e fle c tio n in the ac plane follow ed by an _a/2 glid e in the ac plane
( i i i ) a two fo ld rota tion about the b axis follow ed by a J>/2 g lid e along this a x is.
The fa cto r group, including the id e n tity operation, i s , th erefore, isomorphous with the point group C2k • Group theory demands that
the c e l l wave functions belonging to the k ■ 0 representation must transform lik e the irred u cib le representations o f the fa cto r group and, since a ll the irred u cib le representations o f are one dimensional, symmetry adapted wave functions fo r Jk - 0 can be constructed by u t iliz in g the p ro je ctio n operator
pl “ “ í x V ) R C3.1)
R
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o f R fo r the i- t h irred u cib le representation, and n is the order
o f the group «
The transformations o f the one s it e wave functions under the group operations are !
a ac
V O ) ■ v ° )i
V O ) - “ V o ) •l
V ° ) - “ V ° )c b
L2 V O ) - - * 2CO)
The symmetry adapted functions can be obtained by means o f the p ro je ctio n operator equation (3 ,1 ) in the form
»
$ .(0 ) - P1'i'2 (0)
where i represents any o f the irred u cib le representations o f the C2h point group. The representations Ag and Bg give only vanishing resu lts owing to the odd p a rity o f the molecular wave fu n ction s, and fo r the representations Au and Bu
* <k) . i - <¥ (0) - V O ) )
Jl 1 2
and
_1
M k) “ ¿2 ( V 0) + V 0 )) •
When k * 0 the unit c e l l wave functions are given by _1
V k) " ¿2 ( V k) 4 V k ))
but the symmetries o f $+ (k) or 4>_(k) depend on the group o f the wave vector _k , I f the vector connecting the centres o f molecules 1
- 43 - symmetry o f the crystal
M'1(k) - ^ ( 0 )
and ^ ( k ) - e1- - ^ « ) ) »
The general symmetry adapted wave functions are therefore
$+(k) - I (¥ (0) ± e 1- - ¥,<0)) (3»2)
V2
Thus, when the molecules come together to form the s o lid , each molecular energy le v e l w i l l s p l i t in to two components due to the symmetric and antisymmetric combinations o f the one s it e wave functions in the c e l l giving ris e to two energy states fo r the excess electron and two fo r the holeo
The cry sta l wave function is constructed, in the Bloch
representation, as a lin ear combination o f unit c e l l wave functions» I f the v e cto r loca tin g the origin o f the i - t h unit c e l l is r. then
—l
B±(k) -
l
e.xp(ikojr.) 4>i ± (k) (3»3)where-the summation i runs over a ll c e lls in the crysta l»
Substitution fo r $+(k) and replacing the summation over a l l unit c e lls by a summation over a l l molecules in the cry sta l equation (3 .3 ) becomes
n± (k) (3o4)
where L - 0 i f contains n _b
(n + o
The sin gle s it e functions ^
and L - 1 i f r. contains “ J
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product o f molecular wave functions in which one molecule is represented as e ith er a p o s itiv e or negative io n , The e f f e c t o f molecular vibrations are included by taking the molecular wave functions as being the product o f an e le c tr o n ic part and a v ib ra tio n a l part» This representation corresponds to the weak- coupling lim it o f vib ron ic interaction,, Sym bolically, the wave function corresponding to the electron or hole on molecule is
- A ♦ * (2a) H <M2a)x . (3 ,5 )
jjiJl J 3
where ^ is the lowest unoccupied molecular o r b ita l in the case o f an excess e le ctro n , highest occupied fo r an excess h o le , x* is the
J
ground sta te vib ra tion a l wave function o f the j - t h molecule, A is the antisymmetrizing operator permuting electron s between the molecule and is o f the form (74)
A - {(2 a )2N/(4N a)S )i £ f l ) PP (3 ,6 )
P
where N is the number o f unit c e lls in the c r y s ta l, 3 ,3 Method o f ca lcu la tion
As has been shown in chapter (2) (equation (2 ,5 4 ), page (29)) the energy dependence upon the wave v ector when the e ffe c t s o f interm olecular overlap have been neglected may be w ritten as
E !(k) - l (±1)L c o s (k ,r ) E l where is given by N Et ' { l c, csn <uJ - V R„ l v Ctf p x core s tates
*s~ <u |<u |r,„ Hu >|u0> } 2 o' o ' 12 1 a 1 S
(3 ,7 )
+
45
the synfcols have been defined in chapter (2 ), page (2 9 ), and L ■ 1 i f contains (n + or L - 0 i f j: contains n l> ,
The amount o f labour involved in the numerical calcu lation o f the e le c tr o n ic part o f the transfer in te g ra ls, , can be considerably reduced i f the core electron s are considered as point charges at the nucleus on which they are centred. I f the number of electron s contributed to the pi-system by the centre a is n^ then equation (3 ,8 ) becomes i
Et ■ o, 3l ‘ “ a l - V S J V
+ Pa <u I <u I T~\ I u > I u0 >
— a 1 a 1 12' a 1 S (3 ,9 )
In forthcoming section s this approximation is referred to as the p i-e le c tr o n approximation.
The matrix elements in volving the operator rj^ in equation (38) and equation (3 ,9 ) give ris e to the s o -c a lle d hybrid in te g r a l.
Evaluation o f such in tegrals (discussed in appendix ( l ) t
page (2^2.)) is very involved and considerable s im p lifica tio n can be obtained i f i t is assumed that the charge d istrib u tio n o f the second e le ctro n , lu0 l 2 » can be considered as concentrated at the nucleus a , The problem then reduces to the calcu la tion o f two-centre, one electron in tegra ls which, by comparison, are e a s ily evaluated. In this
approximation equation (3 ,9 ) reduces to
Ea - |<X1lx 0>l 2 e2
l
cac0<ua(- “ r t ) £ 2 “ nJ u8 (r )> (3 o l0 )The in tegrals between the molecule at the o rig in and the
molecules at the com ers and side centres o f the unit c e l l have been calculated using equations ( 3 ,8 ) , (3 ,9 ) and (3 ,1 0 ). This is equivalent to the ca lcu la tion o f the in tegrals between the molecule at p o sitio n
«
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numbered (1 ), figure (3 «1 ), and the molecules in p osition (2) through (1 0 )o Neglecting in teraction s with other molecules the energy dependence on k is
E*(k) ■ 2Ej cos(ko£) + 2Eg cos(k«b) + 2E^(cos Oc0(b+ c)) + cos (k « (hr-c)) ) + 2E^ c o s (k 0<a) + 2Eg c o s (k 0(c+a)) + 2E^(cos (k 0(_a+b)) + c o s (k « (a -b ))) + 2Eg( cos (k o (a+t>+c)) + cos (k « (a-brfc) ) ) ± 2Eg(cos (ko i(a +b,)) + c o s (k 01Ca—b ) ) )
± 2E ^ ( cos (k «( J(a+b) + c)) + c o s (k 0(J(a-b)+£>)) « (3 b ll)
The energy bands can be more rea d ily v isu a lized i f the sp e cia l cases when the wave v e cto r, lc , is p a ra lle l to a re cip roca l la t t ic e vector
a“ 1, b-2 or £ ~ 3 are considered« The relation sh ips between energy and wave vector is then
E+(k| I T 1) “ 2 (e2 + E3 + 2E4) + 2(E5 + Eg + 2E? + 2Eg) cos (koj|) ± 4 (Eg + E10) c o s (k « j/2 )
E±(k| l b " 1) - 2(E2 + E5 + E6) + 2(E3 + 2E4 + 2E? + 2Eg) cos(k«lj ± 4(Eg + E^0)cos(Jc«b/2)
E±(k | Ic“ 1) - 2(E3 + E5 + Ey ± 2E9)
+ 2(E2 + 2E4 + Eg + 2Eg ± 2E10)cos(k ,jc) « (3 «12)
3«4 Numerical Calculations
The f i r s t step in numerical ca lcu la tio n o f the transfer
in tegrals is choice o f a su itab le wave function fo r the p o sitiv e or negative io n 0 Following the example o f Le Blanc and Katz the excess e lectron or hole is assigned to the lowest unoccupied, highest
48
occupied, molecular o r b ita l o f the neutral m olecule» The molecular o r b ita ls o f the neutral molecule are approximated by a lin ea r
confeination o f neutral carbon 2p wave functions u. . The
z r
a n a ly tica l form o f the e lectron and hole wave functions are therefore
*<£ ~ " l 4 ui (3.13)
i
where c? are the Hueckel c o e ffic ie n t s , calcu lated without the in clu sion o f overlap, fo r the lowest unoccupied and highest occupied molecular o r b ita l respectively,. The neutral carbon 2p wave fu n ction s, u. , are taken to be sin gle Slater 2p z functions characterized by the screening parameter
ç »
u.
l ex p C -^ r) (3.14)
where n. is the unit vector defining the d ire ctio n o f the 2p o r b it a l. The two centre in tegrals can be sim p lifie d by expanding in the form
- ( n . . R . . ) ( n . . R . . )
<u. If lu s> ■ J ■■■ rv ""1i1 <p If Ip >
l 1 op1 j r#>2
r0 l
«"I «op'9 (]li i ) (n# oR«
“ —i ° nj ---? a 3 - 1 ,1 R .. > <PJ % I Ptt> (3.15)
where , n^ are unit vectors defin in g the d ire ctio n o f the o rb ita ls u. and u. , R .. is the vector connecting atoms i and i , F
i j J op
represents e ith er the nuclear a ttra ction or electron repulsion operator, p^ and p^ are S later 2pff and 2pQ atomic o rb ita ls r e sp e ctiv e ly .
The one-electron in tegrals were evaluated in closed form by expanding the in tegra ls in p rola te spheroidal coordinates. The
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An(a ), ®n (3 ), (34) which are e a s ily calculated» The two electron in tegrals are calculated using the Zeta function expansion method o f Coulson and Barnett, which has been discussed in appendix ( I ) ,
page (* » )« For a given basis se t o f in tern u c le a r distance
R - 4»5 (0.125)16»0 au. the values o f the in teg ra l fo r a p a rticu la r in tem u clea r distan ce. are obtained by in terp ola tion using A itken's method. I t is found that by using a large basis s e t in R very high accuracies are obtained fo r the in terpolated in tegrals and
that the order o f polynomial used in the in terp ola tion has l i t t l e
e f f e c t on the accuracy o f the re s u lt. Since the one electron in tegrals are r e la tiv e ly simple to evaluate these are determined fo r each
in dividual R^j . I t should be noted that, fo r the tw o-electron in te g ra ls , once a set o f in tegrals have been calculated fo r basis s e t R and screening parameter £ further sets o f in tegrals o f
basis R' and screening parameter £ can be obtained using the rela tion
I ( c t . ) I ' ( a . ) ' '■ "'■J— m ... . ■ £ ~ a. 3 £R. 3 - £'R!3 (3.16)
where the elements o f the new basis se t R* is given by
Rj£ £ ’ where R’ . and R.
J J
and R resp ectiv ely
are the j - t h elements o f the basis sets R"
- 50 - N a p h th a le n e sh ow in g t h e n u m b erin g o f t h e atom s I n t h e m o l e c u l e . x y z
A
0 .7 1 6
0.092
2 .8 1 6
B
0 . 9 3 40 .9 6 0
1 .892
C
0 . 3 9 00.611
0 . 2 9 7D
0 . 6 1 41
.4 8 3-O .685
E
0 .0 8 3
1.121
-
2 . 1 9 5a
b
c
8.2 3 5
6.003
8 .6 58
Table(3 .1 )
A to m ic c o - o r d i n a t e s * and u n i t b e t a122
55
c e l l c o n s t a n t s * o f n a p h t h a le n e . * u n i t s : 1 » -1 nm- 51 -
3 »5 Numerical results and band structure
The crysta l data fo r naphthalene was taken from Abrahams et al (90) and xs lis t e d in table (3<>l)o The transfer and overlap in teg ra ls,
calculated between the molecules at p o sitio n 1, the com er o f the
unit c e l l , and the remaining molecules w ithin the unit c e l l , fo r various values o f the screening parameter, C , are given in tables (3<>2) and
(3»3) respectivelyo As was expected a decrease in screening parameter, £ , resu lts in an increase in magnitude o f the transfer in tegral due to the slower rate at which the Slater functions f a l l o f f with
interm olecular distance» In table (3»2) the vibration overlap in te g ra l is taken as unity» The p lots o f the excess electron and hole band stru ctu re, fo r screening parameter 24»57 nm“ 1, along the re cip ro ca l cry sta l axes are given in figures (3»3) and (3»4)» The shapes o f the energy bands fo r any other cases are not shown as va ria tion o f the screening parameter, in general, alters only the band widths, the shape o f the band remaining unaltered» I f needed these can be calculated using the resu lts o f table (3»2) and
equation (3 »1 2 )0 The electron repulsion, nuclear a ttra ction and tra n sfer in tegrals calculated in the p i-e le c tr o n and lo c a lis e d core approximations (equation (3»9) and equation (3»10) re sp e ctiv e ly ) are given in table (3o4)o I t can be seen that these approximations are quite good when the centres o f the in tera ctin g molecules are separated by large distan ces, unfortunately, these in teraction s contribute l i t t l e to the band structure» There is an o v e ra ll e rro r o f about 25% fo r the p i-e le c tr o n approximation and over 30% fo r the lo c a lis e d core
approximation which although large are to be expected since in the p i-e le c tr o n approximation a large proportion o f the neutral p oten tia l is neglected w hile the lo c a lis e d core approximation amounts to a
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