- Email: [email protected]
Non-additivity in intermolecular forces. A comparison of results obtained by different methods
Volume Il. number 4
kHiMICALPHYSIC!S UXTERS.
.I_
.’ : 1.
‘,
,_I ‘. :
1 Noveaber 1971
‘.
NON-ADDITIiITY
IN INTERMOLECULAR
FORCES.
A COMPARISON OF RliZSULTS OBTAINED BY DKFFEREWZ MmODS R. BLOCK, R. R&L
and G. TER MATEN
ReQived 14 September 1971
~i~u~t~~ns of tree-~dy ~ntera~~onsbetween atoms on the b&sol the Hs-modei and the Eel-model were carried out in first order of exchangep~t~bation theory, using Shter fut dons. Comparison with &n&z C&Z& tions withgaussian functions shows that they are essentially the Same. Re mt rentIts obtained with the SCF LCAO MO approximation are quite comparablewith those offirsf.order perturbation tbeery.
turbation theory reported in the literature have started from gaussian ground-state functions far the system of three atoms. We have repeated these calculations in first order, using Slater functions in the Hsmodel, quartet state. As regards (ii), we have compared firstorder results in the H,- and He,-models, again using Sbtter functions. To obtain an impre~ion regarding (iii), we have compared our resuIts with those published recently by Bader et al. [6) with the SCF LCAO MO approximation. The system of refer* ence is in a! cases that of three atoms, forming an isosceies trkngie with opening angle 8 at the central atom. The two equal sides of the triangle have length R, measurkd in atomic units.
t . Introduction
Renewed interest has recently arisen concerning the importace of non-adding, ~tera~tions between closed-shell atoms or ions [l-6], e.g., in connection with crystal stability [ 1,2], third vi&i coefficients [4] and elastic &nstaots of ionic solids [7]. These anaIyses are based on exchange perturbation theory [1,2,7], perturbation expansions in the region of small overlap [3,4], or on the SCF LCAO MO ape proximatioh [6]. The systems studied were three closed-shell at+ or ions, in a one- or two-effect&e electron fper atom) model &-model, quartet State and Heg-model, singlet state, respectively), for differ= ent geometricaf confiurations. Since discrepancies have arisen‘in the literature regarding the order of 2. W1+to&l, Stater flmctions magnitude of three-body interactions [6f, further study of this problem is important. .’ We,used a $jater (fs)ground-state function Among the possible causes df these discre&kes, ,$&) = ‘Pt2.1P2 exp(-k) for each of the three efwe mention: (i) the choke of the a&lytieal form for ground; L fective electrons (spins pan&l), with t the distance from the effective electran’to its nucleus and X a vastate wavefunctions in perturbation excisions, riable pkame~er. The d&n& of X was chosen sb as @) &e &ice of the number of kffective elections, : *i yield t&e same ekpec‘tali6& %&&$‘5f F(OrOfF2, peiatom, . . .. .,:, (ii) inadequacy of the $CF’$.CA&W3 approxi-. &i&h gi%s almost the same reklts) calculated with a _’ I mation in a’ljrrdted basis; applied to wkakly interact- ., sirnpte gaussian function QsGr @‘2rr”3” exp(-&J2r!), - .... for a ‘&J&Inuige,of values for ffia’gaussian parameter ing systems. .‘. Con~er~n~‘(~)* calculations bati on exchange per.’ f3.The relation’ thus obtained between X and B reads ,.
: ; _’ .:
.
_‘..
I .:.
‘.
:
.,” ‘,:
‘:, : ..: .;,-: :, (,._ .’ : ‘.,: .’ : :. .. .._:‘, _ : : r ., ,:
,.
,,, ,. ..’ ._’
;
_:
... . _,,
::.
: :
.’ .’ ;. .: ‘, ..‘.. . ,.., . ... ‘. I, :. .‘; ,.,. ,. . . ‘.. ,, ,:
.:. ‘._ :‘. .’
‘, . .
42s
,,
..
vc+&l,n~~_r 4. .y., .
,: :
,,
O.i804 Ci.iSSS ,‘.
60 ,73>5 ‘+3.9 .:.
j063, ,.. r2&3.I ‘. ., ‘, -i&J : .’
..
: ^ ,,,::’
‘,
._
‘-.
: ,’
-O&$6?
,,‘0,1915’
-0;24i : ‘.--K&3’ -6.173
--CL095 Lo.006
-0.5193
-0:oolz +0.0168 .i,
I
”
0.00561 .0.00558 .,
”
~:00003 . ‘,. 0.00000~
.’
O,OO
+5@004
0.00&S
:.
_’ ,_ -5.585
AgJooir
‘0.05837 0.05613 ff.005 70
,~.088
+opo‘?
,)....
:
,(‘,.j ’.;,
:
‘.+ /:‘-I Nc&Wr.l9il.
(.
.. -0.0685~ -0.0532.
0.2122 “. &2026< ,’ 0.1955
,I
;
‘,,_
.: .”
:_
:’,.
)
.,
.._
1
._‘.’
_’ .’
: .
.
,:
,.
..,
,”
_,
..
:
Table2 (UI au) ~thckes-&%I, for different -valuesof.&R (;1& ihe tirbi&&&ant) and different VlWs‘Of the opm@+ngte B at tQe ixntral atom of ihe i+keIes triangfe;R I 3 au is the length of the two’equaf sides. Results [email protected] foi th$stotaIpair energy Efo’, the d&&ion from a~difi~ty MI” ard therehtive three*itom eneigy *-r/JC~o) :
‘. First-order ihr&tom’interaction~
_,
. . ‘.’ : .: .,. ”
‘. BO,
q: .60. 73.8
‘.
.’
‘Ml
,:, ~,~6~~ a;7308
-0mo -0.1621
,;,8%.9: 97.2
0.6594. 0.6368
)’ ‘: -‘-OS149 .--0.0906
106.J ff6.S .
0.6250’ 0.6077 :
-0.0658 -.‘. “-+0409
123.3 "
:‘,,
0.5985’ ..
143,l
0.5920 '.
180 .‘_
O.SWc1.. . . .
8”
AR =’ 3.0 .
‘,’
)..
,: :, ,’
:
‘M-2- 2.1 ,.&l/qol
‘fij”’
-0.240 .-a222
-0.174 -ai 42 -5.106 -0.067
” -0.027 -0.0160 *0+0087 ;_, +0.015 +0.0328 : *AM mq3.p
‘1
..
+El’
0.3352 U.2695 0.2419 0.2351 ‘0.2308 ‘, 0.2280. .iJ.2261
0.2252 5.2245.
,’
..
.:.
..I ., hE’;/Ef@
~-0.0686 -0.0~65 -0.0278~
,-c3.20$ tC1.173 -0.115
.. -5.5198 ,-0.512% -O&OS? --O,oof4
‘-0.584. -0.05s ‘-$I.029 '~0.006
+0.1)031
+0,014’
.I 1. : : +0.0068 .’ ‘, ‘,, _.+&f
-‘.
,. .’
+&030
.”
: )” ‘.
.
Volume 1l;nqber
4
tXQM!CAL PHYSICS FERS
1 November 1971
A = atitf2 p (equating the expectation values of r2 gives K = 2”2fl). with 2.0 G’pR < 3.4 as determined’. in ref. [l] for rare-gas solids (R is the nearest-neighb,our,djstauce) we have 2.7 < M 4 4S for such sys terns.
Consider now k triplet (A, B, Cl ofatoms, with in total three effective electrons 1,2,3.. The proundstate Gavefunctions for the atoms are denoted by da(l), @u(2)aud tit(3), respectively. Let H' be the
perturbation hamiltonian for this system and P the projection operator for three parallel spins (P belongs. to the [l? ] -representation of the permutation group
S,). With the notation $ = $I,( l)dn(2)@&3), it can be shown [S,9] that in good approximation the firstorder perturbation energy is given by: q=
W~~‘WWV)
*
0)
The fust-order perturbation energy for the pair of atoms (A, B) is obtained by removing atom C to infinity, and similarly for the other pairs. The sum of these pair interactions is denoted by E$O);the relative three-body component is then simply AE,/tiT)
= (El -ti;+/q
.
(2)
The calculations were carried out for values M7 = 2.1 and 5.1. The results for q”), A,?, and A,!ZI/~~~,with different
values of the
opening
angle
8, are given
in
Eu”
-0
table 1. From the table we can draw the conduion, in comparing the,results with those obtained with gaussian wavefunctions [ 1J , that the values of the reIative first-order three-atom energy with Slater-type orbit&, as a function of the angle 8, show the same general behaviour as those obtained with gaussian
Fig. 1. Relati~: flrstdrder threeatom interzctions A&,&*) for isosceks oonfiguntions, as a function of the open’- 6 (R = 3 au). Chives A, B;C and D give the perturbation results in the Hes-modelwith 1sSbter functions (orbitalexponents 1.7; 1.3; 1.0; 0.7. respebkely); curw E represents the SCF IKAO MO rcsuIts of Bader et af. [6I, and F is a comparable curve obtained in the perturbation approach of the I-Is-model with gausrian chzge densities (@Z= 2.3) [ 1J.
functions; AEl,5$‘) is negative for small angles and positive for large 8.
3. He+odeI,
Rater functions
The curves A, B, C a&J give the resuhs for AR = S-1,3.9,3.0 and 2.1, respectively. In addition, curve E gives’thc values CCAE/E@I obtained by Bader et al. [6] in an SCF LCAO MO approach. Here, AE is the deviation from pair-wise additivity and E(O) is the total two-body Inferactiorrenergy’. FiGtIy, curve F
6.
The same S later-type functions were used for the six effective electrons, with each atom in a .singIet state, The corresponding Ijrojection’operator can readily, be constructed; it belongs to the [23] -representation of the permutation group :S.6. The _. results are reported in table i.. To make a cotiparign with diffeient methods ,:’ used’in the literature, werplot jri fig. 1 the values of $ZI/qo). in the si+ectrou model,as a furktion of. ,.. -. ,_ ..’ :
._
represents the results of Jansen [l] for a value OR = ., ‘2.9, where /3is the gaussian parameter. -,The ;esults’obt~,ed.by ‘these different methods a& syg to be tisatIsfacto$ agreement witheach
:,
.,.. other. We‘note also fseeXables 1 and.2) that the ML :
‘,.’
‘; :.
.,
: .
:
;..
427
,. .’ .“l”, ;.i:,:.-; ;, ‘. ‘, ‘..:‘: ane;‘ai;ii.tw~effe‘ctive’ciectr*n $+&&&i*n gq& ,,, i ,’ _.. [I J L;.jansen;Phys.‘Rev.‘135 (1984}.Al’Zk& : “.’ I. : e~~~e~t,~e~rn~~t, imp@!g that&e nunjbki of kf-: [2] ~.‘~mb~i~n~ I;:‘Jz&ssn, @~yi. Ijev..136 (1X6) ++lOil, .”., :: ~fpc$$,efectr&s-per atom is of x%5 signifi~anc+? f&r the :_ ‘. .) ” :Hnd subssqtinip+ers; .,‘:value_sof rela,tiG th+&mi “~teradtip;ts. . 131: J.N,@xrelT, M. -#and D.R. Wilikms, Pro!+ Roy. . .-it isr&$&?@,~iriparticu$r, that ~e’,I&AQ~MO ; .. ‘. ‘Sac.A284.61966)566:,.. ; : .’ : met&&$as followed bq’Bader et, al, Ieads td iesults 141 D.R. Wilt&s, ~:~scha&. and J+: Ikirreli~ J. Chk. _ qiuite compkrable with t+se obt@ned in a.. Fhys..47(1967)4916.’ ’ ” ,‘ ..‘,which’& [S) ?I.E. Sherudod; &G.,de Rocck..and EA. Mason;3.Chem,’ ,’f7.,@.x~er elichange,~t?urbation’tri?atment. On this .. .’ :‘. -%% ‘%[email protected]: @as@, Pidwever;IXI.-lid ck&x$onscart be,reached [6] ,R.F.W.Bader,OA. Novjiro and.V.?eItr~n-Lbpez, ctrem. : regarding e.g. &ystai stability of &e-gas solids, for I’hys,titters8 (1971)$68. : :.u;hich kcond-order retributions are &,sential(appa_ [7j .fz. Lombardi;L, &mm and R, Rittdr, &ys. Rev. 185 (1969) 1150.. -‘.‘. .: : ..’rent&, n0 van der YaaIs minimum ii, obtain+ by’ 181 L. Jamen, Phys. f&z&_ 152 ~196’0 63. " @aderetaf.‘for the pair energy). A probation meth- . 191V. ByersBrown,Churn;Phys.Letters2 ~~9~~)103. : od extended 'to second order isthus ~et3XQf)! t0 es&&$‘j ev&fi~&& &n af the nkadditiw inter~c;tions, : .‘,
.42&i
kHiMICALPHYSIC!S UXTERS.
.I_
.’ : 1.
‘,
,_I ‘. :
1 Noveaber 1971
‘.
NON-ADDITIiITY
IN INTERMOLECULAR
FORCES.
A COMPARISON OF RliZSULTS OBTAINED BY DKFFEREWZ MmODS R. BLOCK, R. R&L
and G. TER MATEN
ReQived 14 September 1971
~i~u~t~~ns of tree-~dy ~ntera~~onsbetween atoms on the b&sol the Hs-modei and the Eel-model were carried out in first order of exchangep~t~bation theory, using Shter fut dons. Comparison with &n&z C&Z& tions withgaussian functions shows that they are essentially the Same. Re mt rentIts obtained with the SCF LCAO MO approximation are quite comparablewith those offirsf.order perturbation tbeery.
turbation theory reported in the literature have started from gaussian ground-state functions far the system of three atoms. We have repeated these calculations in first order, using Slater functions in the Hsmodel, quartet state. As regards (ii), we have compared firstorder results in the H,- and He,-models, again using Sbtter functions. To obtain an impre~ion regarding (iii), we have compared our resuIts with those published recently by Bader et al. [6) with the SCF LCAO MO approximation. The system of refer* ence is in a! cases that of three atoms, forming an isosceies trkngie with opening angle 8 at the central atom. The two equal sides of the triangle have length R, measurkd in atomic units.
t . Introduction
Renewed interest has recently arisen concerning the importace of non-adding, ~tera~tions between closed-shell atoms or ions [l-6], e.g., in connection with crystal stability [ 1,2], third vi&i coefficients [4] and elastic &nstaots of ionic solids [7]. These anaIyses are based on exchange perturbation theory [1,2,7], perturbation expansions in the region of small overlap [3,4], or on the SCF LCAO MO ape proximatioh [6]. The systems studied were three closed-shell at+ or ions, in a one- or two-effect&e electron fper atom) model &-model, quartet State and Heg-model, singlet state, respectively), for differ= ent geometricaf confiurations. Since discrepancies have arisen‘in the literature regarding the order of 2. W1+to&l, Stater flmctions magnitude of three-body interactions [6f, further study of this problem is important. .’ We,used a $jater (fs)ground-state function Among the possible causes df these discre&kes, ,$&) = ‘Pt2.1P2 exp(-k) for each of the three efwe mention: (i) the choke of the a&lytieal form for ground; L fective electrons (spins pan&l), with t the distance from the effective electran’to its nucleus and X a vastate wavefunctions in perturbation excisions, riable pkame~er. The d&n& of X was chosen sb as @) &e &ice of the number of kffective elections, : *i yield t&e same ekpec‘tali6& %&&$‘5f F(OrOfF2, peiatom, . . .. .,:, (ii) inadequacy of the $CF’$.CA&W3 approxi-. &i&h gi%s almost the same reklts) calculated with a _’ I mation in a’ljrrdted basis; applied to wkakly interact- ., sirnpte gaussian function QsGr @‘2rr”3” exp(-&J2r!), - .... for a ‘&J&Inuige,of values for ffia’gaussian parameter ing systems. .‘. Con~er~n~‘(~)* calculations bati on exchange per.’ f3.The relation’ thus obtained between X and B reads ,.
: ; _’ .:
.
_‘..
I .:.
‘.
:
.,” ‘,:
‘:, : ..: .;,-: :, (,._ .’ : ‘.,: .’ : :. .. .._:‘, _ : : r ., ,:
,.
,,, ,. ..’ ._’
;
_:
... . _,,
::.
: :
.’ .’ ;. .: ‘, ..‘.. . ,.., . ... ‘. I, :. .‘; ,.,. ,. . . ‘.. ,, ,:
.:. ‘._ :‘. .’
‘, . .
42s
,,
..
vc+&l,n~~_r 4. .y., .
,: :
,,
O.i804 Ci.iSSS ,‘.
60 ,73>5 ‘+3.9 .:.
j063, ,.. r2&3.I ‘. ., ‘, -i&J : .’
..
: ^ ,,,::’
‘,
._
‘-.
: ,’
-O&$6?
,,‘0,1915’
-0;24i : ‘.--K&3’ -6.173
--CL095 Lo.006
-0.5193
-0:oolz +0.0168 .i,
I
”
0.00561 .0.00558 .,
”
~:00003 . ‘,. 0.00000~
.’
O,OO
+5@004
0.00&S
:.
_’ ,_ -5.585
AgJooir
‘0.05837 0.05613 ff.005 70
,~.088
+opo‘?
,)....
:
,(‘,.j ’.;,
:
‘.+ /:‘-I Nc&Wr.l9il.
(.
.. -0.0685~ -0.0532.
0.2122 “. &2026< ,’ 0.1955
,I
;
‘,,_
.: .”
:_
:’,.
)
.,
.._
1
._‘.’
_’ .’
: .
.
,:
,.
..,
,”
_,
..
:
Table2 (UI au) ~thckes-&%I, for different -valuesof.&R (;1& ihe tirbi&&&ant) and different VlWs‘Of the opm@+ngte B at tQe ixntral atom of ihe i+keIes triangfe;R I 3 au is the length of the two’equaf sides. Results [email protected] foi th$stotaIpair energy Efo’, the d&&ion from a~difi~ty MI” ard therehtive three*itom eneigy *-r/JC~o) :
‘. First-order ihr&tom’interaction~
_,
. . ‘.’ : .: .,. ”
‘. BO,
q: .60. 73.8
‘.
.’
‘Ml
,:, ~,~6~~ a;7308
-0mo -0.1621
,;,8%.9: 97.2
0.6594. 0.6368
)’ ‘: -‘-OS149 .--0.0906
106.J ff6.S .
0.6250’ 0.6077 :
-0.0658 -.‘. “-+0409
123.3 "
:‘,,
0.5985’ ..
143,l
0.5920 '.
180 .‘_
O.SWc1.. . . .
8”
AR =’ 3.0 .
‘,’
)..
,: :, ,’
:
‘M-2- 2.1 ,.&l/qol
‘fij”’
-0.240 .-a222
-0.174 -ai 42 -5.106 -0.067
” -0.027 -0.0160 *0+0087 ;_, +0.015 +0.0328 : *AM mq3.p
‘1
..
+El’
0.3352 U.2695 0.2419 0.2351 ‘0.2308 ‘, 0.2280. .iJ.2261
0.2252 5.2245.
,’
..
.:.
..I ., hE’;/Ef@
~-0.0686 -0.0~65 -0.0278~
,-c3.20$ tC1.173 -0.115
.. -5.5198 ,-0.512% -O&OS? --O,oof4
‘-0.584. -0.05s ‘-$I.029 '~0.006
+0.1)031
+0,014’
.I 1. : : +0.0068 .’ ‘, ‘,, _.+&f
-‘.
,. .’
+&030
.”
: )” ‘.
.
Volume 1l;nqber
4
tXQM!CAL PHYSICS FERS
1 November 1971
A = atitf2 p (equating the expectation values of r2 gives K = 2”2fl). with 2.0 G’pR < 3.4 as determined’. in ref. [l] for rare-gas solids (R is the nearest-neighb,our,djstauce) we have 2.7 < M 4 4S for such sys terns.
Consider now k triplet (A, B, Cl ofatoms, with in total three effective electrons 1,2,3.. The proundstate Gavefunctions for the atoms are denoted by da(l), @u(2)aud tit(3), respectively. Let H' be the
perturbation hamiltonian for this system and P the projection operator for three parallel spins (P belongs. to the [l? ] -representation of the permutation group
S,). With the notation $ = $I,( l)dn(2)@&3), it can be shown [S,9] that in good approximation the firstorder perturbation energy is given by: q=
W~~‘WWV)
*
0)
The fust-order perturbation energy for the pair of atoms (A, B) is obtained by removing atom C to infinity, and similarly for the other pairs. The sum of these pair interactions is denoted by E$O);the relative three-body component is then simply AE,/tiT)
= (El -ti;+/q
.
(2)
The calculations were carried out for values M7 = 2.1 and 5.1. The results for q”), A,?, and A,!ZI/~~~,with different
values of the
opening
angle
8, are given
in
Eu”
-0
table 1. From the table we can draw the conduion, in comparing the,results with those obtained with gaussian wavefunctions [ 1J , that the values of the reIative first-order three-atom energy with Slater-type orbit&, as a function of the angle 8, show the same general behaviour as those obtained with gaussian
Fig. 1. Relati~: flrstdrder threeatom interzctions A&,&*) for isosceks oonfiguntions, as a function of the open’- 6 (R = 3 au). Chives A, B;C and D give the perturbation results in the Hes-modelwith 1sSbter functions (orbitalexponents 1.7; 1.3; 1.0; 0.7. respebkely); curw E represents the SCF IKAO MO rcsuIts of Bader et af. [6I, and F is a comparable curve obtained in the perturbation approach of the I-Is-model with gausrian chzge densities (@Z= 2.3) [ 1J.
functions; AEl,5$‘) is negative for small angles and positive for large 8.
3. He+odeI,
Rater functions
The curves A, B, C a&J give the resuhs for AR = S-1,3.9,3.0 and 2.1, respectively. In addition, curve E gives’thc values CCAE/E@I obtained by Bader et al. [6] in an SCF LCAO MO approach. Here, AE is the deviation from pair-wise additivity and E(O) is the total two-body Inferactiorrenergy’. FiGtIy, curve F
6.
The same S later-type functions were used for the six effective electrons, with each atom in a .singIet state, The corresponding Ijrojection’operator can readily, be constructed; it belongs to the [23] -representation of the permutation group :S.6. The _. results are reported in table i.. To make a cotiparign with diffeient methods ,:’ used’in the literature, werplot jri fig. 1 the values of $ZI/qo). in the si+ectrou model,as a furktion of. ,.. -. ,_ ..’ :
._
represents the results of Jansen [l] for a value OR = ., ‘2.9, where /3is the gaussian parameter. -,The ;esults’obt~,ed.by ‘these different methods a& syg to be tisatIsfacto$ agreement witheach
:,
.,.. other. We‘note also fseeXables 1 and.2) that the ML :
‘,.’
‘; :.
.,
: .
:
;..
427
,. .’ .“l”, ;.i:,:.-; ;, ‘. ‘, ‘..:‘: ane;‘ai;ii.tw~effe‘ctive’ciectr*n $+&&&i*n gq& ,,, i ,’ _.. [I J L;.jansen;Phys.‘Rev.‘135 (1984}.Al’Zk& : “.’ I. : e~~~e~t,~e~rn~~t, imp@!g that&e nunjbki of kf-: [2] ~.‘~mb~i~n~ I;:‘Jz&ssn, @~yi. Ijev..136 (1X6) ++lOil, .”., :: ~fpc$$,efectr&s-per atom is of x%5 signifi~anc+? f&r the :_ ‘. .) ” :Hnd subssqtinip+ers; .,‘:value_sof rela,tiG th+&mi “~teradtip;ts. . 131: J.N,@xrelT, M. -#and D.R. Wilikms, Pro!+ Roy. . .-it isr&$&?@,~iriparticu$r, that ~e’,I&AQ~MO ; .. ‘. ‘Sac.A284.61966)566:,.. ; : .’ : met&&$as followed bq’Bader et, al, Ieads td iesults 141 D.R. Wilt&s, ~:~scha&. and J+: Ikirreli~ J. Chk. _ qiuite compkrable with t+se obt@ned in a.. Fhys..47(1967)4916.’ ’ ” ,‘ ..‘,which’& [S) ?I.E. Sherudod; &G.,de Rocck..and EA. Mason;3.Chem,’ ,’f7.,@.x~er elichange,~t?urbation’tri?atment. On this .. .’ :‘. -%% ‘%[email protected]: @as@, Pidwever;IXI.-lid ck&x$onscart be,reached [6] ,R.F.W.Bader,OA. Novjiro and.V.?eItr~n-Lbpez, ctrem. : regarding e.g. &ystai stability of &e-gas solids, for I’hys,titters8 (1971)$68. : :.u;hich kcond-order retributions are &,sential(appa_ [7j .fz. Lombardi;L, &mm and R, Rittdr, &ys. Rev. 185 (1969) 1150.. -‘.‘. .: : ..’rent&, n0 van der YaaIs minimum ii, obtain+ by’ 181 L. Jamen, Phys. f&z&_ 152 ~196’0 63. " @aderetaf.‘for the pair energy). A probation meth- . 191V. ByersBrown,Churn;Phys.Letters2 ~~9~~)103. : od extended 'to second order isthus ~et3XQf)! t0 es&&$‘j ev&fi~&& &n af the nkadditiw inter~c;tions, : .‘,
.42&i