Collision-induced light scattering spectra of the inert gases and hydrogen at high densities provide evidence of nonadditive three-body interaction effects, for which a quantitative theory is needed. In this work, we derive and evaluate the three-body polarizability Δα(3) for interacting molecules with negligible electronic overlap. Our results, based on nonlocal response theory, account for dipole-induced-dipole (DID) interactions, quadrupolar induction, dispersion, and concerted induction-dispersion effects. The contribution of leading order comes from a DID term that scales as α3d−6 in the molecular polarizability α and a representative distance d between the molecules in a cluster. Quadrupolar induction effects are also large, however, ranging from ∼35% to 104% of the leading DID terms for equilateral triangular configurations of the species studied in this work, at separations approximately 1 a.u. beyond the van der Waals minima in the isotropic pair potentials. For the same configurations, the dispersion terms range from 2% to 7% of the total Δᾱ(3). The dispersion and induction-dispersion contributions are derived analytically in terms of integrals over imaginary frequency, with integrands containing the polarizability α(iω) and the γ hyperpolarizability. For H, He, and H2, the integrals have been evaluated accurately by 64-point Gauss–Legendre quadrature; for heavier species, we have developed approximations in terms of static polarizabilities, static hyperpolarizabilities, and van der Waals interaction energy coefficients (C6 and C9). In the isotropic interaction-induced polarizability Δᾱ, the three-body terms are comparable in magnitude to the two-body terms, due to a cancellation of the first-order, two-body DID contributions to Δᾱ. For the heavier species in this work (Ar, Kr, Xe, N2,CH4, and CO2) in the configurations studied, the three-body contributions to Δᾱ range from −7 to −9% of the two-body terms for equilateral triangular arrays and from 35% to 47% of the two-body terms for linear, centrosymmetric systems.

1.
S.
Pestelli
,
U.
Bafile
,
L.
Ulivi
, and
M.
Zoppi
,
Phys. Rev. A
49
,
4602
(
1994
).
2.
F.
Barocchi
,
M.
Celli
, and
M.
Zoppi
,
Phys. Rev. A
38
,
3984
(
1988
).
3.
J.
van der Elsken
and
R. A.
Huijts
,
J. Chem. Phys.
88
,
3007
(
1988
).
4.
U.
Bafile
,
L.
Ulivi
,
M.
Zoppi
, and
F.
Barocchi
,
Chem. Phys. Lett.
138
,
559
(
1987
).
5.
U.
Bafile
,
L.
Ulivi
,
M.
Zoppi
,
M.
Moraldi
, and
L.
Frommhold
,
Phys. Rev. A
44
,
4450
(
1991
).
6.
M.
Moraldi
,
M.
Celli
, and
F.
Barocchi
,
Phys. Rev. A
40
,
1116
(
1989
).
7.
B. M.
Ladanyi
,
A.
Barreau
, and
B.
Dumon
,
Mol. Phys.
77
,
735
(
1992
).
8.
U.
Bafile
,
L.
Ulivi
,
M.
Zoppi
, and
S.
Pestelli
,
Mol. Phys.
79
,
179
(
1993
).
9.
P. R.
Certain
and
P. J.
Fortune
,
J. Chem. Phys.
55
,
5818
(
1971
).
10.
A. D.
Buckingham
,
P. H.
Martin
, and
R. S.
Watts
,
Chem. Phys. Lett.
21
,
186
(
1973
).
11.
A. D.
Buckingham
and
K. L.
Clarke
,
Chem. Phys. Lett.
57
,
321
(
1978
).
12.
K. L. C.
Hunt
,
B. A.
Zilles
, and
J. E.
Bohr
,
J. Chem. Phys.
75
,
3079
(
1981
).
13.
P. D.
Dacre
,
Mol. Phys.
47
,
193
(
1982
);
P. D.
Dacre
,
Can. J. Phys.
60
,
913
(
1982
);
P. D.
Dacre
and
L.
Frommhold
,
J. Chem. Phys.
76
,
3447
(
1982
).
14.
P. W.
Fowler
,
K. L. C.
Hunt
,
H. M.
Kelly
, and
A. J.
Sadlej
,
J. Chem. Phys.
100
,
2932
(
1994
).
15.
D. M.
Bishop
and
M.
Dupuis
,
Mol. Phys.
88
,
887
(
1996
).
16.
R.
Moszynski
,
T. G. A.
Heijmen
,
P. E. S.
Wormer
, and
A.
van der Avoird
,
J. Chem. Phys.
104
,
6997
(
1996
);
T. G. A.
Heijmen
,
R.
Moszynski
,
P. E. S.
Wormer
, and
A.
van der Avoird
,
Mol. Phys.
89
,
81
(
1996
);
R.
Moszynski
,
T. G. A.
Heijmen
,
P. E. S.
Wormer
, and
A.
van der Avoird
,
Adv. Quantum Chem.
28
,
119
(
1997
).
17.
B.
Fernandez
,
C.
Hättig
,
H.
Koch
, and
A.
Rizzo
,
J. Chem. Phys.
110
,
2872
(
1999
).
18.
J. J.
Pérez
,
J. H. R.
Clarke
, and
A.
Hinchliffe
,
Chem. Phys. Lett.
104
,
583
(
1984
).
19.
J.
Waite
and
M. G.
Papadopoulos
,
Theor. Chim. Acta
75
,
53
(
1989
).
20.
A. D.
Buckingham
and
I. D.
Hands
,
Chem. Phys. Lett.
185
,
544
(
1991
).
21.
C.
Jarque
and
A. D.
Buckingham
,
J. Chem. Soc., Faraday Trans.
88
,
1353
(
1992
).
22.
C.
Aleman
,
J. J.
Pérez
, and
A.
Hinchliffe
,
Int. J. Mass Spectrom. Ion Processes
122
,
331
(
1992
).
23.
P. A.
Egelstaff
and
A.
Teitsma
,
Phys. Rev. Lett.
43
,
503
(
1979
);
A.
Teitsma
and
P. A.
Egelstaff
,
Phys. Rev. A
21
,
367
(
1980
);
P. A.
Egelstaff
,
Can. J. Chem.
66
,
598
(
1988
);
J. J.
Salacuse
and
P. A.
Egelstaff
,
Phys. Rev. A
38
,
4313
(
1988
);
P. A.
Egelstaff
,
Phys. Scr.
T29
,
288
(
1989
).
24.
W. J.
Meath
and
R. A.
Aziz
,
Mol. Phys.
52
,
225
(
1984
);
W. J.
Meath
and
M.
Koulis
,
J. Mol. Struct.: THEOCHEM
72
,
1
(
1991
).
25.
J. A.
Barker
,
Phys. Rev. Lett.
57
,
230
(
1986
).
26.
L.
Reatto
and
M.
Tau
,
J. Chem. Phys.
86
,
6474
(
1987
);
A.
Meroni
,
L.
Reatto
, and
M.
Tau
,
Mol. Phys.
80
,
977
(
1993
).
27.
D.
Levesque
,
J. J.
Weis
, and
J.
Vermesse
,
Phys. Rev. A
37
,
918
(
1988
);
D.
Levesque
and
J. J.
Weis
,
Phys. Rev. A
37
,
3967
(
1988
).
28.
J.
Cao
and
B. J.
Berne
,
J. Chem. Phys.
97
,
8628
(
1992
).
29.
C. A.
Parish
and
C. E.
Dykstra
,
J. Chem. Phys.
98
,
437
(
1993
).
30.
M. J.
Elrod
,
D. W.
Steyert
, and
R. J.
Saykally
,
J. Chem. Phys.
94
,
58
(
1991
);
M. J.
Elrod
,
D. W.
Steyert
, and
R. J.
Saykally
,
J. Chem. Phys.
95
,
3182
(
1991
);
N.
Pugliano
and
R. J.
Saykally
,
Science
257
,
1937
(
1992
);
M. J.
Elrod
and
R. J.
Saykally
,
Chem. Rev.
94
,
1975
(
1994
);
M. J.
Elrod
,
J. G.
Loeser
, and
R. J.
Saykally
,
J. Chem. Phys.
98
,
5352
(
1993
);
M. J.
Elrod
,
R. J.
Saykally
,
A. R.
Cooper
, and
J. M.
Hutson
,
Mol. Phys.
81
,
579
(
1994
).
31.
J. M.
Hutson
,
J. A.
Beswick
, and
N.
Halberstadt
,
J. Chem. Phys.
90
,
1337
(
1989
);
A. R.
Cooper
,
S.
Jain
, and
J. M.
Hutson
,
J. Chem. Phys.
98
,
2160
(
1993
);
A.
Ernesti
and
J. M.
Hutson
,
Faraday Discuss.
97
,
119
(
1994
).
32.
R.
Moszynski
,
P. E. S.
Wormer
,
B.
Jeziorski
, and
A.
van der Avoird
,
J. Chem. Phys.
103
,
8058
(
1995
);
T.
Korona
,
R.
Moszynski
, and
B.
Jeziorski
,
J. Chem. Phys.
105
,
8178
(
1996
);
V. F.
Lotrich
and
K.
Szalewicz
,
J. Chem. Phys.
106
,
9668
(
1997
);
V. F.
Lotrich
and
K.
Szalewicz
,
J. Chem. Phys.
106
,
9688
(
1997
);
V. F.
Lotrich
and
K.
Szalewicz
,
Phys. Rev. Lett.
79
,
1301
(
1997
);
R.
Moszynski
,
P. E. S.
Wormer
,
T. G. A.
Heijmen
, and
A. D.
van der Avoird
,
J. Chem. Phys.
108
,
579
(
1998
);
V. F.
Lotrich
,
P.
Jankowski
, and
K.
Szalewicz
,
J. Chem. Phys.
108
,
4725
(
1998
);
E. M.
Mas
,
V. F.
Lotrich
, and
K.
Szalewicz
,
J. Chem. Phys.
110
,
6694
(
1999
).
33.
B.
Guillot
,
S.
Bratos
, and
G.
Birnbaum
,
Phys. Rev. A
22
,
2230
(
1980
).
34.
G.
Tarjus
,
D.
Kivelson
, and
V.
Friedrich
,
J. Mol. Struct.
223
,
253
(
1990
).
35.
S. M.
El Sheik
,
G. C.
Tabisz
, and
L.
Ulivi
,
Mol. Phys.
72
,
345
(
1991
).
36.
F.
Barocchi
,
L.
Carraresi
, and
M.
Celli
,
Phys. Rev. A
46
,
R3598
(
1992
).
37.
H.
Stassen
and
W. A.
Steele
,
J. Chem. Phys.
103
,
4408
(
1995
).
38.
L.
Ulivi
,
N.
Meinander
, and
F.
Barocchi
,
Phys. Rev. Lett.
75
,
3094
(
1995
).
39.
V.
Teboul
and
Y.
Le Duff
,
J. Chem. Phys.
107
,
10415
(
1997
);
V.
Teboul
,
Mol. Phys.
92
,
127
(
1997
);
V.
Teboul
,
Mol. Phys.
96
,
1637
(
1999
).
40.
U. Bafile, S. Pestelli, L. Ulivi, and M. Zoppi, in Collision- and Interaction-Induced Spectroscopy, NATO ASI Series C 452, edited by G. C. Tabisz and M. Neuman (Kluwer, Dordrecht, 1995), pp. 31–40.
41.
M.
Celli
,
L.
Carraresi
,
F.
Barocchi
, and
M.
Neumann
,
Europhys. Lett.
21
,
825
(
1993
).
42.
G.
Chalasiński
and
M. M.
Szczȩśniak
,
Mol. Phys.
63
,
205
(
1988
);
G.
Chalasiński
,
S. M.
Cybulski
,
M. M.
Szczȩśniak
, and
S.
Scheiner
,
J. Chem. Phys.
91
,
7048
(
1989
);
M. M.
Szczȩśniak
,
G.
Chalasiński
, and
P.
Pieuch
,
J. Chem. Phys.
99
,
6732
(
1993
);
G.
Chalasiński
and
M. M.
Szczȩśniak
,
Chem. Rev.
94
,
1723
(
1994
);
R.
Moszynski
,
S. M.
Cybulski
, and
G.
Chalasiński
,
J. Chem. Phys.
100
,
4998
(
1994
);
R.
Moszynski
,
B.
Jeziorski
,
S.
Rybak
,
K.
Szalewicz
, and
H. L.
Williams
,
J. Chem. Phys.
100
,
5080
(
1994
).
43.
R.
Magli
,
H.
Fredrikze
,
P.
Chieux
, and
F.
Barocchi
,
Europhys. Lett.
15
,
609
(
1991
).
44.
F.
Formisano
,
F.
Barocchi
, and
R.
Magli
,
Phys. Rev. E
58
,
2648
(
1998
);
F.
Formisano
,
F.
Barocchi
, and
R.
Magli
,
Physica B
241
,
958
(
1997
).
45.
J. M.
Bomont
,
N.
Jakse
, and
J. L.
Bretonnet
,
Phys. Rev. B
57
,
10217
(
1998
).
46.
See,
A.
DiCicco
,
A.
Filipponi
,
J. P.
Itie
, and
A.
Polian
,
Phys. Rev. B
54
,
9086
(
1996
), and references therein.
47.
B.
Jeziorski
and
W.
Kolos
,
Int. J. Quantum Chem.
(Suppl. 1)
12
,
91
(
1977
);
K.
Szalewicz
and
B.
Jeziorski
,
Mol. Phys.
38
,
191
(
1979
);
B.
Jeziorski
,
R.
Moszynski
, and
K.
Szalewicz
,
Chem. Rev.
94
,
1887
(
1994
).
48.
X.
Li
and
K. L. C.
Hunt
,
J. Chem. Phys.
105
,
4076
(
1996
).
49.
X.
Li
and
K. L. C.
Hunt
,
J. Chem. Phys.
107
,
4133
(
1997
).
50.
W. J. A.
Maaskant
and
L. J.
Oosterhoff
,
Mol. Phys.
8
,
319
(
1964
).
51.
T.
Keyes
and
B. M.
Ladanyi
,
Mol. Phys.
33
,
1271
(
1977
).
52.
J. E.
Sipe
and
J.
Van Kranendonk
,
Mol. Phys.
35
,
1579
(
1978
).
53.
K. L. C.
Hunt
,
J. Chem. Phys.
78
,
6149
(
1983
).
54.
K. L. C.
Hunt
,
J. Chem. Phys.
80
,
393
(
1984
).
55.
H.
Ishihara
and
K.
Cho
,
Phys. Rev. B
48
,
7960
(
1993
).
56.
J. K.
Jenkins
and
S.
Mukamel
,
J. Chem. Phys.
98
,
7046
(
1993
);
V.
Chernyak
and
S.
Mukamel
,
Phys. Rev. B
48
,
2470
(
1993
);
M.
Hartmann
,
V.
Chernyak
, and
S.
Mukamel
,
Phys. Rev. B
52
,
2528
(
1995
);
A.
Takahashi
and
S.
Mukamel
,
J. Chem. Phys.
100
,
2366
(
1994
);
S.
Mukamel
,
A.
Takahashi
,
H. X.
Wang
, and
G.
Chen
,
Science
266
,
250
(
1994
);
T.
Wagensreiter
and
S.
Mukamel
,
J. Chem. Phys.
104
,
7086
(
1996
).
57.
D. E.
Stogryn
,
Phys. Rev. Lett.
24
,
971
(
1970
);
D. E.
Stogryn
,
J. Chem. Phys.
52
,
3671
(
1970
);
D. E.
Stogryn
,
Mol. Phys.
22
,
81
(
1971
).
58.
P.
Piecuch
,
Chem. Phys. Lett.
110
,
496
(
1984
);
P.
Piecuch
,
Mol. Phys.
59
,
1067
(
1986
);
P.
Piecuch
,
Mol. Phys.
59
,
1085
(
1986
);
P.
Piecuch
,
Mol. Phys.
59
,
1097
(
1986
).
59.
B. J.
Orr
and
J. F.
Ward
,
Mol. Phys.
20
,
513
(
1971
).
60.
D. M.
Bishop
,
J. Chem. Phys.
100
,
6535
(
1994
).
61.
The integral of ααβA(r,r) over all space with respect to r and r gives the polarizability ααβA. For isotropic systems, ααβ=αδαβ. Moment integrals that contain ααβA(r,r) multiplied by products such as (r−RA)γ(r−RA)δ⋯(r−RA)ω or by (r−RA)γ(r−RA)δ⋯(r−RA)ω are related to the dipole-quadrupole polarizability AA, the dipole-octopole polarizability EA, or to the higher-multipole susceptibilities. All of these tensors (A, E, …) vanish identically for isotropic systems. Moment integrals that contain ααβA(r,r) multiplied by (r−RA)γand by (r−RA)δ are related to the quadrupole polarizability tensor CA, as shown in Ref. 54.
62.
A. D.
Buckingham
,
Adv. Chem. Phys.
12
,
107
(
1967
).
63.
H. B.
Callen
and
T. A.
Welton
,
Phys. Rev.
83
,
34
(
1951
).
64.
K. L. C.
Hunt
and
J. E.
Bohr
,
J. Chem. Phys.
83
,
5198
(
1985
).
65.
K. L. C.
Hunt
and
J. E.
Bohr
,
J. Chem. Phys.
84
,
6141
(
1986
).
66.
D. M.
Bishop
and
J.
Pipin
,
Chem. Phys. Lett.
236
,
15
(
1995
).
67.
I.
Waller
,
Z. Phys.
38
,
635
(
1926
).
68.
C. A.
Coulson
,
Proc. R. Soc. Edinburgh, Sect. A: Math. Phys. Sci.
61
,
20
(
1941
).
69.
A. D.
Buckingham
,
C. A.
Coulson
, and
J. T.
Lewis
,
Proc. Phys. Soc., London, Sect. A
69
,
639
(
1956
).
70.
G. L.
Sewell
,
Proc. Cambridge Philos. Soc.
45
,
678
(
1949
).
71.
D. M.
Bishop
and
S. M.
Cybulski
,
Chem. Phys. Lett.
211
,
255
(
1993
).
72.
A.
Kumar
and
W. J.
Meath
,
Can. J. Chem.
63
,
1616
(
1985
).
73.
G.
Maroulis
and
A. J.
Thakkar
,
J. Chem. Phys.
89
,
7320
(
1988
).
74.
D. M.
Bishop
and
J.
Pipin
,
Int. J. Quantum Chem.
45
,
349
(
1993
).
75.
D. M.
Bishop
and
J.
Pipin
,
J. Chem. Phys.
97
,
3375
(
1992
);
D. M.
Bishop
and
J.
Pipin
,
J. Chem. Phys.
99
,
4875
(
1993
).
76.
G.
Maroulis
and
A. J.
Thakkar
,
J. Chem. Phys.
88
,
7623
(
1988
);
α=11.61 a.u. was obtained by
S. R.
Langhoff
,
C. W.
Bauschlicher
, Jr.
, and
D. P.
Chong
,
J. Chem. Phys.
78
,
5287
(
1983
).
77.
G.
Maroulis
,
Chem. Phys. Lett.
226
,
420
(
1994
).
78.
G.
Maroulis
and
A. J.
Thakkar
,
J. Chem. Phys.
93
,
4164
(
1990
).
79.
A.
Dalgarno
,
Adv. Chem. Phys.
12
,
143
(
1967
).
80.
R. M.
Glover
and
F.
Weinhold
,
J. Chem. Phys.
66
,
191
(
1977
).
81.
L. J.
Bartolotti
and
J.
Tyrrell
,
Mol. Phys.
36
,
97
(
1978
).
82.
W.
Rijks
and
P. E. S.
Wormer
,
J. Chem. Phys.
88
,
5704
(
1988
);
W.
Rijks
and
P. E. S.
Wormer
,
J. Chem. Phys.
90
,
6507
(
1989
);
A. J.
Thakkar
,
H.
Hettema
, and
P. E. S.
Wormer
,
J. Chem. Phys.
97
,
3252
(
1992
);
P. E. S.
Wormer
and
H.
Hettema
,
J. Chem. Phys.
97
,
5592
(
1992
);
H.
Hettema
,
P. E. S.
Wormer
, and
A. J.
Thakkar
,
Mol. Phys.
80
,
533
(
1993
);
P. E. S.
Wormer
,
H.
Hettema
, and
A. J.
Thakkar
,
J. Chem. Phys.
98
,
7140
(
1993
);
H.
Hettema
,
P. E. S.
Wormer
,
P.
Jorgensen
,
H. J. A.
Jensen
, and
T.
Helgaker
,
J. Chem. Phys.
100
,
1297
(
1994
).
83.
B.
Kundu
and
D.
Mukherjee
,
Chem. Phys. Lett.
179
,
468
(
1991
).
84.
C.
Hättig
and
B. A.
Hess
,
J. Phys. Chem.
100
,
6243
(
1996
).
85.
A.
Salam
and
T.
Thirunamachandran
,
J. Chem. Phys.
104
,
5094
(
1996
).
86.
V. P.
Osinga
,
S. J. A.
van Gisbergen
,
J. G.
Snijders
, and
E. J.
Baerends
,
J. Chem. Phys.
106
,
5091
(
1997
).
87.
G. H. F.
Diercksen
,
W. P.
Kraemer
,
T. N.
Rescigno
,
C. F.
Bender
,
B. V.
McKoy
,
S. R.
Langhoff
, and
P. W.
Langhoff
,
J. Chem. Phys.
76
,
1043
(
1982
);
J. A.
Stephens
and
V.
McKoy
,
J. Chem. Phys.
88
,
1737
(
1988
).
88.
L.
Veseth
,
Phys. Rev. A
44
,
358
(
1991
);
L.
Veseth
,
Phys. Rev. A
49
,
939
(
1994
);
L.
Veseth
and
H. P.
Kelly
,
Phys. Rev. A
45
,
4621
(
1992
).
89.
N.
Stein
,
C.
Hättig
, and
B. A.
Hess
,
Chem. Phys.
225
,
309
(
1997
).
90.
W.
Byers Brown
and
D. M.
Whisnant
,
Mol. Phys.
25
,
1385
(
1973
).
91.
J. E.
Bohr
and
K. L. C.
Hunt
,
J. Chem. Phys.
86
,
5441
(
1987
);
J. E.
Bohr
and
K. L. C.
Hunt
,
J. Chem. Phys.
87
,
3821
(
1987
);
K. L. C.
Hunt
,
Y. Q.
Liang
, and
S.
Sethuraman
,
J. Chem. Phys.
89
,
7126
(
1988
);
X.
Li
and
K. L. C.
Hunt
,
J. Chem. Phys.
100
,
7875
(
1994
);
K. L. C. Hunt and X. Li, in Collision- and Interaction-Induced Spectroscopy, NATO ASI Series C, edited by G. C. Tabisz and M. N. Neuman (Kluwer, Dordrecht, 1995), Vol. 452, pp. 61–76.
92.
D. J.
Margoliash
,
T. R.
Proctor
,
G. D.
Zeiss
, and
W. J.
Meath
,
Mol. Phys.
35
,
747
(
1978
).
93.
A.
Kumar
and
W. J.
Meath
,
Mol. Phys.
54
,
823
(
1985
).
94.
D. J.
Margoliash
and
W. J.
Meath
,
J. Chem. Phys.
68
,
1426
(
1978
).
95.
S. A. C.
McDowell
and
W. J.
Meath
,
Mol. Phys.
90
,
713
(
1997
).
96.
B. L.
Jhanwar
and
W. J.
Meath
,
Chem. Phys.
67
,
185
(
1982
).
97.
W.
Kolos
and
L.
Wolniewicz
,
J. Chem. Phys.
43
,
2429
(
1965
);
W.
Kolos
and
L.
Wolniewicz
,
J. Chem. Phys.
46
,
1426
(
1967
).
98.
A. K.
Dham
,
A. R.
Allnatt
,
W. J.
Meath
, and
R. A.
Aziz
,
Mol. Phys.
67
,
1291
(
1989
).
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