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Intermolecular interaction coefficients C8 and C10 using point-charge models
Volume 61, number 3
CHEMICAL PHYSICS LETI-ERS
1 MlL’CIl1379
LNTERhlOLECULAR INTERACTION COEFFICIENTS C, AND Cl0 USlNG POINT-CffARGE J-A_ YOFFE Departntenr of .~fat~~enrarics, Unir*ersirv of Notringham,
Norttigham.
MODELS
1VG7 ZRD. IYK
Received 19 October I978
Values of the two-body mtemction coefficients CS and Cl0 betrreen like species are obtained from point-charge-model formulae_ I_exxisset results using Frost wavefunctrom may be compared with rttomicullycentred \iarefunction ualues 3, well as results obtained using an experimental pointdmr~e model. Results for bolh C8 and C;u can be improtcd b) normduation using theoretical .md experimentedstatic poiarizabilit) flues, providing excellent results for Cs_
1. introduction Long-range interaction coefficients, such as the twobody interaction coefficient C,, have been calculated using dipole oscdIator strength distnbutions (DOSD’s) constructed from extensive experimental information [I]_ Calculations for interaction coefficients can be greatly simplified by using a discrete represenration of these DOSD’s, called pseudo dipole oscrhator strength diitrrbutions (PDOSD’s) (hlargoliash and Meath 121). and values for two-body interaction coefficients C, .md iv4 [2], as well as three-body coefficients -y3 131, can be computed without difficulty. These results provide “experiment< vah~es with which to compare ab initio resuks, and values of a(o), C6, da and ~3 using three different point-charge models have been found to be in good agreement with these values for a number of molecules [4] _ Indeed the PDOSD’s themselves can be thought of in terms of a point-charge model [4] and the ab initio point-charge models would seem to provide a good method to estimate these properties and interaction coefficients for larger molecules, where up till now few or no estimates are avaiIabIe. For the two-body interaction, not only are the calculation of C6 and Wa (or d4), a relativistic correction term, important, but terms C8 and Cl0 in the original series [5], AE=
-C&+
C8jRg - CrO/RJO - ___,
may also be of importance
molecular separation [C;] _Estimates of C, and C,, have mainly been confined to atoms and small molecules and are often simply ignored through lack of reliable results It is the purpose of this note to compare estimates for C, and Cl0 using ab initio point-charge model values wnh those obtained using the “experimental” point-charge model of Margolizh and hfeath.
2. Point-charge
model fonnulne for C, and Cl0
We consider the general case of point chdrges
Table i Model
frequencies
(1)
depending on the inter593
Volume 61. number 3
CHEMICAL
1 March 1979
PHYSICS LETTERS
and gaussian exponents&j_ The Shipman model (denoted by S) reallocates off-centre charges in the HalI model onto gaussian centres whilst the Amos-Yoffe mode1 (AY), unlike the Hall and Shipman models, is restricted to Lewis basis sets and allocates charges of 2 to each gaussian centre- Each molecuie in the hfargoliash-hleath model (MM) is represented by ten point charges with associated frequencies that reproduce expcrimentaI.Iy known dipole oscillator strength sums S(I;-) where
(3-I and where, for instance S(0) = 2, conserving the total charge, and S(--2) = a(O), conserving the experimental static polarizability. The formuIae for interaction coefficients can now be given for the general pointcharge case and the specific model resuhs are obtained simply by substituting in the appropriate charges and frequencies given in tabIe I_ The expressions for C, and Cl0 using pointcharge modeIs can be derived from the oscilIator model [5 J and may be written
and “A RB
crO=y
cc i= 1i=l “A “B
_+.I$ I I f&)2(&2(& I I
-I- &) 3
I
1 E
where there are lzA p oint charges for species A and rzs for species B.
3_ ResuIts Results for the coefficients C&A-A) and Clo(A-A) are given in tables 2 and 3 respectively for Frost-model Lewis set wavefunctions [ 1 I-131 given previously using the Amos-Yoffe (AY) formulae [6] _ These are compared with values obtained from atomically centred wavefunctions, whose exponents are given by PopIe et al- [ 141, in conjunction with the Hall (H) and Shipman (S) mode1 for_muIae. The exponents uskd for these atomically centred wavefunctions were the unconstrained set but the contraction coefficients were not used and the density matriv was found using all the gaussians from OPIT [IS] _Two gaussians for each of the Is, 2s. 2p,. 2pu and 2p, orbitals, making a total of ten, were centred on every heavy atom, whilst two
Tabfe 2 Values of Ca(_A-A)
in atomic units3)
_Atomfmokcuie _A__.He 0 HZ Nt ol Hz0 NH3 CH4
-_- ___________
PopIe wavefunction =) (CS)H
Gt)s
(C8)hflM
(14.35) 1l-40 (190.8) 190.8 (224.4) 368.6 915.2 (1035) (882-3) 1911 (730.6) 5992 (1634) 1256 (2282) 3395
11.96 (143.5) 221.1 (192.2) 4022 (223.1) 925.2 (1034) 1590 (873.6) 719.8 (731.9) 1445 (1641) 3728 (2264)
1454 192.5 224.0 1069 8895 730.7 1660 2290
5.42 (14.29) I82 333 I130 2040
‘) Normabed values in parentheses_
594
“Experiment’*
Frost wavefunctior.b) G),Y
(221)
(840) (1592) (2243) b, Refs. [ll--13]_
=) Exponents of ret [ 14]_
CHEMICAL
Volume 61, number 3 Table 3 Vahte~ of Cto(A-A) Atom/molecule
in atomic units a)
43.3 3500 -
W? Na 02 Hz0 NH3
“Experiment” fClO)S
(Clo)H 129.4 2978 10877 i 3444 44296 12107 26125 83470
(145.4)
For helium,
however, two gaussians with exponents 4.0978 and 0.532 I, found using OPIT, were placed on the atomic centre- AI1 results can then be compared with the “experimental” results of hfargoliash and Meath @ihI). Table 2 gives vahtes of C,(A-A) with normalised values in brackets according to the ratio of the theoretical and experimental polarizabriities_ We express o(0) as cr(0) = k/W? )
(5)
where k is a constant and W the average (valence) exponent for the molecule or atom in question_ Then, for the Hall model. for instmce, we fiid
and the corresponding results for Ca and C,, may be normahsed by taking their dependence on WH as (&)-4 and (aH)-5 respectively_ For C, improved results are obtained using this normalisation procedure for all three pointcharge models producing values in excellent agreement with the “experimental” values given in table 3 results again are im-
cClo)MM
137.0
(172.0)
(3114) (5788) (15642) (15128) (16421) (38747) (5 1254) _- --~
(4456)
gaussians were centred on every hydrogen_
For ClO(A-A)
--
Pople wavefunction
3720 (11824) 17100 (26249) 35800 (40298) CH4 -.-_ a) Normalised vfues in parentheses.
(c8 )&Ihi -
1 hfarch 1979
-----~
Frost wavefunction (ClO)AY
He 0
PHYSICS LETTERS
3841 12088 13621 31978 16082 33049 95609
(172.5)
(2978) (5848) (15686) (16858) (15513) (36301) (50809)
183.7
3417 5196 10366 18464 16267 42626 53028
proved by normalisation but are not in as good agreeresults in table 2 ment with (C~O)A~XIas the c8(A-A) are with (C8)51hI _Table 3 does show that the unnormalised Frost wavefunction results provide only about 23% for the (ClO)hf,I value for He and Hz0 whilst for NH3 only about 40% In fact for the Pople wavefunctions normalisation according to (G)-6 dependence for Cl0 would provide better results except for O2 and CH1. Of course the exponents of Pople are not the only values that can be used and atomically optimised wavefunctions may well overemph.Kze the core electrons when set in a molecular context. Several different sets of esponents optimised in a molecular situation for a variety of molecules for H, C, N and 0 are to be published [ 161 and the resultmg LXavefunctrons should provide an improved valence description. Using exponents from these wavefunctions, again ignoring contraction coefficients and obtaining density matri., elements using all the gaussians from OPIT, we obtain the x alues given in table 4 for C8 and Clo_ Generally improved raw results over the Pople values are obtained except for C8 for N2, and once again normalised
results are in
Tabie 4
Results for CatA-A)
and Cre(A-A)
using exponents optimised in .I molecular environment [ 161 in atomrc umts a)
hfolecule
CC&
G3)s
(CS)MM
Ha
220 6 (224.4) 613.7 (849.7) 601.1 (711.9) 1600 (1622)
241.6 (223.1) 616.3 (849.5) 660.0 (711.0) 1775 (1614)
124.0 889.5 730.7 1660
2434 1330
2683 1341
2290 1069
02 Ha0 NH3 ‘34
W? a) Normal&d
(2278) (1044)
(2266) (1043)
5589
7454 9973 33777 55189 21477
(clohIM
(ClO)S
(c,o)H
(5709)
(11195) (11311) (34364) (50815) (15871)
6258 7508 11481 38975 63072 21749
(5664) (11214) (12600) (34613) (51080) (15E91)
5196 18464 16267 42626 52028
20366
values in parentheses. 595
Volume 61, number 3
CHEMICAL PHYSICS LETTERS
better agreemera with (Cs>st;rr_Vrdues for Cxo, however, are r~~de worse by norm&isation for Hz and N7 and once again rest&s for Cl0 are not as cfose to Go)s1.\t as cs is to CC&st-
Acknowledgement I would like to thank Professor R-E. Christoffersen for &owing me to use exponents for wavefunctions prior to pubfiration nnd the S.R.C_ for the award of a Postdoctoral feifowsbip.
References [I j G-D. Zeiss orrd WJ. .lfecth, I\iol. Phys. 33 (1977) 1155. [2] DJ. Yaqolkrsh zurd W-f. bfexh. 1..Chem- Phys_ 68 (1978) 14Z6_
596
1 hfarch 1979
I3 j DJ_ Mq@osb, T-R_ Proctor. G-D_ Zeiss and WJ. hfeath. MOL Phyr 35 (i97S) 747. I41 J-A. Yoffe, Theoret. Chim. AC&X. to be published. -[5I H. hfsrSen;tu end N.R.. Kestner, Theory of intermoIecuku forces (Pergrmon, New York* 1969). [6] A-T. Amos and J.A. Yoffe. Theoret. Chim. Acm 42 (1976) 247_ 171 G-G_ Hell, Chem. Phys. Letters 6 (1973) 501.. [SJ A-D_ Teft and G-G. Hclf. Theoret. Chfm. Acta 31(1973) 311_ [9I LL- Shipmen, Chem. Phys. Letters 31 (1975) 361. [ 101 A-T. Amos and J-A. Yoffe, Theoret. Chim. Acto 40 f197S ZZl_ [ 111 A-A- Frost. J. Chem. Phys. 47 (1967) 3307_ [I21 A-A. Frost and R-A. Rouse,J_ Am_ Chem- Sot. 90 (1968) I965 [13J A-A. Frost, J. Phys. Chem. 72 (1968) 1389. (141 R- Ditchtiefd. W-f. Hehre and J-A. Pople. J. Chem. Phys. 52 (1970) SOOL [IS] B. Ford, G-G. Hall and J.C. Packer. Intern. J. Quruttum Chem 4 (1970) 533. IId] D- Span&x ad R-E. Christoffersen, to be published.
CHEMICAL PHYSICS LETI-ERS
1 MlL’CIl1379
LNTERhlOLECULAR INTERACTION COEFFICIENTS C, AND Cl0 USlNG POINT-CffARGE J-A_ YOFFE Departntenr of .~fat~~enrarics, Unir*ersirv of Notringham,
Norttigham.
MODELS
1VG7 ZRD. IYK
Received 19 October I978
Values of the two-body mtemction coefficients CS and Cl0 betrreen like species are obtained from point-charge-model formulae_ I_exxisset results using Frost wavefunctrom may be compared with rttomicullycentred \iarefunction ualues 3, well as results obtained using an experimental pointdmr~e model. Results for bolh C8 and C;u can be improtcd b) normduation using theoretical .md experimentedstatic poiarizabilit) flues, providing excellent results for Cs_
1. introduction Long-range interaction coefficients, such as the twobody interaction coefficient C,, have been calculated using dipole oscdIator strength distnbutions (DOSD’s) constructed from extensive experimental information [I]_ Calculations for interaction coefficients can be greatly simplified by using a discrete represenration of these DOSD’s, called pseudo dipole oscrhator strength diitrrbutions (PDOSD’s) (hlargoliash and Meath 121). and values for two-body interaction coefficients C, .md iv4 [2], as well as three-body coefficients -y3 131, can be computed without difficulty. These results provide “experiment< vah~es with which to compare ab initio resuks, and values of a(o), C6, da and ~3 using three different point-charge models have been found to be in good agreement with these values for a number of molecules [4] _ Indeed the PDOSD’s themselves can be thought of in terms of a point-charge model [4] and the ab initio point-charge models would seem to provide a good method to estimate these properties and interaction coefficients for larger molecules, where up till now few or no estimates are avaiIabIe. For the two-body interaction, not only are the calculation of C6 and Wa (or d4), a relativistic correction term, important, but terms C8 and Cl0 in the original series [5], AE=
-C&+
C8jRg - CrO/RJO - ___,
may also be of importance
molecular separation [C;] _Estimates of C, and C,, have mainly been confined to atoms and small molecules and are often simply ignored through lack of reliable results It is the purpose of this note to compare estimates for C, and Cl0 using ab initio point-charge model values wnh those obtained using the “experimental” point-charge model of Margolizh and hfeath.
2. Point-charge
model fonnulne for C, and Cl0
We consider the general case of point chdrges
Table i Model
frequencies
(1)
depending on the inter593
Volume 61. number 3
CHEMICAL
1 March 1979
PHYSICS LETTERS
and gaussian exponents&j_ The Shipman model (denoted by S) reallocates off-centre charges in the HalI model onto gaussian centres whilst the Amos-Yoffe mode1 (AY), unlike the Hall and Shipman models, is restricted to Lewis basis sets and allocates charges of 2 to each gaussian centre- Each molecuie in the hfargoliash-hleath model (MM) is represented by ten point charges with associated frequencies that reproduce expcrimentaI.Iy known dipole oscillator strength sums S(I;-) where
(3-I and where, for instance S(0) = 2, conserving the total charge, and S(--2) = a(O), conserving the experimental static polarizability. The formuIae for interaction coefficients can now be given for the general pointcharge case and the specific model resuhs are obtained simply by substituting in the appropriate charges and frequencies given in tabIe I_ The expressions for C, and Cl0 using pointcharge modeIs can be derived from the oscilIator model [5 J and may be written
and “A RB
crO=y
cc i= 1i=l “A “B
_+.I$ I I f&)2(&2(& I I
-I- &) 3
I
1 E
where there are lzA p oint charges for species A and rzs for species B.
3_ ResuIts Results for the coefficients C&A-A) and Clo(A-A) are given in tables 2 and 3 respectively for Frost-model Lewis set wavefunctions [ 1 I-131 given previously using the Amos-Yoffe (AY) formulae [6] _ These are compared with values obtained from atomically centred wavefunctions, whose exponents are given by PopIe et al- [ 141, in conjunction with the Hall (H) and Shipman (S) mode1 for_muIae. The exponents uskd for these atomically centred wavefunctions were the unconstrained set but the contraction coefficients were not used and the density matriv was found using all the gaussians from OPIT [IS] _Two gaussians for each of the Is, 2s. 2p,. 2pu and 2p, orbitals, making a total of ten, were centred on every heavy atom, whilst two
Tabfe 2 Values of Ca(_A-A)
in atomic units3)
_Atomfmokcuie _A__.He 0 HZ Nt ol Hz0 NH3 CH4
-_- ___________
PopIe wavefunction =) (CS)H
Gt)s
(C8)hflM
(14.35) 1l-40 (190.8) 190.8 (224.4) 368.6 915.2 (1035) (882-3) 1911 (730.6) 5992 (1634) 1256 (2282) 3395
11.96 (143.5) 221.1 (192.2) 4022 (223.1) 925.2 (1034) 1590 (873.6) 719.8 (731.9) 1445 (1641) 3728 (2264)
1454 192.5 224.0 1069 8895 730.7 1660 2290
5.42 (14.29) I82 333 I130 2040
‘) Normabed values in parentheses_
594
“Experiment’*
Frost wavefunctior.b) G),Y
(221)
(840) (1592) (2243) b, Refs. [ll--13]_
=) Exponents of ret [ 14]_
CHEMICAL
Volume 61, number 3 Table 3 Vahte~ of Cto(A-A) Atom/molecule
in atomic units a)
43.3 3500 -
W? Na 02 Hz0 NH3
“Experiment” fClO)S
(Clo)H 129.4 2978 10877 i 3444 44296 12107 26125 83470
(145.4)
For helium,
however, two gaussians with exponents 4.0978 and 0.532 I, found using OPIT, were placed on the atomic centre- AI1 results can then be compared with the “experimental” results of hfargoliash and Meath @ihI). Table 2 gives vahtes of C,(A-A) with normalised values in brackets according to the ratio of the theoretical and experimental polarizabriities_ We express o(0) as cr(0) = k/W? )
(5)
where k is a constant and W the average (valence) exponent for the molecule or atom in question_ Then, for the Hall model. for instmce, we fiid
and the corresponding results for Ca and C,, may be normahsed by taking their dependence on WH as (&)-4 and (aH)-5 respectively_ For C, improved results are obtained using this normalisation procedure for all three pointcharge models producing values in excellent agreement with the “experimental” values given in table 3 results again are im-
cClo)MM
137.0
(172.0)
(3114) (5788) (15642) (15128) (16421) (38747) (5 1254) _- --~
(4456)
gaussians were centred on every hydrogen_
For ClO(A-A)
--
Pople wavefunction
3720 (11824) 17100 (26249) 35800 (40298) CH4 -.-_ a) Normalised vfues in parentheses.
(c8 )&Ihi -
1 hfarch 1979
-----~
Frost wavefunction (ClO)AY
He 0
PHYSICS LETTERS
3841 12088 13621 31978 16082 33049 95609
(172.5)
(2978) (5848) (15686) (16858) (15513) (36301) (50809)
183.7
3417 5196 10366 18464 16267 42626 53028
proved by normalisation but are not in as good agreeresults in table 2 ment with (C~O)A~XIas the c8(A-A) are with (C8)51hI _Table 3 does show that the unnormalised Frost wavefunction results provide only about 23% for the (ClO)hf,I value for He and Hz0 whilst for NH3 only about 40% In fact for the Pople wavefunctions normalisation according to (G)-6 dependence for Cl0 would provide better results except for O2 and CH1. Of course the exponents of Pople are not the only values that can be used and atomically optimised wavefunctions may well overemph.Kze the core electrons when set in a molecular context. Several different sets of esponents optimised in a molecular situation for a variety of molecules for H, C, N and 0 are to be published [ 161 and the resultmg LXavefunctrons should provide an improved valence description. Using exponents from these wavefunctions, again ignoring contraction coefficients and obtaining density matri., elements using all the gaussians from OPIT, we obtain the x alues given in table 4 for C8 and Clo_ Generally improved raw results over the Pople values are obtained except for C8 for N2, and once again normalised
results are in
Tabie 4
Results for CatA-A)
and Cre(A-A)
using exponents optimised in .I molecular environment [ 161 in atomrc umts a)
hfolecule
CC&
G3)s
(CS)MM
Ha
220 6 (224.4) 613.7 (849.7) 601.1 (711.9) 1600 (1622)
241.6 (223.1) 616.3 (849.5) 660.0 (711.0) 1775 (1614)
124.0 889.5 730.7 1660
2434 1330
2683 1341
2290 1069
02 Ha0 NH3 ‘34
W? a) Normal&d
(2278) (1044)
(2266) (1043)
5589
7454 9973 33777 55189 21477
(clohIM
(ClO)S
(c,o)H
(5709)
(11195) (11311) (34364) (50815) (15871)
6258 7508 11481 38975 63072 21749
(5664) (11214) (12600) (34613) (51080) (15E91)
5196 18464 16267 42626 52028
20366
values in parentheses. 595
Volume 61, number 3
CHEMICAL PHYSICS LETTERS
better agreemera with (Cs>st;rr_Vrdues for Cxo, however, are r~~de worse by norm&isation for Hz and N7 and once again rest&s for Cl0 are not as cfose to Go)s1.\t as cs is to CC&st-
Acknowledgement I would like to thank Professor R-E. Christoffersen for &owing me to use exponents for wavefunctions prior to pubfiration nnd the S.R.C_ for the award of a Postdoctoral feifowsbip.
References [I j G-D. Zeiss orrd WJ. .lfecth, I\iol. Phys. 33 (1977) 1155. [2] DJ. Yaqolkrsh zurd W-f. bfexh. 1..Chem- Phys_ 68 (1978) 14Z6_
596
1 hfarch 1979
I3 j DJ_ Mq@osb, T-R_ Proctor. G-D_ Zeiss and WJ. hfeath. MOL Phyr 35 (i97S) 747. I41 J-A. Yoffe, Theoret. Chim. AC&X. to be published. -[5I H. hfsrSen;tu end N.R.. Kestner, Theory of intermoIecuku forces (Pergrmon, New York* 1969). [6] A-T. Amos and J.A. Yoffe. Theoret. Chim. Acm 42 (1976) 247_ 171 G-G_ Hell, Chem. Phys. Letters 6 (1973) 501.. [SJ A-D_ Teft and G-G. Hclf. Theoret. Chfm. Acta 31(1973) 311_ [9I LL- Shipmen, Chem. Phys. Letters 31 (1975) 361. [ 101 A-T. Amos and J-A. Yoffe, Theoret. Chim. Acto 40 f197S ZZl_ [ 111 A-A- Frost. J. Chem. Phys. 47 (1967) 3307_ [I21 A-A. Frost and R-A. Rouse,J_ Am_ Chem- Sot. 90 (1968) I965 [13J A-A. Frost, J. Phys. Chem. 72 (1968) 1389. (141 R- Ditchtiefd. W-f. Hehre and J-A. Pople. J. Chem. Phys. 52 (1970) SOOL [IS] B. Ford, G-G. Hall and J.C. Packer. Intern. J. Quruttum Chem 4 (1970) 533. IId] D- Span&x ad R-E. Christoffersen, to be published.