- Email: [email protected]
Study of localizability for some small molecules
Volume
7 2, number 2
CHEMICAL
STUDY OF LOCALIZABILITY
PHYSICS
LETJYERS
1 June 1980
FOR SOME SMALL MOLECULES
Ede KAPUY, Corneha KOZMUTZA Quantum i71wry Group, Polytechnical
Umrersrty, Budapest, fiungary
and Raymond DAUDEL Centre de hlkanrque Recerred
2 October
Ondulatorre Appltquek du C-N R S . Parts, France 1979;
III fiial form 7 March 1980
locehzed charge denntres of some small molecules at theu experimental and calculated equiliirium geometries have been studred using the 6-31G and 6-31G/d basis sets. introducing polarization functions it has been found that the bond-pair cfuuge minima drstnbutions are “more”, while the lone-pau ones are “less” locahzed at their theoretically determmed totakrtergy than at the correspondmg expertmental geometries.
1. Introduction It IS well known that the independent-particle model provrdes systematically shorter bond lengths and larger bond angles at the calculated equdibrium geometnes than at the experimental ones. This is implied by the increase of the nuclear potential in going from the expenmental to the corresponding theoretical values. For HF, H20 and NH3, usmg the so-called 6-31G/d basis set [l] , the changes in the total energres and nuclear potentrals are shown in table 1. The experimental geometry data were taken from ref. [2], while the theoretical values are given in ref. [3] _ As the nuclear potential increases, the electton denarty also mcreases. Therefore, it 1s expected that at short bond lengths the sum of the localized interactionenergy contnbutions wrll be larger than at large bond Table 1 Changes III the total energy and nuclear potentrals (m hartree) for HF, Hz0 and NHs, using the 6-31G/d basis set Total energy
Nuclear potential
Hz0
-0.00007 -0.00026
NH3
-0.00023
+0.04419 +0.10774 +0.10897
HF
lengths. (Although the bond angles are slightly larger at the theoretically determined minima, the effect of short length dominates, as can be seen from the vahres of the increased nuclear potentials.) It is interesting, however, to investigate: (1) Whether all interactionenergy contriiutions are increasing, or only a certain part of them in going to shorter bond lengths. (2) If the sum of diagonal terms (self-repulsions, the socalled localization sum) is increasing, whether it is valid or not for each type of localized orbitals. (3) Whether the conclusions are general for both (sp) and (sp + polarization functions) basis sets. (4) Whether or not the results depend on the moIecular system (number of bonds or lone pairs, locaked orbrtals, etc.).
2. Geometry dependence of localizability In this paper we investigate the localizability of eIectroncharge distributions of some smal.I systems both at the experimental and the calculated equiliirium geometnes, using the Edmiston-Ruedenberg energy-locrdizatron criterion [4]_ The study was done with the use of the basis sets 6-31G [S] and 6-31G/d [I]. The expet-i-
Volume 7 2. number 2
Table 2 Sum of Iocahzed*,rbltal energy contrlbutlons usmg a 6-31G basis (III hartrre)
HF Hz0
exp. cak e\p CdC.
hW,
exp CA.
D~gonaI terms
Offdl;lgorul
9.50171 9 49980 8 ‘6731 8 27060 7.19139 7.18605
17.97132 17 96466 14.75702 14.79460 12 09822 12 23019
HF
exp. CdC.
Hz0
exp CdC.
NH3
exp. Wk.
276
OffdwonaJ
9.49717 9 50045 8 26782 8.27704 7.20942 7 21989
17.98804 18 OOllZ 14.78305 14 82144 12.11214 12.15347
=w CdC.
Hz0
elp. CdC
NH3
elp. CdC
Table 3 Sum of locahzed+rbltal energy contributions using a 6-31G/d bans (m bartree) Diagonal terms
Table 4 Coulomb energy contnbutlons obtamed from the indlndual locahzed orbltak usmg a 6-31G basis (in hartree)
terms
nxntal geometnes were taken from ref. [6]. the calculated values from ref [5] (basis 6-31G) and from ref. [3] (basis 6-31G/d)_ The correspondmg theorettcal values for HF are R = 1.7403 au and R = 1.7 183 au, respectively In table 2 are gwen the sums of diagonal and off-diagonal terms of energy contrrbutlons obtained from locahzed orbltals, usmg the 6-31G basis. It can be seen that the shorter the bond length the larger the sum of contnbutrons for HF. Both &agonal and off-diagonal terms are larger at the theoretical than at the ehpenmental geometry for H20. It IS not the same, however, for NH, _ the sum of the off-diagonal terms IS larger at the calculated equihbnum geometry whde the sum of diagonal terms is smaller Thus may certamly be due to the very large HNH bond angle (117’) of NH, at the theoretlcally obtamed total energjr mmunum. As to the results obtatned by ustng the 6-31G/d basis set. we have found similar regularities for ali systems mbestigated the values are given m table 3 The sums are larger at the calculated than at the e\perunenral equdrbnum geometries. These results are m agreement with the mcieasmg values of nuclear potentrals m gomg from experimentally to theorettcaily determined mmima. The energy contrlbutlons from
HF
1 June 1980
CHEMICAL PHYSICS LETTERS
Core
Bond
5.49468 5 49471 4 85079 1.84978 4.20680 4.20324
0.90227 0.90012 0 82048 0 82574 0.74678 0.75623
pa
Lone pau 1.03492 1.03499 0.88778 0 88467 0.74525 0 71412
each type of locahzed orbltals, obtamed by the use of the 6-31G basis are presented separately III table 4. The general conclusion holds that the bond and lonepair charge densities are always chdngtng m the oppoate dlrectlon. The contnbutlon of a lone-Parr orbltal 1s smaller at the expertmental equihbrium, whtle the bond-pair ones at the calculated equtibrmm are smaller for the HF molecule. As to the case of H,O and NH3, the opposite relation holds. It is general. however, that the core orbital contrrbutron does not change much (less than 0.152). Introducmg d-type functrons on the heavy atoms (table 5). theoretical equlhbnum geometnes approach the expenmental ones and the results become umform for the molecules studled. It should be noted, however. that the expression “approach to the experimental val-
ues” IS not used m a strict sense (for inore details see e.g ref. [7]). Namely, approaching the Hartree-Fock hmit, the bond-pair energy contnbutlons are shghtly larger at the calculated equ&briu_m geometnes whde the core ones and the lone-pair ones are smaller at the evpenmental equlllbnum posltlon of the nuclei. In order to confirm the above results we mvestlgated the same quantities usmg polanzatlon functions Tabk 5
Coulomb energy contrlbutlons obtamed from the indlvldual localued orblrals ~slng a 6-31G/d basis (m hartree)
terms HF
exp. C&T.
H20
+\p. CaIC
NH3
elp. cak.
Core
Bond pair
Lone pau
5.48787 5.48780 4.81412 4.84384 4.20048 4.20024
0.91894 0.92292 0.83325 0.83907 0 757.50 0.76 160
1.03012 I.02991 0.87860 0.87753 0 73644 0.73485
Volume 72, number 2 Table 6 Properties
CHEMICAL
studred mcluding p-type functions
on the hydrogens
6-31Gld
dragonai terms offdngonal terms core energy contnbution bond-pan energy cantrrbution lone-pau energy contriiutron
a)
See
PHYSICS
1 June 1980
LETTERS
for Hz0
(ii hartree)
+ p basrs
DUMmg’S
ev
UlC.
exp.
8.27041 14.79166 4.84435 0.83613 0 87690
8.28280 14.84270 4.84394 0.84392 0.87551
8.26399 14.79396 4.85003 0.83627 0.8707 1
basis Se
a)
CdC.
8.27237 14.83043 4.8497 1 0.84167
0.86966
te\t.
even on the hydrogen for H20. The values are grven in table 6. ihey have been computed by using the 6-31G/ d + p basis [l] , or a Dunning basrs set [S] , (9 s 5 p 1 d/ 4 s 1 p) contracted to 4 s 3 p I d/2 s 1 p gausslans. The calculated equrhbrra were as follows. R = 1.7807 au, (Y =106.10”for6-31G/d+pandR=1.7883au,ar = 105 82” for Dunning’s basrs set. The differences betlveen the energy contrrbutions at the experimental and those at the theoretrcal equrlibrium geometnes are sunilar to those found usmg a 6-3 1 G/d basis set. The results show that while the presence of one d-type basis function on the heavy atom is necessary to descrrbe suitably the main drfferences between core, bond and lone-pair orbital densities at varrous geometries, the p-type functrons on the hydrogens do not cause large changes in comparison to a (spd/s) type basrs set.
3. Conclusion It has been shown - using the EdmistonRuedenberg locahzation cnterium -that in going from the experrmental to the calculated equihbnum geometries, some regularities could be found which reflect suitably the main behaviour of drfferent types of locahzed orbrtals. It is also remarkable that the results of this work are in agreement with those obtained u-ra basrs set dependence study [9]. As Lhe conrnbutron of locahzed orbrtals show similar regulanties as do the total molecular propertres on approaching the Hartree-Fock hmrt, it 1s expected that the locahzedorbrtal contnbutrons do also converge to certain linutmg values. In thrs case - usmg the transferabrlrty prop-
of locabzed-orbrtal contributions [lo] - the contnbution determined m a similar system could be used for an analysis of larger related molecules. Having investigated the energy contnbutions of localized charge densrties in this paper, a similar analysis for the electric moment contributions will be the subject of onz of our next works.
erties
Acknowledgement Thanks are due to Zs. Ozoroczy and M-E. Stephens for theu help in the calculations.
References 111 P-C. Hariharan and J-A. Popie, Theoret. Chim. Ada 18 (1973) 213. 121 E. Kapuy. C. Kozmutza and M-E. Stephens, Theoret. Chun. Acta 43 (1976) 175. 131 PC. Hariharan and J-A. Popie. Mol. Phys. 27 (l974) 209_ 1-41 C. Edmiston and K. Ruedenberg. Rev. Mod. Phys. 35 (1963) 457. [S] WJ. Heihue, R. Ditchtield and LA. Pople, i. Chem. Phys. 56 (1972) 2257. [6] Interatomic distances, supplement. Spekal pubkation No. 18 (The Chemical Society, London, 1965). [7] S. Bell, J. Chem. Phys. 68 (1978) 3014. [B] T.H Dunning. J. Chem. Phys_ 53 (197012833; 55 (1971) 716.3958. [9] E. Kapuy, C. Kozmutza. R. Daudel and ME. Stephens, Theoret. Chim. Acta 50 (1976) 31. [ 101 E. Kapuy, C. Kozmutza. R. Daudel and M-E. Stephens. Theoret. Chim. Acta, to be pubhshed.
277
7 2, number 2
CHEMICAL
STUDY OF LOCALIZABILITY
PHYSICS
LETJYERS
1 June 1980
FOR SOME SMALL MOLECULES
Ede KAPUY, Corneha KOZMUTZA Quantum i71wry Group, Polytechnical
Umrersrty, Budapest, fiungary
and Raymond DAUDEL Centre de hlkanrque Recerred
2 October
Ondulatorre Appltquek du C-N R S . Parts, France 1979;
III fiial form 7 March 1980
locehzed charge denntres of some small molecules at theu experimental and calculated equiliirium geometries have been studred using the 6-31G and 6-31G/d basis sets. introducing polarization functions it has been found that the bond-pair cfuuge minima drstnbutions are “more”, while the lone-pau ones are “less” locahzed at their theoretically determmed totakrtergy than at the correspondmg expertmental geometries.
1. Introduction It IS well known that the independent-particle model provrdes systematically shorter bond lengths and larger bond angles at the calculated equdibrium geometnes than at the experimental ones. This is implied by the increase of the nuclear potential in going from the expenmental to the corresponding theoretical values. For HF, H20 and NH3, usmg the so-called 6-31G/d basis set [l] , the changes in the total energres and nuclear potentrals are shown in table 1. The experimental geometry data were taken from ref. [2], while the theoretical values are given in ref. [3] _ As the nuclear potential increases, the electton denarty also mcreases. Therefore, it 1s expected that at short bond lengths the sum of the localized interactionenergy contnbutions wrll be larger than at large bond Table 1 Changes III the total energy and nuclear potentrals (m hartree) for HF, Hz0 and NHs, using the 6-31G/d basis set Total energy
Nuclear potential
Hz0
-0.00007 -0.00026
NH3
-0.00023
+0.04419 +0.10774 +0.10897
HF
lengths. (Although the bond angles are slightly larger at the theoretically determined minima, the effect of short length dominates, as can be seen from the vahres of the increased nuclear potentials.) It is interesting, however, to investigate: (1) Whether all interactionenergy contriiutions are increasing, or only a certain part of them in going to shorter bond lengths. (2) If the sum of diagonal terms (self-repulsions, the socalled localization sum) is increasing, whether it is valid or not for each type of localized orbitals. (3) Whether the conclusions are general for both (sp) and (sp + polarization functions) basis sets. (4) Whether or not the results depend on the moIecular system (number of bonds or lone pairs, locaked orbrtals, etc.).
2. Geometry dependence of localizability In this paper we investigate the localizability of eIectroncharge distributions of some smal.I systems both at the experimental and the calculated equiliirium geometnes, using the Edmiston-Ruedenberg energy-locrdizatron criterion [4]_ The study was done with the use of the basis sets 6-31G [S] and 6-31G/d [I]. The expet-i-
Volume 7 2. number 2
Table 2 Sum of Iocahzed*,rbltal energy contrlbutlons usmg a 6-31G basis (III hartrre)
HF Hz0
exp. cak e\p CdC.
hW,
exp CA.
D~gonaI terms
Offdl;lgorul
9.50171 9 49980 8 ‘6731 8 27060 7.19139 7.18605
17.97132 17 96466 14.75702 14.79460 12 09822 12 23019
HF
exp. CdC.
Hz0
exp CdC.
NH3
exp. Wk.
276
OffdwonaJ
9.49717 9 50045 8 26782 8.27704 7.20942 7 21989
17.98804 18 OOllZ 14.78305 14 82144 12.11214 12.15347
=w CdC.
Hz0
elp. CdC
NH3
elp. CdC
Table 3 Sum of locahzed+rbltal energy contributions using a 6-31G/d bans (m bartree) Diagonal terms
Table 4 Coulomb energy contnbutlons obtamed from the indlndual locahzed orbltak usmg a 6-31G basis (in hartree)
terms
nxntal geometnes were taken from ref. [6]. the calculated values from ref [5] (basis 6-31G) and from ref. [3] (basis 6-31G/d)_ The correspondmg theorettcal values for HF are R = 1.7403 au and R = 1.7 183 au, respectively In table 2 are gwen the sums of diagonal and off-diagonal terms of energy contrrbutlons obtained from locahzed orbltals, usmg the 6-31G basis. It can be seen that the shorter the bond length the larger the sum of contnbutrons for HF. Both &agonal and off-diagonal terms are larger at the theoretical than at the ehpenmental geometry for H20. It IS not the same, however, for NH, _ the sum of the off-diagonal terms IS larger at the calculated equihbnum geometry whde the sum of diagonal terms is smaller Thus may certamly be due to the very large HNH bond angle (117’) of NH, at the theoretlcally obtamed total energjr mmunum. As to the results obtatned by ustng the 6-31G/d basis set. we have found similar regularities for ali systems mbestigated the values are given m table 3 The sums are larger at the calculated than at the e\perunenral equdrbnum geometries. These results are m agreement with the mcieasmg values of nuclear potentrals m gomg from experimentally to theorettcaily determined mmima. The energy contrlbutlons from
HF
1 June 1980
CHEMICAL PHYSICS LETTERS
Core
Bond
5.49468 5 49471 4 85079 1.84978 4.20680 4.20324
0.90227 0.90012 0 82048 0 82574 0.74678 0.75623
pa
Lone pau 1.03492 1.03499 0.88778 0 88467 0.74525 0 71412
each type of locahzed orbltals, obtamed by the use of the 6-31G basis are presented separately III table 4. The general conclusion holds that the bond and lonepair charge densities are always chdngtng m the oppoate dlrectlon. The contnbutlon of a lone-Parr orbltal 1s smaller at the expertmental equihbrium, whtle the bond-pair ones at the calculated equtibrmm are smaller for the HF molecule. As to the case of H,O and NH3, the opposite relation holds. It is general. however, that the core orbital contrrbutron does not change much (less than 0.152). Introducmg d-type functrons on the heavy atoms (table 5). theoretical equlhbnum geometnes approach the expenmental ones and the results become umform for the molecules studled. It should be noted, however. that the expression “approach to the experimental val-
ues” IS not used m a strict sense (for inore details see e.g ref. [7]). Namely, approaching the Hartree-Fock hmit, the bond-pair energy contnbutlons are shghtly larger at the calculated equ&briu_m geometnes whde the core ones and the lone-pair ones are smaller at the evpenmental equlllbnum posltlon of the nuclei. In order to confirm the above results we mvestlgated the same quantities usmg polanzatlon functions Tabk 5
Coulomb energy contrlbutlons obtamed from the indlvldual localued orblrals ~slng a 6-31G/d basis (m hartree)
terms HF
exp. C&T.
H20
+\p. CaIC
NH3
elp. cak.
Core
Bond pair
Lone pau
5.48787 5.48780 4.81412 4.84384 4.20048 4.20024
0.91894 0.92292 0.83325 0.83907 0 757.50 0.76 160
1.03012 I.02991 0.87860 0.87753 0 73644 0.73485
Volume 72, number 2 Table 6 Properties
CHEMICAL
studred mcluding p-type functions
on the hydrogens
6-31Gld
dragonai terms offdngonal terms core energy contnbution bond-pan energy cantrrbution lone-pau energy contriiutron
a)
See
PHYSICS
1 June 1980
LETTERS
for Hz0
(ii hartree)
+ p basrs
DUMmg’S
ev
UlC.
exp.
8.27041 14.79166 4.84435 0.83613 0 87690
8.28280 14.84270 4.84394 0.84392 0.87551
8.26399 14.79396 4.85003 0.83627 0.8707 1
basis Se
a)
CdC.
8.27237 14.83043 4.8497 1 0.84167
0.86966
te\t.
even on the hydrogen for H20. The values are grven in table 6. ihey have been computed by using the 6-31G/ d + p basis [l] , or a Dunning basrs set [S] , (9 s 5 p 1 d/ 4 s 1 p) contracted to 4 s 3 p I d/2 s 1 p gausslans. The calculated equrhbrra were as follows. R = 1.7807 au, (Y =106.10”for6-31G/d+pandR=1.7883au,ar = 105 82” for Dunning’s basrs set. The differences betlveen the energy contrrbutions at the experimental and those at the theoretrcal equrlibrium geometnes are sunilar to those found usmg a 6-3 1 G/d basis set. The results show that while the presence of one d-type basis function on the heavy atom is necessary to descrrbe suitably the main drfferences between core, bond and lone-pair orbital densities at varrous geometries, the p-type functrons on the hydrogens do not cause large changes in comparison to a (spd/s) type basrs set.
3. Conclusion It has been shown - using the EdmistonRuedenberg locahzation cnterium -that in going from the experrmental to the calculated equihbnum geometries, some regularities could be found which reflect suitably the main behaviour of drfferent types of locahzed orbrtals. It is also remarkable that the results of this work are in agreement with those obtained u-ra basrs set dependence study [9]. As Lhe conrnbutron of locahzed orbrtals show similar regulanties as do the total molecular propertres on approaching the Hartree-Fock hmrt, it 1s expected that the locahzedorbrtal contnbutrons do also converge to certain linutmg values. In thrs case - usmg the transferabrlrty prop-
of locabzed-orbrtal contributions [lo] - the contnbution determined m a similar system could be used for an analysis of larger related molecules. Having investigated the energy contnbutions of localized charge densrties in this paper, a similar analysis for the electric moment contributions will be the subject of onz of our next works.
erties
Acknowledgement Thanks are due to Zs. Ozoroczy and M-E. Stephens for theu help in the calculations.
References 111 P-C. Hariharan and J-A. Popie, Theoret. Chim. Ada 18 (1973) 213. 121 E. Kapuy. C. Kozmutza and M-E. Stephens, Theoret. Chun. Acta 43 (1976) 175. 131 PC. Hariharan and J-A. Popie. Mol. Phys. 27 (l974) 209_ 1-41 C. Edmiston and K. Ruedenberg. Rev. Mod. Phys. 35 (1963) 457. [S] WJ. Heihue, R. Ditchtield and LA. Pople, i. Chem. Phys. 56 (1972) 2257. [6] Interatomic distances, supplement. Spekal pubkation No. 18 (The Chemical Society, London, 1965). [7] S. Bell, J. Chem. Phys. 68 (1978) 3014. [B] T.H Dunning. J. Chem. Phys_ 53 (197012833; 55 (1971) 716.3958. [9] E. Kapuy, C. Kozmutza. R. Daudel and ME. Stephens, Theoret. Chim. Acta 50 (1976) 31. [ 101 E. Kapuy, C. Kozmutza. R. Daudel and M-E. Stephens. Theoret. Chim. Acta, to be pubhshed.
277