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Long-range dipoles, quadrupoles, and hyperpolarizabilities of interacting inert-gas atoms
Volme
CHEMICAL
70. number 2
PHYSICS
1 March 1980
LETTERS
LONG-RANGE DIPOLES, QUADRUPOLES, AND HYl’ERPOLAlUZAEtILITXES OF INTERACTING K.L
INERT-GAS ATOMS
c. HUNT
Department
of Chemrrtry. Michigan State Umvewty.
Rec?elved 3 November
1979,
East Lansng. Mtchigan 48824,
UI final form 17 December
USA
1979
A model relatmg the long-range dipoles of inert-gas heterodlatoms to the distortion of the electromc structure assonated urith dlsperslon forces is presented. Expressions are pven for the contriiutlons to the dipole varymg as the utverse seventh power of the atornx separahon R. The model 1s ako used to determine the long-range quadrupoles of homonuclear atomic paus Pau hyperpolanzabihtIes are obtamed m the model by adding contriiutions of classical multipolar mteractions and contributrons associated WIUI the distortion of the electronic structure. An independent test shows the consistency of the model and provides values for the effectwe mean-square dipoles and quadrupoles of the inert-gas atoms.
1. Introduction Due to the distortion of electronic structure which occurs during a colhslon of Inert-gas atoms, a heteronuclear pair has a nonvanishmg dipole moment, which IS mamfested in far-infrared absorption by Inert-gas mixtures 11,2] At long range, the electronic distortion may be related to the dispersion forces by a model proposed by Buckmgham for calcula~~g pair polanzablhtres of spherical atoms [3]. The lowest-order contrrbution to the induced dipole , where R ISthe internuclear separation along the t axis [4] varies as R rq =D7R-7
+D,R-’
+ ... .
(1)
The Interaction of a homonuclear atom pair gves a nonvamshmg quadrupole, wtuch vatles as RM6 at Iong range. The interaction of two spherical atoms also affects the atomic hyperpolarizablhties y, thus contnbuting to the second Kerr virlal coefficient of inert-gas samples [5,6]. In calculating long-range pau hyperpolarizabihtles. classica! multipolar interactions must be taken into account, in addltlon to the distortion associated with dispersion forces. Interactions alter the pan hyperpolarizablhty tensor elements yzzzz, rzzXX, and r,,, by terms varying as Rm3, but the first nonvanishing change m the mean pan hyperpolarizabi lity LLT(AT = EAT,,& varies as R -6. The model used here has already been apphed to calculate the effects of long-range interactions on the trace 131 and amsotropy [7] of the polarizabllity of a pair of atoms. For H, and He*, the model was found to give good agreement with the accurate values for both parallel and perpendicular components of the polarizabllity varying as R-6, and It also gives good values for the contnbution to the mean pair polarizabMy of Hz varying as R-8.
and hyperpolanzaAccurate quantum mechanical calculation of the coefficients D7, Dg, and the quadrupole bility coefficients is difficult and may not be practicable for pairs heavier than Ne-Ar. A simple model relatmg these coefficients to the known dispersion energy coefficients and to atomic polarizabilities is therefore useful. Although the expressions for D, obtained with the model are not as accurate as those given by the best approximation developed by Byers Brown and Whisnant [8,9], the model used here has a clear physical interpretation. At short range where overlap IS Important, there are exchange [IO] and orbital Qstortion effects on the pair dipoles, quadrupoles, and hyperpolanzablhties which are not mcluded m the model. At the conventional collision 336
Volume 70,
CHEMKAL
2
I
LETIERS
i980
drameter for inert-gas heterodiatoms,the overlap dipoles are somewhat larger than the dispersion dipoles and are generally opposed in sign [9], so the model is not directly useful in calculating collision-induced absorption spec-
tra. Nevertheless, there is contmued mterest in the long-range dipole coefficients
[l I].
2. Application of the model in calculating the long-range dipole of heteronuckar
atom pairs
The dipole mduced in a heteroatom pair by long-range mteractions may be related to the multipole polarizabilities of the isolated atoms. The fluctuating electric fields which cause the polarization are estimated from the coefficients C,, Cg . . . of the long-range dispersion energy [3,7]
udiiersio*(R)=
ne3 C2nR-2n_
(2)
The dipole mduced m an isolated atom by an electrrc field F, field-gradient F',and higher gradient F” may be wntten as a power senes [ 121 P?‘(F)
= QP~
+ &ap&&Fs
+T&~~~~,,&#&F&
* --- + %p&P&
+ :p&a~Fs~;sF&
+ -3)
Because of the random onentation of the fluctuatmg charge distrrbution of the neighboring atom which determines the field and field gradients, terms having an odd product of F and its gradients vanish when the mean induced di-
pole IS computed. pFd(F) = $ BaDraFpF& + & SLyprti+FxG
+ ..- .
(4)
For spherical atoms B IS isotropic, and if dupersion is negligrble B has only one independent
BNW = ;B[3(6,.&,
-t-6 c1&)
component:
- 2%Yp$ I -
4-Q
The B tensor gwes the energy of a system quadratic in an applied field FL and linear in the field-gradient F,i (F,‘, = -2F& = -28&J [ 121 Ezzzz = -$Bi$
F; .
(6)
The mean product of the field and field-gradient at atom A due to the randomly oriented charge distribution of atom B at distance R in the positrve z direction is (FpF.s)A
= [T&Qu
+$Tp,o(@Eo>8
+---I
[-T’~p(clp)B- fTr+JOpo)B- ---I = -$TkT,&
- --- 9
01
where Tap= V&V, (R-l), Tap7= VaV8Vv (R-l), . _. and BB and eB are the fluctuating multipole moments of a tom B. By changing the sign of R, which changes the signs of T tensors with an odd number of indices, and interchanging labels A and B, the mean product of the field and field-gradrent at atom B due to the fhrctuating multiis
poles of atom A may be evaluated directly from (7). Thus the sum of the mean induced dipole moments (pFd)A
* (A’:~)B = ; W&,,
)&-
@+j
),&
&T.se
-I-- - - -
(88)
Assging that the effective mean-square moments of an atoms the same rn homonuclear and heteronuclear pairs, f12 may be related to the homonuclear C, coefficrent (and 0 2 = O,O, may be related to C, and C,), as in the earher work, by taking Ut$ers,on
(R) = --&
- &,F&F&
- _. . ,
@I
where C is the quadrupole polarizabrlitfli 21, and 3 and F&F’ are &hemean-square field and field-gradient at one A atom due to the other. Relating F2 and F&F&e to f13 and%? A gives 337
;zA =
-C;A(2”&1
Srmphfymg the T iensor, (/,rp”)A
,
g;
= -(C,AA
(8) using (5)
- 5C;AC/(rA)(2c@’
the expression
= ~(C6uB,/aA
+ (#“d)B
1 March 1980
CHEMICAL PHYSICS LETTERS
Volume 70, number 2
for Ffrom
- C’tBBA/~g)Rm7
.
(10)
(13), and the explictt
expressrons
for the components
+ . .. .
of (11)
If (&“d)A + (~5~)~ IS posrtrve, the sign of the Induced drpole 1s A-B +. Clearly the drpole vanishes If atoms A and B are rdentrcal. The expressron for D, obtamed from (11) can be evaluated exactly for the case that atom B 1s a hydrogen atom and atom A IS a one-electron atom with a hydrogemc wavefunctron scaled by a factor J‘_The multrpole polanzabrlity tensors and drspersion-energy coefficrents for the scaled hydrogen atom A are related to those for the unscaled atom by [S] ,
C,AA = <-$j
“A = <-4a ,
BA = cm8B.
(12)
Thus D, = $(c,B/01)5-2(1
- c-6) _
(13)
For the H atom, C, = -6.499 au [13], and the exact values of the multrpole polanzabrlities are (Y= 5 and B =- y(m au) The resulting values of D, for 5 in the range 1 0 to 2 0 are m table 1, along wrth accurate perturbation results for D, obtained in a fimte basis set calculation [8] _The agreement between approximate and accurate results is good for 5 near 1.0, but poor for larger < values. The discrepancres between the accurate values for D, and those obtained from (13) may reflect drfferences between the effective mean-square dipoles for homonuclear H-H pairs and those for heteronuclear H-Hs. paus. Assummg that the effectE mean-ware drpole for the scaled H atom A m the pan IS always related to that for the unscaled H atom by & = c-2~2 , the C6 coefficients for H-H, interactions can be used to give pau-specrfic values of 7 for an H--H< pax 1.12 =
-c@-H~)~‘/cr(l
The effectrve
-
+ C-2) .
(14)
p2 for an H atom (14) decreases
monotonically
from 3
= 0 7221 at 5 = 1.0 to 7
= 0.4677
at
Table 1 The long-range dipole ioeffiaents 07 for a hydrogen atom mteractmg 1~1tha hydrogen-hke atom scaled by a factor 3‘. The sign of the dipole IS Ht*+
5‘
a) Ref. [8]
338
07 eq (13)
eq (15)
accurate a)
1.0 1.1 12 1.3 1.4 1.5 1.6 1.7 18 1.9
0 83.0 106 6 108.2 102.1 93 5 84 7 76 5 69.1 62.6
0 82.32 103 20 101.32 9 1.65 80 08 68 90 58.91 50 28 42 95
2.0
56.8
36.77
0 85.65 106 29 102 87 91.48 78 49 66.29 55 64 46 65 39.17 33 00
Volume 70, number 2 r = 2.0. From (8)
CHEMICAL PHYSICS LETTERS
(14)
and the assumed
D, = 3BC@-H&l-
S-6)/cu(l
1 March 1980
scaling relation,
+ r2),
cw
whrch can be evaluated given accurate values for the c6 coefficients from [8]_ As shown in table I, using (15) significantly improves agreement between approximate and accurate values for the D, coefficients of H-Hr pairs, when 5 is greater than 1.S_ Parr-specific values of p2 are also needed for the paus H-He and H-Ne to reduce the relative error in the model D7 coefficrents below 25%; however for the mert-gas heterodlatoms and for H-Ar, H-I&, and H-Xe pairs, the effective mean-square moments are very close to the homonuclear pair values (lo), as shown in section 5 below. Hence the expression for D, obtarned from (11) IS expected to be a good approximation for these species; and the only quanhty wluch remams to be determined accurately IS 6. Uncoupled Hartree-Fock calculatrons of B for the Inert-gas atoms [ 141 have been performed, but there are large uncertainties in the reported B values, znd these lead to even larger uncertamties in D7 values, smce D, ISfound as the difference of two terms. More reliable ab mltio calculations of B for Isolated atoms should be feasible at present; but B is known for ab initio calculation only for He [14,15]. Expenmental values of B could be obtamed from measurements of the brrefringence induced by apphcation of a field-gradient to mert-gas samples [16]. 3. Long-range
quadrupole
moments
atom pairs
of homonuclear
There are two contrrbutions to the quadrupole moment of a homonuclear atom pair, the first from the separated dipole moments induced in each atom, equal m magnitude but opposite in sign, and the second from the quadrupole moments induced 111each by interactlons. The mean interactlon-induced atomic quadrupole is emd
= 1
@Ed)*
= :B,,
== T_ T, ,y28 = -_(BCJ&Y)R-~
The total quadrupole
moment
_
of the parr IS
- R ($qA
+ R (p>B
OAB zz = (@~d)A
+ (@gd)S
on substituting
values for ,uyd from the prevrous
azy
;
section,
+ __ .
= ; (RC6/&)R-6
(19)
For H2 in the tnplet state, the long-range pair quadrupole [15] gves 0,. = 17.~5R-~. BHe = -6.587 4. Long-range
(W
hyperpolarizabilities
of homonuclear
is O,,
inert-gas
= 384SR-6
in thus model;
for He, the value
atom pairs
The classrcal contribution to the pair hyperpolarizability comes from the dipole induced in each atom by the hyperpolarized &pole and higher multipoles on the other. Taking F m eq. (3) to be the sum of the external field and the field due to the moments induced in the neighboring molecule by the external field, and solving the equatrons self-consistently for the induced atomic dipoles cubic in the external field gives the c&zsic~~ contnbution to the pair hyperpolanzabrlity Ay A7zzzr Akcxx
= l&-/R-3
+ 32aQR-5
= -8ol-/R-3
A-yxxzz = $-yR-3
+ 12aQR-5 -
- 36B2R-5 - y@R-5
~~I-YQR-~ + 18B2R-5
+ ~OCX~~R-~ + . . . , + 20&R-6 +!&x2rR-6
+ ... , + _._ ,
(204
Wb) (2Q4 339
Volume
70, number
2
CHEMICAL
where the Isolated-atom
hyperpolanzabrlity
tensor
PHYSICS LETTERS
1 March 1980
IS (21)
The effect of correlatron the Y hyperpolarizabihty +@FR-6,
A&Z &;xx_r A&Z
is to modify the Intrinsic hyperpolartzabrlity via the *tensor [eq. (3)]. The change m due to the mean-square field of the fluctuating dipole of a neighbormg molecule IS GW
= %&2-6 Z
.
Wb)
=%&FR-~,
(23~)
where Q = @ZZzrzz. Addmg (20a)-(20c) tributron to the hyperpolarrzablhty AT = ~~~~~~~ = (+
to (23a)-(23c),
cy2y - $#C6cr1)R-6
using (10) for2
and contracting
to give the mean pair con-
+ ... .
(24)
Thus the R dependence of AT for atoms at long range 1s the same as that of the square of the pau polarizabihty anisotropy. Consequently the contrrbutron to the second Kerr virial coefficrent BK from the pair hyperpolanzabrhty could be srgnrficant. For the H atom, cr = ;, .y = 10665/8, $J = 112346055/32 [17] and C, = -6.499 [13]; for atoms wtth these same relattve values, BK (at T = 300 K) would be doubled by including the pair hyperpolarizabrlity. However, for Inert gas paus the classrcal pau hyperpolarizabrhty (Fa2rRw6) modifies BK by less than 5% [ 181, and values of the @ tensor are not known.
5. Test of the model for effective mean-square dipoles and quadrupoles hi the model, the c6 and cs coefficients poles and quadrupoles by c6 AB- - -A&$ c8 AB
2
-
&+A2
= -CYA@B 2 -“BOA2
340
pairs are related
to the effective
mean-square
di-
(25)
, - 5cA@ 2
Table 2 Values of the effectrve mean-square were obtaiid from the model (10) (AJS) and O* (AB) were determmed The ground-state expectatron values
He Ne AI Kr Xe
for heteronuclear
- SC,&2
dipoles 7 and values from (25) (01 rr* 10)
(26)
.
and quadrupoles 3 for the mert-gas atoms The quantrtres 2 (AA) and 2 (AA) of the homonuclear drspersion energy coefficrents Cp and C@ 119 J, while 2 and (26) by werghted least-squares fit of ail the pair dispersron energy coefficients. [9] are also grven for comparrson. All values are grven IIIatonuc umts
&AA)
ll(AB)
tolLI* IO)
&AA)
3
(AB)
0533+0004 129 %008 3 03 f 0.16 3.98 f 0 27 4.78 f 0.15
0533 1.17 2.96 3 88 4.50
2 37 6.08 16 42 23 63 -
272+007 6502 383 33.8 f 15.4 48.3 f 22.2 69 6 f 29.7
2.74 6.71 33.1 46.4 66.7
CHEMICAL
Volume 70, number 2
PHYSICS
LETTERS
L Mar&
II980
The utility of the model for heteronuclear pa& depends upon p and 02 for atom A being independent of the nei&boring atom B. For an H atom interacting wi”h scaled H atoms, this is not the case, and pair-g@cific values of ~2 are required in cakulatmg the D7 coefficients. Fortunately, the effective moments cc2 and 6@ for inert gas-atoms seem to be nearly the same 111homonucilear or heteronuclear pairs, as shown by tables 2,3, and 4Values of pz and SA determined from the homonuclear dispersion energy coefficients CGM and Cg$ using (10) are given in table 2 for comparison with values obtamed by a weighted ieast-squares fit to the C, and C, coefEcients of ah the inert-gas pairs using (25) and (26). The quantihes Ce, C,, or, and C were taken from the work of Table 3 Dupersron energy coefticrents Ce for inert-gas atom pairs as obtained wrth the optimized mean-square dipoles p2 are given first. and the most accurate values reported by Tang et al. [ 19 ] are grven immediately below, with error bounds_ The agreement can be improved by excluding from the least-squaresfit Ce values for ah pairs with xenon, as shown by resultsgiven in parentheses. AK values are m atomrc umts
Ne
He
Kr
Ar
He
-1.46 -1.47 + 0 01 (-1.47)
Ne
-3.02 -3.13 f 0.08 (-3.12)
Ar
-9.97 -9 82 f 0.35 (--IO 00)
-20.8 -20 7 f 1.3 (-21.6)
-65.6 -67.2 f 3.6 (-65.9)
Kr
- 14.2 -13 6 + 0.6 (-14.1)
-29.8 -28 7 * 2.1 (-30.7)
-92.4 -94.3 * 5.7 (-918)
Xe
-20.7 -18 3 t 0.6
Xe
-6.20 -6.87 * 0 40 (-6.56)
-43.8 -37.8
-_I30 -133 * 9 (-127)
-131 -129 rt 5
r 2.0
-181 -184 = 9
-246 -2615
8
Table 4
Drspersron energy coefficients Ca for inert-gas atom paus as obtained with the oprimlzed mean-squaredipoIes 3 and quadrupofes ?? are given above the most accurate values (with error bounds) reported by Tang et aL 1191. Al1 values are in atomicunits
5 AB He
Ne
He
-14.02 - 14.02 it 0 20
Ne
-32.2 -32.7
Ar Kr Xe
+ 4.5
Ar
Kr
-73.2 -76 0 f 20.4
-158 -153.5
t 27.1
-351 -344
i 89
-1450 -1480
r 342
-238 -226.4
i 43.7
-528 -504
t 132
-2120 -2170
* 510
f 78.4
-807 -744
t 213
-3 140 -3170 f 770
-364 -336.6
Xe
-3080 -3180
* 740
4510 4670
rt 1110
-6520 -6860
f f620
CHEMICAL
Volume 70. number 2
PHYSICS
LETTERS
1 March 1980
Table 5
Dispersion energy coefficrents c6 and ca for H atom-mertsas atom pairs as calculated untb the model [eqs (25) and (26)] g7 and 3 values deduced from homonuclcar dlsperslon energy coefficients are hsted along with accurate values [19] A
C6(H-A)
He Ne Ar Kr Xe
Cs(H-A)
model
accurate
model
accurate
-3 -7 -21 -30 -412
-282*001 -5.71 f 0.07 -20.0 f 0 3 -285 +06 -40 9 f 0 5
-47.4 -110 0 -439 -639 -937
-41.8 f 0 3 -919*54 -418 f 38 -621 f 64 -927 f 180
39 73 6 0
usmg
Tang et al. [ 191 with weights assigned according
to the square of the error &n.mds reported there; l.c2 values optrmized to fit the C, coeffictents were treated as fixed m findmg values of O2 from C,. Tables 3 and 4 grve the fitted values of C6 and C, together wrth the best theoretical values and show that the model is consistent to hrgh accuracy for the inert-gas atom pairs. In table 5, model values of the C6 and C, coefficients for H atom-inert gas atom pairs areg_len for comparrson with accurate values [ 191. The model values were obtained from (25) and (26) usmg lr2 and O2 calculated from homonuclear dispersion energy coefficients The agreement between model and accurate results IS good for HAr, H-Kr, and H-Xe, but not for H-He or H-Ne. It is mterestmg to compare the expectation values of p-z for the ground state of the inert-gas atoms with the effectrve mean-square drpoles (table 2), the effective mean-square dipole is only 20 to 25% of the expectation value, a result demonstrated earher for the hydrogen atom [_7]. Acknowledgement I would
like to thank
Professor
A.D. Buckmgham
for many
helpful
drscussrons
and for his Interest
m thus work.
References 111 2-J. KISS and H L Welsh, Phys Rev Letters 2 (1959) 166 [2j D-R Bosomworth and H P Gush, Can 5. Phys 43 (1965) 729,751 131 A.D. Buckingham, Trans Faraday Sot 52 (1956) 1035. [4J A D. Buckingham. Propru%s optlques et acoustlques des fluids cornprim& et actions mtermol6culaues (Centre National de h Recherche Sclentfique, Paris, 1959) p 57 [S] A.D Buckmgham and 3 A Poplc. Proc Phys Sot. (London) A68 (1955) 905 [6] A.D. Buckmgham and D A Dunmur, Trans. Faraday Sot 64 (1868) 1776. [7] A D. Buckmgham and K L. Clarke, Chem. Phys Letters 57 (1978) 321. IS] W. Byers Brown and D.M msnant, hlol Phys. 25 (1973) 1385 191 D M. Whnnant and W Byers Brown, Mol. Phys 26 (1973) 1105 [lo] A.J Lacey and W. Byers Brown, Mol. Phys 27 (1974) 1013. [ll] L-W. Bruch. C-T. Corcoran and F. Wemhold, Mol Phys 35 (1978) 1205; ; R M Berns, P.E S. Wormer, F Molder and A. van der Avond, J. Chem. Phys 69 (1978) 2102. [It] AD Buckmgham, Advan Chem. Phys 12 (1967) 107. 1131 L Pauhng and J-Y. Beach, Phys. Rev. 47 (1935) 686. j14] J D. Lyons, P-W. Langhoff and R.P. Hurst, Phys Rev. 151 (1966) 60. [IS] A K Bhattacharya and P.K. MukherJee, Intern. J. Quantum Chem. 7 (1973) 491. [16] A D. Buckmgham, J. Chem. Phys. 30 (1959) 1580. [17] C-L. SeweU, Proc. CambrIdge PhlL Sot. 45 (1949) 678. 1181 D.A. Dunmur and N.E. Jessup, MoL Phys. 37 (1979) 697, D.A. Dunmur, D.C. Hunt and NE. Jessup, MoL Phys. 37 (1979) 713. [19] K-T. Tang, J.M. Norbeck and P.R. Certam, J Chem. Phys. 64 i1976) 3063.
342
CHEMICAL
70. number 2
PHYSICS
1 March 1980
LETTERS
LONG-RANGE DIPOLES, QUADRUPOLES, AND HYl’ERPOLAlUZAEtILITXES OF INTERACTING K.L
INERT-GAS ATOMS
c. HUNT
Department
of Chemrrtry. Michigan State Umvewty.
Rec?elved 3 November
1979,
East Lansng. Mtchigan 48824,
UI final form 17 December
USA
1979
A model relatmg the long-range dipoles of inert-gas heterodlatoms to the distortion of the electromc structure assonated urith dlsperslon forces is presented. Expressions are pven for the contriiutlons to the dipole varymg as the utverse seventh power of the atornx separahon R. The model 1s ako used to determine the long-range quadrupoles of homonuclear atomic paus Pau hyperpolanzabihtIes are obtamed m the model by adding contriiutions of classical multipolar mteractions and contributrons associated WIUI the distortion of the electronic structure. An independent test shows the consistency of the model and provides values for the effectwe mean-square dipoles and quadrupoles of the inert-gas atoms.
1. Introduction Due to the distortion of electronic structure which occurs during a colhslon of Inert-gas atoms, a heteronuclear pair has a nonvanishmg dipole moment, which IS mamfested in far-infrared absorption by Inert-gas mixtures 11,2] At long range, the electronic distortion may be related to the dispersion forces by a model proposed by Buckmgham for calcula~~g pair polanzablhtres of spherical atoms [3]. The lowest-order contrrbution to the induced dipole , where R ISthe internuclear separation along the t axis [4] varies as R rq =D7R-7
+D,R-’
+ ... .
(1)
The Interaction of a homonuclear atom pair gves a nonvamshmg quadrupole, wtuch vatles as RM6 at Iong range. The interaction of two spherical atoms also affects the atomic hyperpolarizablhties y, thus contnbuting to the second Kerr virlal coefficient of inert-gas samples [5,6]. In calculating long-range pau hyperpolarizabihtles. classica! multipolar interactions must be taken into account, in addltlon to the distortion associated with dispersion forces. Interactions alter the pan hyperpolarizablhty tensor elements yzzzz, rzzXX, and r,,, by terms varying as Rm3, but the first nonvanishing change m the mean pan hyperpolarizabi lity LLT(AT = EAT,,& varies as R -6. The model used here has already been apphed to calculate the effects of long-range interactions on the trace 131 and amsotropy [7] of the polarizabllity of a pair of atoms. For H, and He*, the model was found to give good agreement with the accurate values for both parallel and perpendicular components of the polarizabllity varying as R-6, and It also gives good values for the contnbution to the mean pair polarizabMy of Hz varying as R-8.
and hyperpolanzaAccurate quantum mechanical calculation of the coefficients D7, Dg, and the quadrupole bility coefficients is difficult and may not be practicable for pairs heavier than Ne-Ar. A simple model relatmg these coefficients to the known dispersion energy coefficients and to atomic polarizabilities is therefore useful. Although the expressions for D, obtained with the model are not as accurate as those given by the best approximation developed by Byers Brown and Whisnant [8,9], the model used here has a clear physical interpretation. At short range where overlap IS Important, there are exchange [IO] and orbital Qstortion effects on the pair dipoles, quadrupoles, and hyperpolanzablhties which are not mcluded m the model. At the conventional collision 336
Volume 70,
CHEMKAL
2
I
LETIERS
i980
drameter for inert-gas heterodiatoms,the overlap dipoles are somewhat larger than the dispersion dipoles and are generally opposed in sign [9], so the model is not directly useful in calculating collision-induced absorption spec-
tra. Nevertheless, there is contmued mterest in the long-range dipole coefficients
[l I].
2. Application of the model in calculating the long-range dipole of heteronuckar
atom pairs
The dipole mduced in a heteroatom pair by long-range mteractions may be related to the multipole polarizabilities of the isolated atoms. The fluctuating electric fields which cause the polarization are estimated from the coefficients C,, Cg . . . of the long-range dispersion energy [3,7]
udiiersio*(R)=
ne3 C2nR-2n_
(2)
The dipole mduced m an isolated atom by an electrrc field F, field-gradient F',and higher gradient F” may be wntten as a power senes [ 121 P?‘(F)
= QP~
+ &ap&&Fs
+T&~~~~,,&#&F&
* --- + %p&P&
+ :p&a~Fs~;sF&
+ -3)
Because of the random onentation of the fluctuatmg charge distrrbution of the neighboring atom which determines the field and field gradients, terms having an odd product of F and its gradients vanish when the mean induced di-
pole IS computed. pFd(F) = $ BaDraFpF& + & SLyprti+FxG
+ ..- .
(4)
For spherical atoms B IS isotropic, and if dupersion is negligrble B has only one independent
BNW = ;B[3(6,.&,
-t-6 c1&)
component:
- 2%Yp$ I -
4-Q
The B tensor gwes the energy of a system quadratic in an applied field FL and linear in the field-gradient F,i (F,‘, = -2F& = -28&J [ 121 Ezzzz = -$Bi$
F; .
(6)
The mean product of the field and field-gradient at atom A due to the randomly oriented charge distribution of atom B at distance R in the positrve z direction is (FpF.s)A
= [T&Qu
+$Tp,o(@Eo>8
+---I
[-T’~p(clp)B- fTr+JOpo)B- ---I = -$TkT,&
- --- 9
01
where Tap= V&V, (R-l), Tap7= VaV8Vv (R-l), . _. and BB and eB are the fluctuating multipole moments of a tom B. By changing the sign of R, which changes the signs of T tensors with an odd number of indices, and interchanging labels A and B, the mean product of the field and field-gradrent at atom B due to the fhrctuating multiis
poles of atom A may be evaluated directly from (7). Thus the sum of the mean induced dipole moments (pFd)A
* (A’:~)B = ; W&,,
)&-
@+j
),&
&T.se
-I-- - - -
(88)
Assging that the effective mean-square moments of an atoms the same rn homonuclear and heteronuclear pairs, f12 may be related to the homonuclear C, coefficrent (and 0 2 = O,O, may be related to C, and C,), as in the earher work, by taking Ut$ers,on
(R) = --&
- &,F&F&
- _. . ,
@I
where C is the quadrupole polarizabrlitfli 21, and 3 and F&F’ are &hemean-square field and field-gradient at one A atom due to the other. Relating F2 and F&F&e to f13 and%? A gives 337
;zA =
-C;A(2”&1
Srmphfymg the T iensor, (/,rp”)A
,
g;
= -(C,AA
(8) using (5)
- 5C;AC/(rA)(2c@’
the expression
= ~(C6uB,/aA
+ (#“d)B
1 March 1980
CHEMICAL PHYSICS LETTERS
Volume 70, number 2
for Ffrom
- C’tBBA/~g)Rm7
.
(10)
(13), and the explictt
expressrons
for the components
+ . .. .
of (11)
If (&“d)A + (~5~)~ IS posrtrve, the sign of the Induced drpole 1s A-B +. Clearly the drpole vanishes If atoms A and B are rdentrcal. The expressron for D, obtamed from (11) can be evaluated exactly for the case that atom B 1s a hydrogen atom and atom A IS a one-electron atom with a hydrogemc wavefunctron scaled by a factor J‘_The multrpole polanzabrlity tensors and drspersion-energy coefficrents for the scaled hydrogen atom A are related to those for the unscaled atom by [S] ,
C,AA = <-$j
“A = <-4a ,
BA = cm8B.
(12)
Thus D, = $(c,B/01)5-2(1
- c-6) _
(13)
For the H atom, C, = -6.499 au [13], and the exact values of the multrpole polanzabrlities are (Y= 5 and B =- y(m au) The resulting values of D, for 5 in the range 1 0 to 2 0 are m table 1, along wrth accurate perturbation results for D, obtained in a fimte basis set calculation [8] _The agreement between approximate and accurate results is good for 5 near 1.0, but poor for larger < values. The discrepancres between the accurate values for D, and those obtained from (13) may reflect drfferences between the effective mean-square dipoles for homonuclear H-H pairs and those for heteronuclear H-Hs. paus. Assummg that the effectE mean-ware drpole for the scaled H atom A m the pan IS always related to that for the unscaled H atom by & = c-2~2 , the C6 coefficients for H-H, interactions can be used to give pau-specrfic values of 7 for an H--H< pax 1.12 =
-c@-H~)~‘/cr(l
The effectrve
-
+ C-2) .
(14)
p2 for an H atom (14) decreases
monotonically
from 3
= 0 7221 at 5 = 1.0 to 7
= 0.4677
at
Table 1 The long-range dipole ioeffiaents 07 for a hydrogen atom mteractmg 1~1tha hydrogen-hke atom scaled by a factor 3‘. The sign of the dipole IS Ht*+
5‘
a) Ref. [8]
338
07 eq (13)
eq (15)
accurate a)
1.0 1.1 12 1.3 1.4 1.5 1.6 1.7 18 1.9
0 83.0 106 6 108.2 102.1 93 5 84 7 76 5 69.1 62.6
0 82.32 103 20 101.32 9 1.65 80 08 68 90 58.91 50 28 42 95
2.0
56.8
36.77
0 85.65 106 29 102 87 91.48 78 49 66.29 55 64 46 65 39.17 33 00
Volume 70, number 2 r = 2.0. From (8)
CHEMICAL PHYSICS LETTERS
(14)
and the assumed
D, = 3BC@-H&l-
S-6)/cu(l
1 March 1980
scaling relation,
+ r2),
cw
whrch can be evaluated given accurate values for the c6 coefficients from [8]_ As shown in table I, using (15) significantly improves agreement between approximate and accurate values for the D, coefficients of H-Hr pairs, when 5 is greater than 1.S_ Parr-specific values of p2 are also needed for the paus H-He and H-Ne to reduce the relative error in the model D7 coefficrents below 25%; however for the mert-gas heterodlatoms and for H-Ar, H-I&, and H-Xe pairs, the effective mean-square moments are very close to the homonuclear pair values (lo), as shown in section 5 below. Hence the expression for D, obtarned from (11) IS expected to be a good approximation for these species; and the only quanhty wluch remams to be determined accurately IS 6. Uncoupled Hartree-Fock calculatrons of B for the Inert-gas atoms [ 141 have been performed, but there are large uncertainties in the reported B values, znd these lead to even larger uncertamties in D7 values, smce D, ISfound as the difference of two terms. More reliable ab mltio calculations of B for Isolated atoms should be feasible at present; but B is known for ab initio calculation only for He [14,15]. Expenmental values of B could be obtamed from measurements of the brrefringence induced by apphcation of a field-gradient to mert-gas samples [16]. 3. Long-range
quadrupole
moments
atom pairs
of homonuclear
There are two contrrbutions to the quadrupole moment of a homonuclear atom pair, the first from the separated dipole moments induced in each atom, equal m magnitude but opposite in sign, and the second from the quadrupole moments induced 111each by interactlons. The mean interactlon-induced atomic quadrupole is emd
= 1
@Ed)*
= :B,,
== T_ T, ,y28 = -_(BCJ&Y)R-~
The total quadrupole
moment
_
of the parr IS
- R ($qA
+ R (p>B
OAB zz = (@~d)A
+ (@gd)S
on substituting
values for ,uyd from the prevrous
azy
;
section,
+ __ .
= ; (RC6/&)R-6
(19)
For H2 in the tnplet state, the long-range pair quadrupole [15] gves 0,. = 17.~5R-~. BHe = -6.587 4. Long-range
(W
hyperpolarizabilities
of homonuclear
is O,,
inert-gas
= 384SR-6
in thus model;
for He, the value
atom pairs
The classrcal contribution to the pair hyperpolarizability comes from the dipole induced in each atom by the hyperpolarized &pole and higher multipoles on the other. Taking F m eq. (3) to be the sum of the external field and the field due to the moments induced in the neighboring molecule by the external field, and solving the equatrons self-consistently for the induced atomic dipoles cubic in the external field gives the c&zsic~~ contnbution to the pair hyperpolanzabrlity Ay A7zzzr Akcxx
= l&-/R-3
+ 32aQR-5
= -8ol-/R-3
A-yxxzz = $-yR-3
+ 12aQR-5 -
- 36B2R-5 - y@R-5
~~I-YQR-~ + 18B2R-5
+ ~OCX~~R-~ + . . . , + 20&R-6 +!&x2rR-6
+ ... , + _._ ,
(204
Wb) (2Q4 339
Volume
70, number
2
CHEMICAL
where the Isolated-atom
hyperpolanzabrlity
tensor
PHYSICS LETTERS
1 March 1980
IS (21)
The effect of correlatron the Y hyperpolarizabihty +@FR-6,
A&Z &;xx_r A&Z
is to modify the Intrinsic hyperpolartzabrlity via the *tensor [eq. (3)]. The change m due to the mean-square field of the fluctuating dipole of a neighbormg molecule IS GW
= %&2-6 Z
.
Wb)
=%&FR-~,
(23~)
where Q = @ZZzrzz. Addmg (20a)-(20c) tributron to the hyperpolarrzablhty AT = ~~~~~~~ = (+
to (23a)-(23c),
cy2y - $#C6cr1)R-6
using (10) for2
and contracting
to give the mean pair con-
+ ... .
(24)
Thus the R dependence of AT for atoms at long range 1s the same as that of the square of the pau polarizabihty anisotropy. Consequently the contrrbutron to the second Kerr virial coefficrent BK from the pair hyperpolanzabrhty could be srgnrficant. For the H atom, cr = ;, .y = 10665/8, $J = 112346055/32 [17] and C, = -6.499 [13]; for atoms wtth these same relattve values, BK (at T = 300 K) would be doubled by including the pair hyperpolarizabrlity. However, for Inert gas paus the classrcal pau hyperpolarizabrhty (Fa2rRw6) modifies BK by less than 5% [ 181, and values of the @ tensor are not known.
5. Test of the model for effective mean-square dipoles and quadrupoles hi the model, the c6 and cs coefficients poles and quadrupoles by c6 AB- - -A&$ c8 AB
2
-
&+A2
= -CYA@B 2 -“BOA2
340
pairs are related
to the effective
mean-square
di-
(25)
, - 5cA@ 2
Table 2 Values of the effectrve mean-square were obtaiid from the model (10) (AJS) and O* (AB) were determmed The ground-state expectatron values
He Ne AI Kr Xe
for heteronuclear
- SC,&2
dipoles 7 and values from (25) (01 rr* 10)
(26)
.
and quadrupoles 3 for the mert-gas atoms The quantrtres 2 (AA) and 2 (AA) of the homonuclear drspersion energy coefficrents Cp and C@ 119 J, while 2 and (26) by werghted least-squares fit of ail the pair dispersron energy coefficients. [9] are also grven for comparrson. All values are grven IIIatonuc umts
&AA)
ll(AB)
tolLI* IO)
&AA)
3
(AB)
0533+0004 129 %008 3 03 f 0.16 3.98 f 0 27 4.78 f 0.15
0533 1.17 2.96 3 88 4.50
2 37 6.08 16 42 23 63 -
272+007 6502 383 33.8 f 15.4 48.3 f 22.2 69 6 f 29.7
2.74 6.71 33.1 46.4 66.7
CHEMICAL
Volume 70, number 2
PHYSICS
LETTERS
L Mar&
II980
The utility of the model for heteronuclear pa& depends upon p and 02 for atom A being independent of the nei&boring atom B. For an H atom interacting wi”h scaled H atoms, this is not the case, and pair-g@cific values of ~2 are required in cakulatmg the D7 coefficients. Fortunately, the effective moments cc2 and 6@ for inert gas-atoms seem to be nearly the same 111homonucilear or heteronuclear pairs, as shown by tables 2,3, and 4Values of pz and SA determined from the homonuclear dispersion energy coefficients CGM and Cg$ using (10) are given in table 2 for comparison with values obtamed by a weighted ieast-squares fit to the C, and C, coefEcients of ah the inert-gas pairs using (25) and (26). The quantihes Ce, C,, or, and C were taken from the work of Table 3 Dupersron energy coefticrents Ce for inert-gas atom pairs as obtained wrth the optimized mean-square dipoles p2 are given first. and the most accurate values reported by Tang et al. [ 19 ] are grven immediately below, with error bounds_ The agreement can be improved by excluding from the least-squaresfit Ce values for ah pairs with xenon, as shown by resultsgiven in parentheses. AK values are m atomrc umts
Ne
He
Kr
Ar
He
-1.46 -1.47 + 0 01 (-1.47)
Ne
-3.02 -3.13 f 0.08 (-3.12)
Ar
-9.97 -9 82 f 0.35 (--IO 00)
-20.8 -20 7 f 1.3 (-21.6)
-65.6 -67.2 f 3.6 (-65.9)
Kr
- 14.2 -13 6 + 0.6 (-14.1)
-29.8 -28 7 * 2.1 (-30.7)
-92.4 -94.3 * 5.7 (-918)
Xe
-20.7 -18 3 t 0.6
Xe
-6.20 -6.87 * 0 40 (-6.56)
-43.8 -37.8
-_I30 -133 * 9 (-127)
-131 -129 rt 5
r 2.0
-181 -184 = 9
-246 -2615
8
Table 4
Drspersron energy coefficients Ca for inert-gas atom paus as obtained with the oprimlzed mean-squaredipoIes 3 and quadrupofes ?? are given above the most accurate values (with error bounds) reported by Tang et aL 1191. Al1 values are in atomicunits
5 AB He
Ne
He
-14.02 - 14.02 it 0 20
Ne
-32.2 -32.7
Ar Kr Xe
+ 4.5
Ar
Kr
-73.2 -76 0 f 20.4
-158 -153.5
t 27.1
-351 -344
i 89
-1450 -1480
r 342
-238 -226.4
i 43.7
-528 -504
t 132
-2120 -2170
* 510
f 78.4
-807 -744
t 213
-3 140 -3170 f 770
-364 -336.6
Xe
-3080 -3180
* 740
4510 4670
rt 1110
-6520 -6860
f f620
CHEMICAL
Volume 70. number 2
PHYSICS
LETTERS
1 March 1980
Table 5
Dispersion energy coefficrents c6 and ca for H atom-mertsas atom pairs as calculated untb the model [eqs (25) and (26)] g7 and 3 values deduced from homonuclcar dlsperslon energy coefficients are hsted along with accurate values [19] A
C6(H-A)
He Ne Ar Kr Xe
Cs(H-A)
model
accurate
model
accurate
-3 -7 -21 -30 -412
-282*001 -5.71 f 0.07 -20.0 f 0 3 -285 +06 -40 9 f 0 5
-47.4 -110 0 -439 -639 -937
-41.8 f 0 3 -919*54 -418 f 38 -621 f 64 -927 f 180
39 73 6 0
usmg
Tang et al. [ 191 with weights assigned according
to the square of the error &n.mds reported there; l.c2 values optrmized to fit the C, coeffictents were treated as fixed m findmg values of O2 from C,. Tables 3 and 4 grve the fitted values of C6 and C, together wrth the best theoretical values and show that the model is consistent to hrgh accuracy for the inert-gas atom pairs. In table 5, model values of the C6 and C, coefficients for H atom-inert gas atom pairs areg_len for comparrson with accurate values [ 191. The model values were obtained from (25) and (26) usmg lr2 and O2 calculated from homonuclear dispersion energy coefficients The agreement between model and accurate results IS good for HAr, H-Kr, and H-Xe, but not for H-He or H-Ne. It is mterestmg to compare the expectation values of p-z for the ground state of the inert-gas atoms with the effectrve mean-square drpoles (table 2), the effective mean-square dipole is only 20 to 25% of the expectation value, a result demonstrated earher for the hydrogen atom [_7]. Acknowledgement I would
like to thank
Professor
A.D. Buckmgham
for many
helpful
drscussrons
and for his Interest
m thus work.
References 111 2-J. KISS and H L Welsh, Phys Rev Letters 2 (1959) 166 [2j D-R Bosomworth and H P Gush, Can 5. Phys 43 (1965) 729,751 131 A.D. Buckingham, Trans Faraday Sot 52 (1956) 1035. [4J A D. Buckingham. Propru%s optlques et acoustlques des fluids cornprim& et actions mtermol6culaues (Centre National de h Recherche Sclentfique, Paris, 1959) p 57 [S] A.D Buckmgham and 3 A Poplc. Proc Phys Sot. (London) A68 (1955) 905 [6] A.D. Buckmgham and D A Dunmur, Trans. Faraday Sot 64 (1868) 1776. [7] A D. Buckmgham and K L. Clarke, Chem. Phys Letters 57 (1978) 321. IS] W. Byers Brown and D.M msnant, hlol Phys. 25 (1973) 1385 191 D M. Whnnant and W Byers Brown, Mol. Phys 26 (1973) 1105 [lo] A.J Lacey and W. Byers Brown, Mol. Phys 27 (1974) 1013. [ll] L-W. Bruch. C-T. Corcoran and F. Wemhold, Mol Phys 35 (1978) 1205; ; R M Berns, P.E S. Wormer, F Molder and A. van der Avond, J. Chem. Phys 69 (1978) 2102. [It] AD Buckmgham, Advan Chem. Phys 12 (1967) 107. 1131 L Pauhng and J-Y. Beach, Phys. Rev. 47 (1935) 686. j14] J D. Lyons, P-W. Langhoff and R.P. Hurst, Phys Rev. 151 (1966) 60. [IS] A K Bhattacharya and P.K. MukherJee, Intern. J. Quantum Chem. 7 (1973) 491. [16] A D. Buckmgham, J. Chem. Phys. 30 (1959) 1580. [17] C-L. SeweU, Proc. CambrIdge PhlL Sot. 45 (1949) 678. 1181 D.A. Dunmur and N.E. Jessup, MoL Phys. 37 (1979) 697, D.A. Dunmur, D.C. Hunt and NE. Jessup, MoL Phys. 37 (1979) 713. [19] K-T. Tang, J.M. Norbeck and P.R. Certam, J Chem. Phys. 64 i1976) 3063.
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