Aug 15, 1986 - with reversed roles for A and B. Use ofEq. (4) is equivalent to the prediction of e (8) from known values of the e ~l and e ~6J coefficients plus the ...
Correlation and prediction of dispersion coefficients for isoelectronic systems A. D. Koutselos and E. A. Mason Citation: The Journal of Chemical Physics 85, 2154 (1986); doi: 10.1063/1.451108 View online: http://dx.doi.org/10.1063/1.451108 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/85/4?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Higher order two- and three-body dispersion coefficients for alkali isoelectronic sequences by a variationally stable procedure J. Chem. Phys. 134, 144110 (2011); 10.1063/1.3577967 Multipolar polarizabilities and two- and three-body dispersion coefficients for alkali isoelectronic sequences J. Chem. Phys. 106, 2298 (1997); 10.1063/1.473089 On determining partial correlation coefficients by the covariance method of linear prediction J. Acoust. Soc. Am. 62, S64 (1977); 10.1121/1.2016301 Cross Correlation in Structural Systems: Dispersion and Nondispersion Waves J. Acoust. Soc. Am. 45, 1118 (1969); 10.1121/1.1911582 Correlation Energy for Atomic Systems. II. Isoelectronic Series with 11 to 18 Electrons J. Chem. Phys. 39, 175 (1963); 10.1063/1.1733998
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Correlation and prediction of dispersion coefficients forisoelectronic systems A. D. Koutselos and E. A. Mason Brown University, Providence, Rhode Island 02912
(Received 24 March 1986; accepted 1 May 1986) Combination rules for dispersion coefficients are greatly extended by way of a parameter corresponding to an effective number of equivalent electrons, the same parameter N that appears in the Slater-Kirkwood formula for the dipole-dipole coefficient. The same N can be used for all members of an isoelectronic sequence, and new formulas are given for higher-order two-body dispersion coefficients and three-body nonadditive coefficients, in which the same N can also be used. The only additional input data needed are static multi pole polarizabilities. A theoretical justification is given using screening-constant wave functions. Extensive empirical testing suggests that the results are usually accurate within about 5% for dipole coefficients and about 5%-10% for higher coefficients. The results apply only to S-state atoms and ions, but should be capable of extension to other systems.
I. INTRODUCTION An excellent way of consolidating and summarizing data on the dipole-dipole London dispersion coefficient e(6) is through the Slater-Kirkwood I formula (in atomic units) e(6) =1. aAa B e2a S AB 2 (aAINA ) 1/2 + (aBINB ) 1/2 0'
(2)
then the same values ofNA andNB can be used in Eq. (1) to calculate e ~~ for the unlike-atom interactions. 2 This procedure is equivalent to the following combination rule3 •4 ; 2e(6) e(6) e(6) _
BB
AA
AB - (a I a )e AA (6) B A
+
(
(6) a A/ a B ) e BB
e(8) _ 15 AB -
4""
(2)
aAaB (aAINA )112 + [(13) + a:>(22) + ?f(31) C AB -(!)AB (!)AB AB'
C i"~
=
?f)!i)
and so on. The last expression, for example, consists of two dipole-octopole terms and one quadrupole-quadrupole term. Each term can be approximated by an expression of the form 4 ,6
a~)a~L)
4(21)I(2L)! (_1_ + _1_), (15) (21 + 2L)! a~) a~L) where a~) is the 2/-pole polarizability of atom A, A~) is a mean excitation energy of A used to obtain closure of a sum arising in second-order perturbation theory, and similarly for index L and atom B. (We will omit these indices on a and a when 1, L = 1.) Since polarizabilities arise as energy perturbations in an external field, it is not surprising that similar expressions hold for the polarizabilities 2 (l: r~) a~) = - - (16) 21 + 1 a~) where the summation runs over all the electrons in atom A. This expression can be regarded as defining a~) by a closure approximation, andEq. (15) for ?f~~) can be regarded as an approximation obtained by replacing a similar closure energy a~~) by the sum A~) + a~L).6,18 Determination of the a's from Eq. (15) via the limiting cases of like-atom interactions leads to a series of combination rules giving ?f ~~) in terms of ?f ~i) and ?f ~i); these are just the generalized combination rules of Kramer and Herschbach. 4 Their disadvantage is lack of knowledge of ?f~i) and ?f~i). To obtain something simpler, we need another way of evaluating the A's. This is most easily accomplished by comparing Eq. (16) with explicit expressions for the a's obtained by a variational calculation. 19 For N equivalent independent electrons, we find the general result to be 4 (l: r 21 )2 aU) (17) - 1(2/+ 1)2 (l:rl-2) , ::z
?f~~)
where the summations run over N terms. For 1= 1 we recover the Kirkwood formula 13 given in Eq. (10), and for 1= 2 we recover the formula for the quadrupole polarizability given by Alvarez-Rizzatti and Mason. 6 Since (l:r 2/ - 2) is equal to N for 1 = 1, we can use Eq. (17) as a recursion relation to find (l:r2/) in terms ofN and thesetofpolarizabilities, aU), aU-I), ... , a. Substitution into Eq. (16) then gives a (/), from which the various dispersion coefficients immediately follow from Eq. (15). In particular, in this way we find a = (N /a)I/2, the same as Eq. (7), and a(2)
=[
9Na ] 114, (a(2»2
(18)
a(3) = (~)1I2 [9Na(a(2»2 2 (a(3»4
]1/8,
etc.
Substituting these expressions for a into Eqs. (14) and ( 15) for the dispersion coefficients, we immediately obtain Eq. ( 1) for C ~6~, Eq. (4) for C ~8~, and the following formula for C )!~) (in atomic units):
(19)
J. Chern. Phys., Vol. 85, No. 4,15 August 1986 Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 195.134.76.38 On: Wed, 09 Mar 2016 10:09:47
2158
A. D. Koutselos and E. A. Mason: Correlation of dispersion coefficients
TABLE IV. Parameters of additional systems used to test formulas for higher dispersion coefficients.· a
Atom
(a~ )
4.500 164.0 292.9 319.2 402.2 37.84 39.1b 71.32 6O.1b 168.7
H Li K Rb Cs Be Mg Ca
a(2)
at3 )
(a~)
(ab)
(e2 a~)
N
1.313(2)C 3.680(4) 1.502(5) 2.127(5) 3.399(5) 4.126(3 ) 5.383(3)b 1.474 ( 4) 1.96(4)b 6.151(4)
6.49 1385 4000 4690 6665 220 250b 634 472b 2785
0.822 0.773 1.13 1.20 1.21 1.59 1.86b 1.97 1.82b 2.87
15.00 1383 4597 5979 9478 302.6 380b 828.0 931b 2717
C(6)
• Data from Standard and Certain (Ref. 11) unless otherwise noted. b Reference 7. cThis notation means 1.313x 102 •
Higher coefficients are easily found. If Eq. (2) is used to eliminate N A and N B in favor of C i,6l and C ~~ in Eq. (4) for C ~~, the result is quite similar to Eq. (30) of Alvarez-Rizzatti and Mason, 6 who started from a variational expression for the dispersion energies rather than from Eq. (15). The present result is simpler, however, and is of comparable accuracy, as will be shown in Tables V and VI. Two comments are in order before proceeding to empirical tests of the formulas for C ~~ and C ~~). The first is that, according to the foregoing derivation, the N's that occur in the formulas for the higher dispersion coefficients are the same ones as used for C ~~; that is, the N 's obtained from Eq. (2) are to be applied to all the dispersion coefficients. The second comment is that application of screening-con-
stant wave functions to the expressions for !1 (/) and a(/) in Eqs. (18) predicts consistently that N is the same for all members of an isoelectronic sequence, as we would expect. That is, !1 (/) varies as (Z - Sj)2 and a(/) as (Z - Sj ) - 2(/ + 1), which gives N independent of (Z - Sj ) for all I. To test the foregoing formulas we have used data on C (8) and C (to) given by Standard and Certain 11 for a number ofSstate atoms, and by Bartolotti7 for some S state ions and atoms. The additional input data needed to predict C (8) and C(tO) for systems not already included in Tables I and II are collected in Table IV. For Be and Mg the two sets of input data are not in very good agreement, and we have therefore used them separately in an internally consistent way (e.g., Bartolotti's input values used with his dispersion coefficients). No results for Na appear in Table IV because the uncertainty of its value of C (6) is so large. 11 Table V summarizes the results for the atom-atom interactions compiled by Standard and Certain, as the ratio of C ~~ from Eq. (4) to the tabulated average C ~~ above the main diagonal of the table, and similarly for C ~~) from Eq. ( 19) below the diagonal. The results are very good; the predictions are within the uncertainty bounds given by Standard and Certain for nearly all the C (8) and most of the C (to). Table VI summarizes the additional results involving ions, as obtained from Bartolotti. 7 Here the agreement is not as good as in Table V, but is nevertheless probably within Bartolotti's uncertainties, as judged by his results on higherorder polarizabilities and dispersion coefficients for atoms. Two ion-atom interactions not included in Table VI are Hwith H and He, for which Davison9 has calculated C ~~ values of limited accuracy. The deviations are 11 % and 23%,
TABLE V. Ratios of predicted to known values of higher dispersion coefficients for atoms. C i"J above the diagonal and C ~~) below the diagonal. a He
Ne
Ar
Kr
Xe
H
Li
K
Rb
Cs
Be
Mg
Ca
He
1.04 1.06
1.06
0.99
0.97
0.93
1.04
1.03
1.04
1.05
1.10
0.93
0.95
1.14
Ne
1.10
1.00
0.98
0.93
1.06
1.04
1.05
1.05
1.10
0.95
0.96
1.14
Ar
0.98
0.97
0.94
1.02
1.03
1.03
1.04
1.09
0.95
0.96
1.10
Kr
1.00
1.00
0.96
1.01
1.03
1.03
1.04
1.08
0.95
0.96
1.10
Xe
0.97
0.97
0.94
0.95
1.03
1.03
1.04
1.09
0.95
0.97
1.08
H
1.06
1.09
1.01
1.02
0.99
1.04
1.04
1.09
0.96
0.96
1.11
Li
1.09
1.10
1.08
1.07
1.06
1.07
1.01
1.02
1.04
1.01
1.00
1.05
K
1.11
1.11
1.09
1.09
1.08
1.08
1.04
1.04
1.02
1.01
1.05
Rb
1.09
1.10
1.08
1.08
1.08
1.08
1.04
1.04
1.04
1.02
1.02
1.06
Cs
1.10
1.11
1.10
1.10
1.10
1.09
1.05
1.06
1.06
1.06
1.05
1.08
Be
1.02
1.04
0.99
0.99
0.97
1.01
1.02
1.04
1.04
1.07
Mg
1.03
1.04
1.01
1.00
0.99
1.01
1.02
1.03
1.03
1.06
Ca
1.08
1.10
1.08
1.07
1.07
1.08
1.06
1.08
1.08
1.09
1.06 0.97
1.04
1.05
1.11 1.09
1.05
• Data from Standard and Certain (Ref. 11).
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A. D. Koutselos and E. A. Mason: Correlation of dispersion coefficients TABLE VI. Ratios of predicted to known values of higher dispersion coefficients involving ions. C.\.."~ above the diagonal and C ~~) below the diagonal."
TABLE VII. Ratios of predicted to known values of three-body nonadditive dispersion coefficients for like atoms.· Atom
Li+
K+
Na+
Li+
0.82
Na+ K+
0.89
B+
1.00
0.96
He Ne Ar
1.04 0.73 0.77 0.94 0.96
0.99 1.13 0.76 0.93 0.93
Be
Mg
B+
He
Ne
Ar
Be
0.96
1.04
0.83
0.79
1.02
0.86
0.93
0.98
0.82
0.80
0.98
0.85
0.91
0.91
0.80
0.80
0.96
0.87
0.98
0.87
0.96
0.98
0.90
0.92 0.75 0.76 0.93 0.93
V. THREE-BODY NONADDITIVE COEFFICIENTS The dispersion interaction among three atoms or ions can be written as the sum of three two-body interactions plus an additional three-body term that represents the deviation from pairwise additivity. Aside from a geometrical factor, three-body coefficients can be written as (following the notation of Standard and Certain II ) roo
Jo
a~)
(iw)
a~L)(iw)a~A.)(iw)dw,
(20)
where a(iw) is the dynamic polarizability as a function of imaginary frequency.18 These coefficients can be approximated in exactly the same way as the two-body coefficients ~~~); the analog ofEq. (15) is4 (/U) - 1 a(/)a(L)a(A.) ZA BC-2AB C X
DDQ
DQQ
DDO
QQQ
He Ne Ar Kr Xe H Li K Rb Cs Be Mg Ca
1.02 1.08 1.06 1.06 1.10 1.01 1.00 1.02 1.03 1.01 1.02 1.01 1.08
1.04 1.13 1.03 1.03 1.03 1.03 1.01 1.03 1.03 1.04 0.99 0.99 1.13
1.07 1.12 0.97 0.97 0.94 1.06 1.02 1.04 1.04 1.07 0.95 0.96 1.16
1.05 1.15 1.02 1.05 1.06 1.03 1.02 1.03 1.03 1.02 1.03 1.02 1.07
1.10 1.08 0.89 0.89 0.83 1.09 1.04 1.07 1.07 1.11 0.90 0.92 1.19
• Data from Standard and Certain (Ref. II). Notation: DQO signifies dipole-quadrupole-octopole, or I = I, L = 2, Ii = 3.
respectively, which is probably satisfactory. On the basis of the foregoing results we hazard the estimate that the present formulas for C ~~ and C ~~) -Eqs. (4) and (19)-are accurate within about 10%.
17'
DDD
Mg
• Data from Bartolotti (Ref. 7).
Z~~) = 1.-
2159
+ a~L> + ag» (a~> + a~L>)(a~L> + ag>)(ag> + a~» a~)a~L>ag>(a~>
,
(21) in which the a's are the same as in Eq. (15). Combination rules giving Z~~~> in terms of ~~2, ~~iL>, ~~~> are straightforwardly obtainable by elimination of the a's; Kramer and Herschbach4 have shown that this yields good results for the dipole--dipole--dipole coefficient. Our result goes somewhat further by using Eqs. (18) to find the a's. In this way we can predict Z ~~> from a~>, a~L>, ag> plus the values of N A , N B , N c , which are determined from ~~1>, ~ M: >, ~ g~ ), respectively (that is, from the C (6) coefficients for like interactions). It is not necessary to know any dispersion coefficients beyond dipole--dipole in order to predict all the higher-order three-body coefficients, as long as the higher-order polarizabilities are known. Results are shown in Table VII comparing the predicted and known values of a number of three-body coefficients for
like atoms. The results are quite good, although roughly half the values in the table fall outside the (rather tight) bounds given by Standard and Certain. Although the interactions in . Table VII involve only like-atom coefficients, we expect Eq. (21) to hold for unlike coefficients as well, because only N values as determined from C (6) coefficients are needed to find all the a values. Because of this dependence only on C(6) coefficients, we expect the transferability of N values within an isoelectronic sequence to apply also to Eq. (21). We estimate that values of Z ~~> can be predicted via Eqs. (18) and (21) within an uncertainty of about 5%-10%, depending on the values of I, L, A.
VI. DISCUSSION The first result of this work is that N values obtained from known C (6) coefficients can be used for isoelectronic atoms and ions to predict other C (6) coefficients. In addition, we have obtained new relations for higher two-body dispersion coefficients and nonadditive three-body coefficients, and shown that the same N values can be used for these coefficients. Thus the only input data needed to predict all the dispersion coefficients are C (6) coefficients for a small number of selected systems, plus the static multipole polarizabilities. In effect, these results greatly extend the useful range of application of combination rules for long-range interactions. Agreement with independently known dispersion coefficients is usually quite good. We estimate that the results are accurate within about 5% for C(6) coefficients, and about 5%-10% for higher coefficients. The main possible exception to this remark is for some transition-metal ions having ~p6d 10 outer electron shells. The main limitation is that the results have been tested only for atoms and ions in S states. The method would be expected to work also for molecules, 4 but needs to be tested. ACKNOWLEDGMENT
This research was supported in part by National Science Foundation Grant No. CHE 85-09416. I J. C. Slater and J. G. Kirkwood, Phys. Rev. 37, 682 (1931). 2J. N. Wilson, J. Chern. Phys. 43, 2564 (1965).
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A. D. Koutselos and E. A. Mason: Correlation of dispersion coefficients
3K. T. Tang, Phys. Rev. 177, 108 (1969). 4H. L. Kramer and D. R. Herschbach, 1. Chern. Phys. 53,2792 (1970). sO. D. Mahan, 1. Chern. Phys. 76, 493 (1982). 6M. Alvarez-Rizzatti and E. A. Mason, 1. Chern. Phys. 59, 518 (1973). 7L. 1. Bartolotti, 1. Chern. Phys. 80, 5687 ( 1984). sA. Dalgarno and A. E. Kingston, Proc. Phys. Soc. London 73, 455 (1959). 9W. D. Davison, Proc. Phys. Soc. London 87, 133 (1966). "'W. E. Donath, J. Chern. Phys. 39, 2685 (1963). "I. M. Standard and P. R. Certain, 1. Chern. Phys. 83, 3002 (1985).
12F. London, Trans. Faraday Soc. 33, 8 (1937). 131. O. Kirkwood, Phys. Z. 33, 57 (1932). 14ft1.. Miiller, Proc. R. Soc. London Ser. A 154, 624 (1936). ISA. Dalgarno, Adv. Chern. Phys. 12,143 (1967). 161. H. Van Vleck, The Theory ofElectric and Magnetic Susceptibilities (Oxford University, London, 1932), p. 91. 171. O. Hirschfelder, C. F. Curtiss, and R. B. Bird, Molecular Theory of Gases and Liquids (Wiley, New York, 1964), p. 954. IRA. Dalgarno and W. D. Davison, Adv. At. Mol. Phys. 2, I (1966). 19A. Dalgarno, Adv. Phys. 11,281 (1962).
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