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A comment on some approximations for the resonance integrals used in semi-empirical calculations
CHEMICAL PHYSICS LETTERS 1 (1968) 667 - 66s. NORTH -HOLLAND PUBLISHING COMPAhY
, ABISTEXDAM
A COMMENT ON SOME APPROXIMATIONS FOR THE RESONANCE INTEGRALS USED IN SEMI-EMPIRICAL CALCULATlONS R. J. WRATTEN Kemisk
Lnboraton’wn
IV.
Universitetspar-km
Tke Univcusit_)r of Copenhagen.
5, Copenhagen
Received
11 March
0,
Dcmnauk
1968
The formula proposed by Linderberg for the evaluation of the resonance integral used in semi-empirical calculations is investigated for applicability to u-systems. It is shown to reproduce the empirical parameters used in CNDO (Complete Neglect Differential Overlap) methods and leads to a link between other formulations for this term.
The most controversial parameter in semiempirical self-consistent field theories is the off-diagonal one electron integral HCLv. The matrix element of the one-electron Hamiltonian including the kinetic energy and the potential energy in the electrostatic field of the core (written as a sum of potentials VA(r) for the various atoms A in the molecule) is Hpv
=
(G,&
f V2-c~V”I*~)
(in a.u.)
Pople suffers from the defect that Scrv has been put equal to zero in the simplification of the S. C. F. equations. Recently, however, a further relationship has been proposed by Linderberg [4] which, in the zero differential overlap approximation, avoids this contradiction. It can be shown that, for a separation Z?(a.u. j between the centres A, B.
(1)
+u-
Greek suffixes span the valence atomic orbitals, *‘Iz. Two methods are currently used to evaluate. this term in the case when + and +v are on dif: ferent atoms A, B. Firstly, %e have the Pople method [l] Iipv = &:gS~v
eV
and quite obviously this is non-vanishing when S.pcrv= 0. It is the purpose of this letter to investigate the correspondance between this approximation and the other less recent approximations_
(2)
A comparison
where &R is a parameter dependent solely on the type of atom and not on the atomic orbitals, sz,
= 0.5(@
and Scrv is the overlap
+pg)
eV
Atom (3)
integral:
S BV = (Q[@v). Alternatively we have the Wolfsberg-Helmholz approximation [2] Hpu = k(HpCr +H,,)SCcv or the Ballhausen-Gray
eV
(4)
method [3]
H p,, = kr(Hpy~H,J~Spv
eV
(5)
Table 1
of empirical mates
667
and semi-empirical
of fi:
(eV)3j
Exponent
H
Bond -LiH
1 .oo
3.00
Li
LiH
0.64
3.00
5.8
Be
BeH
0.96
2.53
10.3
B
BH
1.29
2.32
15.1
C
CH
1.61
2.12
20.7
h’
NH
1.92
1.97
0
OH
2.24
F
FH
2.56
1.83 1.74
26.5 33.3 40.0
R (a.~.)
--
a) Calculated from eq. (10). b) Obtained from ref. [7].
dimensionwhere 1 C k, k’ == 2 is an adjustable, less constant. Of these formulae, the one due to April 1968
dQv -- 27.21 R XeV
,!32
9.0
esti-
R.J.WFUTTEN
668
Linderberg has already shown [4] that for relectron hydrocarbon systems this relationship bereproduces the acceptable valQes for R tween nearest neighbours. We are at pik&ent investigating the generality of eq. (6) to heteroatoms and the results will appear late: [Sj. However, a very interestfr_7 and simple result appears when we consider a-systems under the Fischer-Kjalmara exponential form [S] for the overlap integral. We have SIlv = S,e -P
where,
for orbital
exponents
(8)
cys,
P = 9.5(CYA+Q)R. Inserting
Professors J. Linderberg are thanked for their helpful thor acknowledges financial of a Royal Society European
this in eq. (6),
Comparison
duces the currently acceptable parameters used in CNDO calculations [7] as is shown in the table. (The diatomic bond length data were taken from Interatomic Distances [8], the exponents were those of Clementi and Raimondi [9].) Finally, a comparison with the other approximations for FIpv indicates that the one due to WolfsbergHelmholz is the more acceptable, but it can be seen that all the approximations for Hcrv are intrinsically related. We are now in the process of investigating the applicability of eq. (6) to a-systems [5] without any approximation for the overlap integral, thereby avoiding the orthogonality contradiction. and C. J. Ballhausen comments. The ausupport in the form Fellowship.
with (3) gives &
= - 27.212
eV.
(10)
The curious fact that an atomic parameter & should be dependent on a valence orbital exponent can be explained because aA = (zA - oA)/YZ.. where _“A is the core charge, oA the screening constant, and ‘2~ the principal quantum number associated with orbital acc on A. Hence
(11) The form of the last three equations as functions of R agrees with the tentativ.2 sllggestion of Pople [I]. When put to the test, eq. (9) readily repro-
[I]J.A.Pople,
D.P.SantryandG.A.Segal,
Phys. 43 (1965) S129. [2] M. Wolfsberg and L.Helmholz. (1952) 837. [3] C. J.Ballhausen
J.Chem.
J. Chem. Phys.
20
and H.B.Gray. Inorg. Chem. 1 (1962) III. [d] J. Linderbern. Chem. Phvs. Letters 1 (1967) 39. [5j R. J. Wratten, in preparation. 161 0. Fischer-Hialmars. Arkiv Fvsik 21 (1962) 223. [7J J.A.Pople aid G.A.&gal, J.chem.Phys. 43 (1965) S136; 44 (1966) 3289. [8] Special Publication II, The Chemical Society (London). [9] E. Clementi and D. L.Raimondi. J. Chem. Phys. 38 (1963) 2686.
, ABISTEXDAM
A COMMENT ON SOME APPROXIMATIONS FOR THE RESONANCE INTEGRALS USED IN SEMI-EMPIRICAL CALCULATlONS R. J. WRATTEN Kemisk
Lnboraton’wn
IV.
Universitetspar-km
Tke Univcusit_)r of Copenhagen.
5, Copenhagen
Received
11 March
0,
Dcmnauk
1968
The formula proposed by Linderberg for the evaluation of the resonance integral used in semi-empirical calculations is investigated for applicability to u-systems. It is shown to reproduce the empirical parameters used in CNDO (Complete Neglect Differential Overlap) methods and leads to a link between other formulations for this term.
The most controversial parameter in semiempirical self-consistent field theories is the off-diagonal one electron integral HCLv. The matrix element of the one-electron Hamiltonian including the kinetic energy and the potential energy in the electrostatic field of the core (written as a sum of potentials VA(r) for the various atoms A in the molecule) is Hpv
=
(G,&
f V2-c~V”I*~)
(in a.u.)
Pople suffers from the defect that Scrv has been put equal to zero in the simplification of the S. C. F. equations. Recently, however, a further relationship has been proposed by Linderberg [4] which, in the zero differential overlap approximation, avoids this contradiction. It can be shown that, for a separation Z?(a.u. j between the centres A, B.
(1)
+u-
Greek suffixes span the valence atomic orbitals, *‘Iz. Two methods are currently used to evaluate. this term in the case when + and +v are on dif: ferent atoms A, B. Firstly, %e have the Pople method [l] Iipv = &:gS~v
eV
and quite obviously this is non-vanishing when S.pcrv= 0. It is the purpose of this letter to investigate the correspondance between this approximation and the other less recent approximations_
(2)
A comparison
where &R is a parameter dependent solely on the type of atom and not on the atomic orbitals, sz,
= 0.5(@
and Scrv is the overlap
+pg)
eV
Atom (3)
integral:
S BV = (Q[@v). Alternatively we have the Wolfsberg-Helmholz approximation [2] Hpu = k(HpCr +H,,)SCcv or the Ballhausen-Gray
eV
(4)
method [3]
H p,, = kr(Hpy~H,J~Spv
eV
(5)
Table 1
of empirical mates
667
and semi-empirical
of fi:
(eV)3j
Exponent
H
Bond -LiH
1 .oo
3.00
Li
LiH
0.64
3.00
5.8
Be
BeH
0.96
2.53
10.3
B
BH
1.29
2.32
15.1
C
CH
1.61
2.12
20.7
h’
NH
1.92
1.97
0
OH
2.24
F
FH
2.56
1.83 1.74
26.5 33.3 40.0
R (a.~.)
--
a) Calculated from eq. (10). b) Obtained from ref. [7].
dimensionwhere 1 C k, k’ == 2 is an adjustable, less constant. Of these formulae, the one due to April 1968
dQv -- 27.21 R XeV
,!32
9.0
esti-
R.J.WFUTTEN
668
Linderberg has already shown [4] that for relectron hydrocarbon systems this relationship bereproduces the acceptable valQes for R tween nearest neighbours. We are at pik&ent investigating the generality of eq. (6) to heteroatoms and the results will appear late: [Sj. However, a very interestfr_7 and simple result appears when we consider a-systems under the Fischer-Kjalmara exponential form [S] for the overlap integral. We have SIlv = S,e -P
where,
for orbital
exponents
(8)
cys,
P = 9.5(CYA+Q)R. Inserting
Professors J. Linderberg are thanked for their helpful thor acknowledges financial of a Royal Society European
this in eq. (6),
Comparison
duces the currently acceptable parameters used in CNDO calculations [7] as is shown in the table. (The diatomic bond length data were taken from Interatomic Distances [8], the exponents were those of Clementi and Raimondi [9].) Finally, a comparison with the other approximations for FIpv indicates that the one due to WolfsbergHelmholz is the more acceptable, but it can be seen that all the approximations for Hcrv are intrinsically related. We are now in the process of investigating the applicability of eq. (6) to a-systems [5] without any approximation for the overlap integral, thereby avoiding the orthogonality contradiction. and C. J. Ballhausen comments. The ausupport in the form Fellowship.
with (3) gives &
= - 27.212
eV.
(10)
The curious fact that an atomic parameter & should be dependent on a valence orbital exponent can be explained because aA = (zA - oA)/YZ.. where _“A is the core charge, oA the screening constant, and ‘2~ the principal quantum number associated with orbital acc on A. Hence
(11) The form of the last three equations as functions of R agrees with the tentativ.2 sllggestion of Pople [I]. When put to the test, eq. (9) readily repro-
[I]J.A.Pople,
D.P.SantryandG.A.Segal,
Phys. 43 (1965) S129. [2] M. Wolfsberg and L.Helmholz. (1952) 837. [3] C. J.Ballhausen
J.Chem.
J. Chem. Phys.
20
and H.B.Gray. Inorg. Chem. 1 (1962) III. [d] J. Linderbern. Chem. Phvs. Letters 1 (1967) 39. [5j R. J. Wratten, in preparation. 161 0. Fischer-Hialmars. Arkiv Fvsik 21 (1962) 223. [7J J.A.Pople aid G.A.&gal, J.chem.Phys. 43 (1965) S136; 44 (1966) 3289. [8] Special Publication II, The Chemical Society (London). [9] E. Clementi and D. L.Raimondi. J. Chem. Phys. 38 (1963) 2686.