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Semi-empirical molecular orbital theory. The one-centre quantities for the elements of the first and second transition series
Volume 11, number
CHEMICAL PHYSICS LE’ITFXS
3
15 October
197 1
SE~-E~I~~AL.~OLE~~~ ORBITA_L,THEORY. THE ONE~ENT~ Q~~~~S FOR THE ELEMENTS OF THE FIRST AND SECOND TRANSITION SERIES L. DI SIP10
E. TONDELLO Laboratoriodi Chimicae TecnologiodeiRadioelementide1 C.N.R., Padova,IMy
G. DE MICHELIS Is&to di ChimfcaCeneraIesUniversit2di Venezia.It&y and L. OLEARI Isrihr~odi Chimica Fiska, Univerrira di Purma, &a& Received 15 July 1971
One-centre
are given which OLI be used
quaritities for the elements of the frost and second traadtion meti -ties one-centre integrals in semi-empirical MO calculat~oons.
instead of the conesponding
The main feature of the semi-empirical MO theory is the approx~ation of the one-centre integrals by certain quantities obtained from the valence state energies of the atoms. The valence state energies for an atom can be expressed by the following equation [l] : where C is a constant, necessary because of the different choice of the zero for the energy. Vi*, g$ are qusni i,j#i i titieo determined from the best FItfing of the valence state energ&& where tli are atomic orbital occupation numbers, Vi are core integrals andgii are quantities related to the We propose to use the-quantities 08 andgt instead Coulomb and exchange integrals by the expression of the corresponding integrals &$and g++in semi-empkicai MO caiculations. In fact these quantities fulfrl g[j = Jq t JKil
E=x niUi +4 C
nirtjgij ++ ;r)‘ni(ni-l)gii,
.’ ,. ,. ‘. .. _. .. ,” .. ‘. ,. ,: ,. y” ‘.,
(I)
~,’ ‘.
:, . .
.:: .” .: ;_ ....‘.( . . . . .; . ‘.‘,.,~.‘,_, ‘. ,.. .,., ,...: ~. .. ., :. ; ,.‘,. .’ :_ ,,:y .‘:’:...,,‘,, _‘ ~, ,11 _: : ..
: _, :
:
;.: .: _ ‘-.
iii
..
.,
y.:
;. ,’
; t
.,
. -; -c
:
‘, .iYd us
v.
Ti
r :
= ‘.
.. .’
h3.94
Co
43cl.59
Ni
‘570.43.
’
Cu
7303
-50.36
-61.22
-72.60
-40.37 l4.03 12.90
-49.24 14.70 1352
-58.54 15.31 14;14
12.23
12.82
13.41
14.00.
14.59
15.18 :
,14.25
14.85 15.64 7.96
15.45 16.27 8.25
il.64
gz=,x g&z ‘, ~B~y*z: gs+p
-141.57
-163.18
-84.50
-96.92
-109.86
-68.27 16.04 14.76
-78.43. 16.71
-89.b2 17.38’
15.38
r&o0
11.25 11.86 6.22
‘., ‘11.85 12.49 6.21
12.45 13.12 6.80
13.05 13.75 7.09
13x% 14.38 7.38
8.29 I.09 6.90
8.57
8.85 7.58 7:36
9.13 7.82 .7.59
9.4 1 8.06 7.82
9.69 8.30 8.05
9.97
10.25
7.34
8.54 8.28
8.79 8.51
7.26 7.05 (!LSEU (5.65>
7.50 7.28 (S.80) (s;.SQ) (5.091
7.98 . 7.74 (6.24) (6.103 (5.351
a.22 7.97 (6.46) C6.2S} (5.48)
8.46 8.20 (6.68) (6.40) (5.61)
8.70 8.42 (6.90) (6.55)
(4.96)
7.74 7.51 (6.02) (5.95). (5.22)
7.13
7.02 6.83 (5.361. (5.50) (4.83)
.’
-‘I--‘i~i~l5.
929.51
-31.93 13.36’ 12.28.
‘, ‘-101.92
‘.
-40.02
822,z
‘Semiempiri~
ud w;
315.#2 -83.88
..
Fe’
-23.92.
:
.c
;.
-30.20
8%S. gs.d
Q,P gx,y
218.48.
MO
:-67.03
.11.05
gxy,xs-y?’
: ,,
-51.37
l2.69 11.66
e;,z.x w,xy ‘&x,zy
.
“, :141.20
‘. -3tLsjO’::
Lvp : Bd,d
.’ Cr
15.01 7.67
_~
(s-7+)
Table 2 quaxhtieu (ii eV) for the elementsof the second trarkiti~n metal series from 2k to Ag Ru
w
Nti’
MO
Tc
76.62 :32.26
126.07 -44.41
193.54 -57.66
276.7
378.29
-72.01
-87.46 -66.32 -54.28
-is.sa
-37.25
-46.43
-56.12
-2i.92
-30.10
-37.72
-45.78
Pd
Rh 501.56
Ag
648.30
-104.01 Y-77.03
818.38
-121.66
-140.41
gd,d
10.38
10.96
11.54
f2.13
12.71
13.87
-99.98 -82.42 14.46
w,zx &‘,ay .gu;&
9.55
10.09
10.63
11.17
11.71.
12.25
12.79
13.33
‘, 9.05 9.22
9.56 9.74
10.07 10.26
11.09 Ii.30
11.60 11.82
12.11 12.34
9.72 6.42
kO.27 6.59’
10.82 6.77
10.58 10.78, il.37 6.94
11.92 7.11
12.47 7.29
13.02 $46
12.62 12.86 13.57 7.64
7.96 6.96
8.23 ~
8.49
9.29
7.4%
8.76 7.43
9.02
7.18 6.97 7.11
7.19 :
7.85’ 7.63.
7.84‘.
6.90
:%
..,:
:Sx~-j~,xy~
::&j gi.d
,,gz+ ‘,&2,x. Bzx,z: Sxy,Z’~ .g9;p gp,p
:.
I, 9.43
7.70.
" 6-51
‘6.74
6.32.
6.54
6.45
6.67
6.75 6.85.‘:
6;25
6.47.
6.69
(5.40),.(5,20) :
: ‘-
.’ ~5.53) (5.37f
'(4.55); (4.70)
-I'gx;y',, )
7.33
_
7.12 f6.08)”
-69.22
‘.
.13.29
I,
7.41. 7.56 .,
.. ,;7.33
(6.25) : (5.9E) .” (5.88) :. :’ (6.05) f5.71j .“.(4.85),,. “, (5.00) ‘. : (5.15)’ ,’ : ‘(5.30j _. (5.74)
” (5;54)
-88.25 -72.60
7,78
8.08 ..
7.55, .. ‘(6.42).
8.00’ 7.76 (6.59)
‘.
..(6.39) ':' (6.22)’ -(5.+j .:, : (5.60)
. : _,. ),,.’ .. : . . : ., ,_ ” ,’ ‘; I,.. .;. ., ,. ,.;:__” .,_)_’ ..: Y&tj~“::‘~~,‘,I y ‘.’ .-.,’ ‘. l.-‘.’ _: ‘. ,. .;,:-, ., .,, ‘, ,:;. ,_ ~.,, .” : ,. ;, ,: ,,:.‘.. : . .. ... ,y.,.;;: . . :: ,’ . _’ ..: ,:.;_ ,_(’ .. ‘..‘, ‘._“ .. ‘.. . ....’: ... .‘( .,.’ ._ .’ -: .: .,I : ., ‘ ., :. .’ .:, _. ,: ‘. :’ ‘.., :+:’ : .,..., .;: I’ ( _’ :. ._ ;,: ‘ , ‘ . _. .’ ,’ : :, ,’ ,’ :. ._ ,I _.. ; .‘, .’ .‘. ,. ‘. ., ., .’,.‘;‘._f ‘,‘._,‘, i) ,_._ . :. ,; ‘. .,
Volume ,ll, number 3
CHEMICAL PHYSICS LETTERS
The valence state energies have been calculated by means of the Slater-Condon parameters evaluated in our previous works [2]. The configurations dnB2sp, and dn-.“p2 have been excluded because of the rather poor accuracy of the corresponding Slater-Condon parameters. The calculations have been performed by applying the least squares method and by imposing the following restrictive conditions: (i) the quantities UF vary parabolically with the atomic number; (ii) the quantities g$ vary linearly with the atomic number; (iii) the quantities gr7 behave for axes transformations like the corresponding integrals gjj of eq. (1). The fist two conditions have been introduced, as in the evaluation.of the Slater-Condon parameters [2], in order to avoid irregular trends of the calculated
quantities with respect to the atomic number. Inthis way we have evahrated all the quantities Ur and g$ which are reported in tables 1 and 2 exe-qt
g&vi& and&. The latter (given in parentheses) have been evaluated in a second step by separately fitting the valence state
1.5October 1971
energies of the configurations dn-*sp and dnm2p2 calculated by means of the parameters given in our previous papers [2] ; Also in this calculation a condition of linear variation with atomic number has been . imposed.. The quantities obtained from this work are sufficiently accurate to permit the evaluation of’all‘the valence state energies for a given metal, beIonging to the oxidation‘states 0, +l, and +2 with a standard deviation smaller than 0.5 eV. This work has beerrsupported by C.N.R. (Italian National Research Council) and by NATO Contract No. 403.
References [l] L.Olti, LDi Sipio and G.De hfichelis. hfoI. phys. 10 (1966) 97. [2] E.TocdeUo, G.De Michelis. &Wari and LDiSipio, Coord. Chem. Rev. 2 (1967) 65; L.Di Sipio, E.TondelIo, G.De hfichetis and L.OIeari, Inorg. Chem. 9 (1970) 927. [3] R.Pariser and G.PY~, J. Chem. E%ys.21(1953) 466, 767.
289
CHEMICAL PHYSICS LE’ITFXS
3
15 October
197 1
SE~-E~I~~AL.~OLE~~~ ORBITA_L,THEORY. THE ONE~ENT~ Q~~~~S FOR THE ELEMENTS OF THE FIRST AND SECOND TRANSITION SERIES L. DI SIP10
E. TONDELLO Laboratoriodi Chimicae TecnologiodeiRadioelementide1 C.N.R., Padova,IMy
G. DE MICHELIS Is&to di ChimfcaCeneraIesUniversit2di Venezia.It&y and L. OLEARI Isrihr~odi Chimica Fiska, Univerrira di Purma, &a& Received 15 July 1971
One-centre
are given which OLI be used
quaritities for the elements of the frost and second traadtion meti -ties one-centre integrals in semi-empirical MO calculat~oons.
instead of the conesponding
The main feature of the semi-empirical MO theory is the approx~ation of the one-centre integrals by certain quantities obtained from the valence state energies of the atoms. The valence state energies for an atom can be expressed by the following equation [l] : where C is a constant, necessary because of the different choice of the zero for the energy. Vi*, g$ are qusni i,j#i i titieo determined from the best FItfing of the valence state energ&& where tli are atomic orbital occupation numbers, Vi are core integrals andgii are quantities related to the We propose to use the-quantities 08 andgt instead Coulomb and exchange integrals by the expression of the corresponding integrals &$and g++in semi-empkicai MO caiculations. In fact these quantities fulfrl g[j = Jq t JKil
E=x niUi +4 C
nirtjgij ++ ;r)‘ni(ni-l)gii,
.’ ,. ,. ‘. .. _. .. ,” .. ‘. ,. ,: ,. y” ‘.,
(I)
~,’ ‘.
:, . .
.:: .” .: ;_ ....‘.( . . . . .; . ‘.‘,.,~.‘,_, ‘. ,.. .,., ,...: ~. .. ., :. ; ,.‘,. .’ :_ ,,:y .‘:’:...,,‘,, _‘ ~, ,11 _: : ..
: _, :
:
;.: .: _ ‘-.
iii
..
.,
y.:
;. ,’
; t
.,
. -; -c
:
‘, .iYd us
v.
Ti
r :
= ‘.
.. .’
h3.94
Co
43cl.59
Ni
‘570.43.
’
Cu
7303
-50.36
-61.22
-72.60
-40.37 l4.03 12.90
-49.24 14.70 1352
-58.54 15.31 14;14
12.23
12.82
13.41
14.00.
14.59
15.18 :
,14.25
14.85 15.64 7.96
15.45 16.27 8.25
il.64
gz=,x g&z ‘, ~B~y*z: gs+p
-141.57
-163.18
-84.50
-96.92
-109.86
-68.27 16.04 14.76
-78.43. 16.71
-89.b2 17.38’
15.38
r&o0
11.25 11.86 6.22
‘., ‘11.85 12.49 6.21
12.45 13.12 6.80
13.05 13.75 7.09
13x% 14.38 7.38
8.29 I.09 6.90
8.57
8.85 7.58 7:36
9.13 7.82 .7.59
9.4 1 8.06 7.82
9.69 8.30 8.05
9.97
10.25
7.34
8.54 8.28
8.79 8.51
7.26 7.05 (!LSEU (5.65>
7.50 7.28 (S.80) (s;.SQ) (5.091
7.98 . 7.74 (6.24) (6.103 (5.351
a.22 7.97 (6.46) C6.2S} (5.48)
8.46 8.20 (6.68) (6.40) (5.61)
8.70 8.42 (6.90) (6.55)
(4.96)
7.74 7.51 (6.02) (5.95). (5.22)
7.13
7.02 6.83 (5.361. (5.50) (4.83)
.’
-‘I--‘i~i~l5.
929.51
-31.93 13.36’ 12.28.
‘, ‘-101.92
‘.
-40.02
822,z
‘Semiempiri~
ud w;
315.#2 -83.88
..
Fe’
-23.92.
:
.c
;.
-30.20
8%S. gs.d
Q,P gx,y
218.48.
MO
:-67.03
.11.05
gxy,xs-y?’
: ,,
-51.37
l2.69 11.66
e;,z.x w,xy ‘&x,zy
.
“, :141.20
‘. -3tLsjO’::
Lvp : Bd,d
.’ Cr
15.01 7.67
_~
(s-7+)
Table 2 quaxhtieu (ii eV) for the elementsof the second trarkiti~n metal series from 2k to Ag Ru
w
Nti’
MO
Tc
76.62 :32.26
126.07 -44.41
193.54 -57.66
276.7
378.29
-72.01
-87.46 -66.32 -54.28
-is.sa
-37.25
-46.43
-56.12
-2i.92
-30.10
-37.72
-45.78
Pd
Rh 501.56
Ag
648.30
-104.01 Y-77.03
818.38
-121.66
-140.41
gd,d
10.38
10.96
11.54
f2.13
12.71
13.87
-99.98 -82.42 14.46
w,zx &‘,ay .gu;&
9.55
10.09
10.63
11.17
11.71.
12.25
12.79
13.33
‘, 9.05 9.22
9.56 9.74
10.07 10.26
11.09 Ii.30
11.60 11.82
12.11 12.34
9.72 6.42
kO.27 6.59’
10.82 6.77
10.58 10.78, il.37 6.94
11.92 7.11
12.47 7.29
13.02 $46
12.62 12.86 13.57 7.64
7.96 6.96
8.23 ~
8.49
9.29
7.4%
8.76 7.43
9.02
7.18 6.97 7.11
7.19 :
7.85’ 7.63.
7.84‘.
6.90
:%
..,:
:Sx~-j~,xy~
::&j gi.d
,,gz+ ‘,&2,x. Bzx,z: Sxy,Z’~ .g9;p gp,p
:.
I, 9.43
7.70.
" 6-51
‘6.74
6.32.
6.54
6.45
6.67
6.75 6.85.‘:
6;25
6.47.
6.69
(5.40),.(5,20) :
: ‘-
.’ ~5.53) (5.37f
'(4.55); (4.70)
-I'gx;y',, )
7.33
_
7.12 f6.08)”
-69.22
‘.
.13.29
I,
7.41. 7.56 .,
.. ,;7.33
(6.25) : (5.9E) .” (5.88) :. :’ (6.05) f5.71j .“.(4.85),,. “, (5.00) ‘. : (5.15)’ ,’ : ‘(5.30j _. (5.74)
” (5;54)
-88.25 -72.60
7,78
8.08 ..
7.55, .. ‘(6.42).
8.00’ 7.76 (6.59)
‘.
..(6.39) ':' (6.22)’ -(5.+j .:, : (5.60)
. : _,. ),,.’ .. : . . : ., ,_ ” ,’ ‘; I,.. .;. ., ,. ,.;:__” .,_)_’ ..: Y&tj~“::‘~~,‘,I y ‘.’ .-.,’ ‘. l.-‘.’ _: ‘. ,. .;,:-, ., .,, ‘, ,:;. ,_ ~.,, .” : ,. ;, ,: ,,:.‘.. : . .. ... ,y.,.;;: . . :: ,’ . _’ ..: ,:.;_ ,_(’ .. ‘..‘, ‘._“ .. ‘.. . ....’: ... .‘( .,.’ ._ .’ -: .: .,I : ., ‘ ., :. .’ .:, _. ,: ‘. :’ ‘.., :+:’ : .,..., .;: I’ ( _’ :. ._ ;,: ‘ , ‘ . _. .’ ,’ : :, ,’ ,’ :. ._ ,I _.. ; .‘, .’ .‘. ,. ‘. ., ., .’,.‘;‘._f ‘,‘._,‘, i) ,_._ . :. ,; ‘. .,
Volume ,ll, number 3
CHEMICAL PHYSICS LETTERS
The valence state energies have been calculated by means of the Slater-Condon parameters evaluated in our previous works [2]. The configurations dnB2sp, and dn-.“p2 have been excluded because of the rather poor accuracy of the corresponding Slater-Condon parameters. The calculations have been performed by applying the least squares method and by imposing the following restrictive conditions: (i) the quantities UF vary parabolically with the atomic number; (ii) the quantities g$ vary linearly with the atomic number; (iii) the quantities gr7 behave for axes transformations like the corresponding integrals gjj of eq. (1). The fist two conditions have been introduced, as in the evaluation.of the Slater-Condon parameters [2], in order to avoid irregular trends of the calculated
quantities with respect to the atomic number. Inthis way we have evahrated all the quantities Ur and g$ which are reported in tables 1 and 2 exe-qt
g&vi& and&. The latter (given in parentheses) have been evaluated in a second step by separately fitting the valence state
1.5October 1971
energies of the configurations dn-*sp and dnm2p2 calculated by means of the parameters given in our previous papers [2] ; Also in this calculation a condition of linear variation with atomic number has been . imposed.. The quantities obtained from this work are sufficiently accurate to permit the evaluation of’all‘the valence state energies for a given metal, beIonging to the oxidation‘states 0, +l, and +2 with a standard deviation smaller than 0.5 eV. This work has beerrsupported by C.N.R. (Italian National Research Council) and by NATO Contract No. 403.
References [l] L.Olti, LDi Sipio and G.De hfichelis. hfoI. phys. 10 (1966) 97. [2] E.TocdeUo, G.De Michelis. &Wari and LDiSipio, Coord. Chem. Rev. 2 (1967) 65; L.Di Sipio, E.TondelIo, G.De hfichetis and L.OIeari, Inorg. Chem. 9 (1970) 927. [3] R.Pariser and G.PY~, J. Chem. E%ys.21(1953) 466, 767.
289