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Dipole moments in rare gas interactions
Volume 158, number I,2
DIPOLE MOMENTS
CHEMICAL PHYSICS LETTERS
2 June 1989
IN RARE GAS INTERACTIONS
M. KRAUSS National Institute of Standards and Technology Gaiihersburg, MD 20899, US.4
and B. GUILLOT Laboratoire de Physique ThhPonquedes Liguides, Umveixtt! Pierrre et Marie Curie, 4, Pluce Jussieu, 75252 Pam Cedex 05. France
Received 28 February 1989
A hybrid model is used to calculate the dipole moments of rare gas dimers and trimers. The exchange and polarization interactions are obtained with a single-configuration self-consistent-field (SCF) calculation and the dispersion contribution to the dipole is obtained by perturbation theory. The SCF dipoles for the trimers are found to compare well with values modeled by an exchange quadrupole-induced dipole mechanism. The dispersion dipole for the Arti dimer is found to be significant compared to the exchange contribution even in the region of the equilibrium separation. Values of the B hypcrpolarizability are evaluated for Ne, Ar, Kr, and Xe which allows the perturbation calculation of the dispersion dipole for any pair of rare gas atoms.
1. Introduction Collision-induced absorption in rare gas fluids is due to the dipole moment produced in a binary collision of dissimilar atoms or a ternary collision of the same atoms. Theoretical modeling of the dynamics and radiative absorption in these fluids requires a knowledge of the potential and dipole moment properties of the interacting species. A recent model of the ternary collision problem [ 1 ] has determined that at intermediate distances the dominant mechanism in a pure rare gas fluid is the development, due to the exchange overlap repulsion, of a quadrupole moment in a pair of atoms which induces a dipole in the third atom. They refer to the exchange quadrupoleinduced dipole mechanism (EQID). The calculation of the exchange dipole was made with a relatively simple effective interaction model [ 2,3], This will be tested here with a self-consistent-field calculation of three interacting rare gas atoms. At large distances the dispersion dipole, which does not require overlap of the atomic charge distributions, is estimated to become comparable to EQID. 142
Except for systems containing cithcr hydrogen or helium atoms, there are few ab initio studies of the dispersion dipole moments of interacting rare gas atoms [ 4 1, The calculation of the dipole moments can be done in analogous fashion to the evaluation of the interaction energy. For larger systems a hybrid scheme has long been in fashion. The exchange interaction is represented by the interaction energy evaluated with a Hartree-Fock (HF) wavefunction of the interacting atoms. The effect of correlation on this interaction is neglected for the intermediate range of distances and is considered at long range by a perturbation correction second order in the correlated multipoles induced in the atoms. The hybrid HF and dispersion correction model has been used to model binary absorption [ 51 with qualitative accuracy. Correlated calculations are more difficult to obtain and have been limited to HeAr [ 61 and HeNe [ 71 when superposition errors are taken into consideration. The study of the larger rare gas atoms are presently limited to the hybrid scheme. The magnitude of the dispersion dipole is examined here for interacting diatoms.
volume 158, number 1,2
CHEMICAL
2 June 1989
PHYSICS LETTERS
The hybrid scheme is difficult to evaluate if all of the atomic electrons are considered in the interaction. Unless accurate HF representations of the nonvalence electrons are used, substantial basis set superposition errors occur in both the energy and dipole moment (DBSSE) [ 8 1. The core electrons can be replaced with compact effective core potentials (CEP) [ 91, which substantially reduce the superposition error. Since only the valence electrons are explicitly considered, rather large basis sets can be used in the HF calculations and the limiting values approached for the basis expansion method. The atomic parameters required for the evaluation of the dispersion correction can also be more readily obtained when it is necessary to treat only the valence electrons.
2. Exchange interactions The accuracy of the dipole moment that can be obtained with the present level of HF calculation must be tested first. This was done by examining the dipole moment in a mixed rare gas dimer, ArKr, as a function of the size of the basis set. All calculations consider only the valence electrons and use the CEP to replace the electron cores. The CEP and basis sets for Kr are now being reported [ 10 1, The atomic basis sets are contracted to a triple-zeta (TZ) set of sp functions. In order to ensure that the long-range behavior of the wavefunction is represented well, two diffuse sp functions were added to both Ar and Kr. The polarization was included with a set of three d functions for both Ar and Kr that were optimized by calculating the dipole polarizability of the atoms. The DBSSE dipole moment values are under 10% of the calculated HF dipole moment for internuclear ArKr distances smaller than 8 au as seen in table 1. At 9 au, the DBSSE reaches,about 25% and gradually rises from that point. The behavior of the DBSSE is much more weakly dependent upon the overlap than the dipole moment itself. The absolute accuracy of the dipole moment cannot easily be determined. Increasing the accuracy for retaining integrals to a threshold better than lo-” and improving the orbital convergence in the self-consistent-field procedure has no effect on the values in table 1. The dipole moment curve of ArKr in table 1 is found to
Table 1 Dipole moment of ArKr R (au)
!t WF, au) Ar+Kr-
DBSSE
fl (disp.
6.0 6.5 7.0 1.5 9.0
0.0210 0.0112 0.0059 0.0031 Q.00058
0.0003 0.0003 0.0002 0.0002 0.00017
- 0.0044 -0.0025 -0.0015 -0.0009 -0.0003
1
agree well with earlier results [ 5 ] to distance shorter than 8.5 au. At longer distances, the earlier results suffered from more substantial DBSSE errors. At intermediate and large distances the ternary collision dipole for a mixed rare gas would be dominated by the induction of a dipole in an atom in the presence of a collision dipole or quadrupole of the other two partners. For a pure gas only the collision quadrupole is present and the EQID mechanism dominates. The general formulas for various orders of induced dipoles have been given [ 111. The simple effective collision model was tested by evaluating the dipole moments for Hell, Ar3, and KrX for a 90” isosceles triangle and comparing it to the EQID values in table 2. The agreement for these large distances and small dipoles is good until the distance is so large that the DBSSE becomes significant. The DBSSE is less than lOoh of the total dipole for distances 9 au and smaller for Kr. At 10 au the value is closer to 40% and beyond this distance even the sign of the Table 2 Comparison
of SCFand
EQID dipole moments for He3, Ar,, and
Kr3 R (au)
w (au) SCF
EQID
He
5.0 6.0 7.0
3.2 -5 2.4 -5 5.5 -6
4.4 -4 3.8 -5 2.7 -6
Ar
6.0 7.0 8.0
6.3 -4 9.1 -5 1.1 -5
5.4 -4 4.6 -5 3.1 -6
Kr
7.0 8.0 9.0 10.0
4.9 1.2 4.0 1.0
1.8 2.9 4.2 5.2
-4 -4 -5 -5
-3 -4 -5 -6
143
Volume
158,number 1,2
CHEMICAL
dipole is uncertain and the moments are noise. Nontheless the EQID mechanism is supported and the simple effective collision model can be parametrized with a few ab initio calculations.
3. Dispersion dipole moment Dipole moments reported earlier [S] were required to be accurate in the region of the inner turning point of the energy curve, but the present interest is over a wider range of distances. The dispersion dipole moment has to be considered from distances of the order of and larger than the position of the energy minimum. The theory of the long-range dispersion dipole shows that the leading term, O7 /R ‘, for a diatomic system is due to the correlated coupling of the induced quadrupole on one atom to the induced dipole on the other. Various approximate formulas have been presented for the evaluation of the D, coefficient [ 11,12 1. The dipole-quadrupole hyperpolarizability, B, is the only parameter that is usually not readily available. But the static value for B has been reported recently for Ne [ 131 and Ar [ 141. These quantities have been evaluated here also using the single configuration finite field method for the Ne, Ar, Kr, and Xe atoms. The present values in atomic units are - 13.9, - 142.8, -307.3, and -699.5, respectively, for Ne, Ar, Kr, and Xe. The comparable values in the literature for NC and Ar are - 13.6 and - 141. The correlated value of B for Ar differs significantly from the SCF value. However, the SCF values are used for both the exchange and dispersion dipole to maintain a consistent hybrid model. Correlated values for B will be reported later. Using the assumption of an analogous mean excitation energy to evaluate the dipole polarizability, cy, or the B hyperpolarizability, the excitation energies used to evaluate the C, energy dispersion coefficient
144
2 June 1989
PHYSICS LETTERS
and cy [ 71 +I’then yield a 0, of - 1244.6 au for ArKr. The dispersion dipole is opposite in sign to the SCF exchange dipole and about 25% of its value between 6.0 and 7.5 au and should become predominant above 9 au. The dispersion calculations for the threebody interaction will involve a number of terms [ 11,15 1, but all of the static polarizabilities and excitation energies can be obtained for the approximations developed by Bohr and Hunt [ 111. “’ Effective excitation
energies
for Ar, 0.79084
au and Kr,
0.69364 au.
References [ 11B. Guillot, R.D. Mountain and G. Birnbaum, Mol. Phys. 64 (1988) 747. [2]L. Jansen, Phys. Rev. 125 (1961) 1798. [ 31 L. Jansen, in: Advances in quantum chemistry, Vol. 2, ed. P.-O. LGwdin (Academic Press, New York, 1965) pp. 119194. [4] W. Meyer, m: Phenomena induced by intermolecular interactions, ed. G. Bimbaum, NATO ASI Series B, Vol. 127 (Plenum Press, New York, 1985) pp. 29-48. [S] G. Birnbaum, M. Krauss and L. Frommhold, .I. Chem. Phys. 80 (1984) 2669. [ 61 W. Meyer and L. Frommhold, Phys. Rev. A 34 ( 1986) 2771. [ 71 M. Krauss, R.M. Regan and D.D. Konowalow, J. Phys. Chcm. 92 ( 1988) 4329. [ 81 G. Karlstrom and A.J. Sadlej, Theoret. Chim. Acta 61 (1982) 1. (91 W.J. Stevens, H. Basch and M. Krauss, J. Chem. Phys. 81 (1984) 6026. [IO] W.J. Stevens, H. Basch and M. Krauss, unpublished CEP for K to Kr. [ 1I] J.E. Bohrand K.L.C. Hunt, J. Chem. Phys. 86 (1987) 5441. [ 121 L. Galatry and T. Gharbi, Chem. Phys. Letters 75 ( 1980) 427. [ 131 G. Maroulis and D.M. Bishop, Chem. Phys. Letters 114 (1985) 182. [ 141 G. Maroulis and D.M. Bishop, J. Phys. B I8 ( 1985) 4675; I. Cemusak, G.H.F. Diercksen and A.J. Sadlej, Chem. Phys. Letters 128 (1986) 18. [ 151P.H. Martin, Mol. Phys. 27 (I 974) 129.
DIPOLE MOMENTS
CHEMICAL PHYSICS LETTERS
2 June 1989
IN RARE GAS INTERACTIONS
M. KRAUSS National Institute of Standards and Technology Gaiihersburg, MD 20899, US.4
and B. GUILLOT Laboratoire de Physique ThhPonquedes Liguides, Umveixtt! Pierrre et Marie Curie, 4, Pluce Jussieu, 75252 Pam Cedex 05. France
Received 28 February 1989
A hybrid model is used to calculate the dipole moments of rare gas dimers and trimers. The exchange and polarization interactions are obtained with a single-configuration self-consistent-field (SCF) calculation and the dispersion contribution to the dipole is obtained by perturbation theory. The SCF dipoles for the trimers are found to compare well with values modeled by an exchange quadrupole-induced dipole mechanism. The dispersion dipole for the Arti dimer is found to be significant compared to the exchange contribution even in the region of the equilibrium separation. Values of the B hypcrpolarizability are evaluated for Ne, Ar, Kr, and Xe which allows the perturbation calculation of the dispersion dipole for any pair of rare gas atoms.
1. Introduction Collision-induced absorption in rare gas fluids is due to the dipole moment produced in a binary collision of dissimilar atoms or a ternary collision of the same atoms. Theoretical modeling of the dynamics and radiative absorption in these fluids requires a knowledge of the potential and dipole moment properties of the interacting species. A recent model of the ternary collision problem [ 1 ] has determined that at intermediate distances the dominant mechanism in a pure rare gas fluid is the development, due to the exchange overlap repulsion, of a quadrupole moment in a pair of atoms which induces a dipole in the third atom. They refer to the exchange quadrupoleinduced dipole mechanism (EQID). The calculation of the exchange dipole was made with a relatively simple effective interaction model [ 2,3], This will be tested here with a self-consistent-field calculation of three interacting rare gas atoms. At large distances the dispersion dipole, which does not require overlap of the atomic charge distributions, is estimated to become comparable to EQID. 142
Except for systems containing cithcr hydrogen or helium atoms, there are few ab initio studies of the dispersion dipole moments of interacting rare gas atoms [ 4 1, The calculation of the dipole moments can be done in analogous fashion to the evaluation of the interaction energy. For larger systems a hybrid scheme has long been in fashion. The exchange interaction is represented by the interaction energy evaluated with a Hartree-Fock (HF) wavefunction of the interacting atoms. The effect of correlation on this interaction is neglected for the intermediate range of distances and is considered at long range by a perturbation correction second order in the correlated multipoles induced in the atoms. The hybrid HF and dispersion correction model has been used to model binary absorption [ 51 with qualitative accuracy. Correlated calculations are more difficult to obtain and have been limited to HeAr [ 61 and HeNe [ 71 when superposition errors are taken into consideration. The study of the larger rare gas atoms are presently limited to the hybrid scheme. The magnitude of the dispersion dipole is examined here for interacting diatoms.
volume 158, number 1,2
CHEMICAL
2 June 1989
PHYSICS LETTERS
The hybrid scheme is difficult to evaluate if all of the atomic electrons are considered in the interaction. Unless accurate HF representations of the nonvalence electrons are used, substantial basis set superposition errors occur in both the energy and dipole moment (DBSSE) [ 8 1. The core electrons can be replaced with compact effective core potentials (CEP) [ 91, which substantially reduce the superposition error. Since only the valence electrons are explicitly considered, rather large basis sets can be used in the HF calculations and the limiting values approached for the basis expansion method. The atomic parameters required for the evaluation of the dispersion correction can also be more readily obtained when it is necessary to treat only the valence electrons.
2. Exchange interactions The accuracy of the dipole moment that can be obtained with the present level of HF calculation must be tested first. This was done by examining the dipole moment in a mixed rare gas dimer, ArKr, as a function of the size of the basis set. All calculations consider only the valence electrons and use the CEP to replace the electron cores. The CEP and basis sets for Kr are now being reported [ 10 1, The atomic basis sets are contracted to a triple-zeta (TZ) set of sp functions. In order to ensure that the long-range behavior of the wavefunction is represented well, two diffuse sp functions were added to both Ar and Kr. The polarization was included with a set of three d functions for both Ar and Kr that were optimized by calculating the dipole polarizability of the atoms. The DBSSE dipole moment values are under 10% of the calculated HF dipole moment for internuclear ArKr distances smaller than 8 au as seen in table 1. At 9 au, the DBSSE reaches,about 25% and gradually rises from that point. The behavior of the DBSSE is much more weakly dependent upon the overlap than the dipole moment itself. The absolute accuracy of the dipole moment cannot easily be determined. Increasing the accuracy for retaining integrals to a threshold better than lo-” and improving the orbital convergence in the self-consistent-field procedure has no effect on the values in table 1. The dipole moment curve of ArKr in table 1 is found to
Table 1 Dipole moment of ArKr R (au)
!t WF, au) Ar+Kr-
DBSSE
fl (disp.
6.0 6.5 7.0 1.5 9.0
0.0210 0.0112 0.0059 0.0031 Q.00058
0.0003 0.0003 0.0002 0.0002 0.00017
- 0.0044 -0.0025 -0.0015 -0.0009 -0.0003
1
agree well with earlier results [ 5 ] to distance shorter than 8.5 au. At longer distances, the earlier results suffered from more substantial DBSSE errors. At intermediate and large distances the ternary collision dipole for a mixed rare gas would be dominated by the induction of a dipole in an atom in the presence of a collision dipole or quadrupole of the other two partners. For a pure gas only the collision quadrupole is present and the EQID mechanism dominates. The general formulas for various orders of induced dipoles have been given [ 111. The simple effective collision model was tested by evaluating the dipole moments for Hell, Ar3, and KrX for a 90” isosceles triangle and comparing it to the EQID values in table 2. The agreement for these large distances and small dipoles is good until the distance is so large that the DBSSE becomes significant. The DBSSE is less than lOoh of the total dipole for distances 9 au and smaller for Kr. At 10 au the value is closer to 40% and beyond this distance even the sign of the Table 2 Comparison
of SCFand
EQID dipole moments for He3, Ar,, and
Kr3 R (au)
w (au) SCF
EQID
He
5.0 6.0 7.0
3.2 -5 2.4 -5 5.5 -6
4.4 -4 3.8 -5 2.7 -6
Ar
6.0 7.0 8.0
6.3 -4 9.1 -5 1.1 -5
5.4 -4 4.6 -5 3.1 -6
Kr
7.0 8.0 9.0 10.0
4.9 1.2 4.0 1.0
1.8 2.9 4.2 5.2
-4 -4 -5 -5
-3 -4 -5 -6
143
Volume
158,number 1,2
CHEMICAL
dipole is uncertain and the moments are noise. Nontheless the EQID mechanism is supported and the simple effective collision model can be parametrized with a few ab initio calculations.
3. Dispersion dipole moment Dipole moments reported earlier [S] were required to be accurate in the region of the inner turning point of the energy curve, but the present interest is over a wider range of distances. The dispersion dipole moment has to be considered from distances of the order of and larger than the position of the energy minimum. The theory of the long-range dispersion dipole shows that the leading term, O7 /R ‘, for a diatomic system is due to the correlated coupling of the induced quadrupole on one atom to the induced dipole on the other. Various approximate formulas have been presented for the evaluation of the D, coefficient [ 11,12 1. The dipole-quadrupole hyperpolarizability, B, is the only parameter that is usually not readily available. But the static value for B has been reported recently for Ne [ 131 and Ar [ 141. These quantities have been evaluated here also using the single configuration finite field method for the Ne, Ar, Kr, and Xe atoms. The present values in atomic units are - 13.9, - 142.8, -307.3, and -699.5, respectively, for Ne, Ar, Kr, and Xe. The comparable values in the literature for NC and Ar are - 13.6 and - 141. The correlated value of B for Ar differs significantly from the SCF value. However, the SCF values are used for both the exchange and dispersion dipole to maintain a consistent hybrid model. Correlated values for B will be reported later. Using the assumption of an analogous mean excitation energy to evaluate the dipole polarizability, cy, or the B hyperpolarizability, the excitation energies used to evaluate the C, energy dispersion coefficient
144
2 June 1989
PHYSICS LETTERS
and cy [ 71 +I’then yield a 0, of - 1244.6 au for ArKr. The dispersion dipole is opposite in sign to the SCF exchange dipole and about 25% of its value between 6.0 and 7.5 au and should become predominant above 9 au. The dispersion calculations for the threebody interaction will involve a number of terms [ 11,15 1, but all of the static polarizabilities and excitation energies can be obtained for the approximations developed by Bohr and Hunt [ 111. “’ Effective excitation
energies
for Ar, 0.79084
au and Kr,
0.69364 au.
References [ 11B. Guillot, R.D. Mountain and G. Birnbaum, Mol. Phys. 64 (1988) 747. [2]L. Jansen, Phys. Rev. 125 (1961) 1798. [ 31 L. Jansen, in: Advances in quantum chemistry, Vol. 2, ed. P.-O. LGwdin (Academic Press, New York, 1965) pp. 119194. [4] W. Meyer, m: Phenomena induced by intermolecular interactions, ed. G. Bimbaum, NATO ASI Series B, Vol. 127 (Plenum Press, New York, 1985) pp. 29-48. [S] G. Birnbaum, M. Krauss and L. Frommhold, .I. Chem. Phys. 80 (1984) 2669. [ 61 W. Meyer and L. Frommhold, Phys. Rev. A 34 ( 1986) 2771. [ 71 M. Krauss, R.M. Regan and D.D. Konowalow, J. Phys. Chcm. 92 ( 1988) 4329. [ 81 G. Karlstrom and A.J. Sadlej, Theoret. Chim. Acta 61 (1982) 1. (91 W.J. Stevens, H. Basch and M. Krauss, J. Chem. Phys. 81 (1984) 6026. [IO] W.J. Stevens, H. Basch and M. Krauss, unpublished CEP for K to Kr. [ 1I] J.E. Bohrand K.L.C. Hunt, J. Chem. Phys. 86 (1987) 5441. [ 121 L. Galatry and T. Gharbi, Chem. Phys. Letters 75 ( 1980) 427. [ 131 G. Maroulis and D.M. Bishop, Chem. Phys. Letters 114 (1985) 182. [ 141 G. Maroulis and D.M. Bishop, J. Phys. B I8 ( 1985) 4675; I. Cemusak, G.H.F. Diercksen and A.J. Sadlej, Chem. Phys. Letters 128 (1986) 18. [ 151P.H. Martin, Mol. Phys. 27 (I 974) 129.