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Role of polarization functions in cation binding. H3NLi+ and H2OLi+
Inkl&tiO~ between a I.i+&on and the bases N;H3 and OH, arc .&iied b$ab initio molecular orbital metho& F&is sets range in size from 631G’ to 6-311 i-$3” and in&de also doubk-valence -typescontai&~ two &s of polari&on tictio&_ ._ Intua~ti clltcgics computed with the singlypot&i&d (SP) sets arc tco high, duk in Iaxgepan to ovcreshaws of the~diipole momam of the bases- Both of the errors arc greatlyralucat ~.lun d functions are added ti, fii-row atomsmP)_ The .amtributions of doxron corrdation IO the inceracf.ionsarc included via M@Icr-Pkssct theory to second and .&ird ordak Whaeas the SP basis YIS kad incorrectly to attractivecontributionsof akdation, this component is positive x&h the DP sets .~ u%i& morooy~.involvesubstantially smallerbasis set &rposition errors at both the SCF and corrdatcd _L+s The second set of d functions is responsible &-for cm-r& behavior of both the -ndand third-order comuoncnts of c&elariok. Lithium cation and p&ton affiities calcuh~ed with the DP basis set are in excdknt aoxxd -with vahes me&y+ experimcnmy-
Ab initio mokcular orbital calculations have contributed a great deal to our understanding of molcctdar interactions [lJ]_ These calculations have been successful in characteriziug the nature of the forces involved and in predicting geometries of various complexes prior to their experimental determinatiou. A traditionally popular means of studying these interactions is via the supermolec&e approach wherein the entire compkx is treated. as a single @rge system. .Within this framework, the interaction energy is computed indirectly as a small difference between very large quantities and is hence extremely susceptible to small errors in the subsystems properties such as electric moments and polarizabilities. Despite the.fact that doublezeta basis sets with a single set Of- polarization functions a&z..well. known. to substantially overt on lute fromblstituteof Chemistry. university OFW~w,~ XI-383 wro&~,PohldR&arch’
= Raipimt of NIIi wJ8Ls-o. ..
dipore moments, recent years have witnessed a large number of ab- initio. studies. of molecular interactions using basis sets of this type+ resulting in exaggeration of the electrostatic attraction and associated inaccuracies in the total iuteraction energy [3-S]_ = -The necessity of using basis sets of fkte size leads to a second problem as well. In order ‘to improve the description of its electronic:structure, each molecule may take advantage of the vacant orbitak of the other subsystem and thereby lower its computed energy. ‘Ilkrefore, in addition to-the genuine ~s~tab,ilityof the_ complex relative- to the isolated subsystems, use of an incomplete basis set results in au. additional .and artificial component knoti as the basis set-superposition error (BSSE) (for a selected .J.istof examples see ref.’ [6J)_ In the --case of weak complexes. the magnitude of- the. BSSE tibtained with a- poor basis set may be comparable to the actual interaction energy itself,. making it difficult ~to extract useful informatiou from_ the ~caI&h.kionS [A_-. An .eff&tive-.means .ef corrkcting for_ the BSSE is by use of the-counter~poiseprocedtire where the spurious lowering of .the estimate
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energy of each mokuIe by the presence of the orbit& of the other is explicitly c&x&ted and subtracted from the previousIy computed interaction energy IS]_ One drawback of the counterpoise procedure is that it furnishes OIIIY-~.U upper limit to the BSSE Attempts to scale the counterpoise correction [9] or to include onIy the vacant MOs of the other subsystem flO] have met with moderate success and appear to generally Iead to underestimates of the true BSSE In addition to the primy artificial Iowering of the energy of each subsystem mentioned above, utihzation of orbitaIs of one _moIecuIeby the other can lead to a secondary effect which, as pointed out by KarIsUiim and SadIej [ll], may improve the caIcuIated electronic properties of each moIecuIe and thereby result in a more a ccurate description of the moIecuIar interaction One should therefore not regard the effeets of basis set superposition as purely an error whose influence is to be totaIIy discardedBasis set superposition errors are not limited to the SCF IeveI but may occur in post-Hartree-Fock treatments as well [X2-17)_ Indced. the error in the cot-reIation energy may be Iarger than the SCF BSSE Despite this fact. the infIuence of basis set superposition upon interaction energies at the correlated IeveI has received very IittIe attention to date_ One difficuhy in evaluation of BSSE at correIatcd IeveIs is that the counterpoise procedure is valid onIy for those CI techniques which are size consistent_ Thus, it would not be appropriate to evaluate the BSSE within the framework of the often used Iimited CI method including only singIe and doubIe excitations_ Many-b&y perturbation theory (MBIT) with M#Ier-PIesset partitioning of the hamihonian [18,19] offers an attractive opportunity to study moIecuIar interactions at correlated levels_ The procedure has been demonstrated to provide accurate results in a fraction of the time required by comparable CI techniques [4]_ Moreover9 its size consistency aIIows the evahxation of BSSE within the framework of the counterpoise procedure Previous calculations have underscored the importance of ehmination of the spurious effects of the BSSE for a ccurate resuhs. For example, inch&on of counterpoise corrections at the MP2 and MP3
IeveIs Ied Newton and ICestner 1161to much superior agreement with experimental data for the water dimer_ Our own recent caIcuIations with the 631G** basis set found rather substantial superposition errors for the complexation reactions of H+ and Li* with a number of bases [lfl_ In fac& the BSSE at correlated levels was comparable in magnitude to the actual contribution of electron correlation to the total interaction energy_ The cakzulations reported in the present paper were carried out with a number of goals in mind. Aithough singly polarized double-zeta basis sets ~eneraIIy lead to .exa~erated dipole moments. several workers have recently suggested that addition of a second set of d functions on each fmt-row atom can yield marked improvements in both the electric moments and polarizabihties and therefore much better estimates of interaction energies [20,21]_ We have accordingly undertaken a systematic study of the effects of basis set size upor. the electric properties and interaction energges of a number of complexes. The caIcuIations are carried out at levels ranging from SCF to third-order M$lIer-PIesset theory to analyze the contributions of eIectron correlation to the interactions_ We have also carefully investigated the effects of basis set superposition error with each basis set at correIated as weII as SCF levels. The systems chosen for study are the complexes formed between the Lie cation and the bases NH3 and OH,_ The smaII sixes of these systems allow us to bring to bear fairly large basis sets and their simplicity aIIows a straightforward interpretation of the data For example, use of an electron-deficient Li’ cation leads to the assumption, substantiated below, that the BSSE may be wholly attributed to utihxation of the Li orbitah by the eiectrons of the base with no contribution from the reverse situation- Another objective of the present work is accurate determination of the proton and lithium cation affinities of ammonia and water, incIuding the effects of zero-point vibrational energies and BSSE at both SCF and correlated levels_ ZMechods Ab initio calculations were carried out using the GAUSSIAN 80 package of computer codes [22].
_- :.
.EIectron correlation was studied using .M&erPlesset perturbation theory [19] to second (MP2)_ -and third (MP3) orders, keeping the ‘cores of the fir+row atoms frozen_ Quadrupole and octupole moments were evaluated by the molecular properties pa&age of the MON!$IERGAUSS program WIInteraction energies were computed as the difference in total energy between each complex and the reference subunits at infinite separation_ The counterpoise method [S] was used to evaluate the basis set superposition error-which was computed as the difference in energy between the isolated subunits on one hand and the subunit energies, each calculated within the basis set and geometry of the entire compIex, on the other. A number of different basis sets were 2ppIied to these systems The smallest, 6-31G* is of split-valence type and contains polarization functions of d type on fmt-row atoms [24]; 6-31G** is identical but includes p functions on H atoms as well. The 6-31G’ * basis set-proposed by Nobes et aI_ [25] is simiIar to 6-31G** except that the exponents used for the polarization functions are somewhat different: the p functions on H are more diffuse while the d functions on N and 0 are somewhat more contracted_ Other basis sets used were of triple-vaIence type with polarization functions on first-row atoms (6311G*) and on ah atoms (6-311G**) [26]. To examine the effects of different numbers of sets of polarization functions, the 6-31G** set was modifted as follows_ 6-31G**(lp,2d) contains a second set of d functions on first-row atoms (3 = 032); the sir&e set of p functions on H has been made more diffuse (5 = 015) [20]_ Ah atoms
Table 1 Gcofncuia optimized by FOG0
NH, NH? Li+-NH3 OH2 OH; Li+-OH2
Ct., _-
h
C,, C 2W C3u C,
=’ Refs. [2&29].
-
method *
Lo21
1095
-
1.016 0.958 O-979 0963
106.4 1042 1135 106_9
1.955 1.795
-L.
functions in the’ 6-31G**(2p,2d) basis set. &I which the pre&$I basis has been supplemented by a contracted (S = 1-l) set of p functions on H_ A last b_asis set, &ed only for the H20-LiC system, is denoted as6-311. -t- G**(2d) [273_ It differs -from $i7311G** onIy in that the single set of d functionCon_ oxygen. with exponent l-292 has been repIaced by 2 pair of sets with exponents 2.584 and 0.646; in addition, a set of diffuse s and p functions has been added ‘to first-row atoms The “doubly pohuize8’ terni is used here to refer to the latter three basis sets where first-row atoms contain two sets of d funccontain
two
sets of poIarization
tiOIlS_ The systems studied in this paper are the complexes of NH, and OH, with the cations H* and Li’_ The geometries of the relevant systems were taken from the previous optimizations by the FOG0 method [28,29] and are listed in table L
3. Results 3. I. Monomer properties
it is well known that molecular interactions are frequently strongly influenced by electrostatic considerations This will be particularly true for the interactions of a lithium cation with the two bases [30]_ Since the charo,e distribution of Li* is sphericaI, the components of the electrostatic energy in order of decreasing magnitude will be ion-dipole. ion-quadrupole, and ion-octupole. It is therefore extremely important that any basis set used for systems of this type accura tely represent the electric moments of the bases Moments calculated with each of the basis sets are presented in table 2 along with experimental data where availableBecause of the particular importance of the dipole moments, the basis set dependence of this property has been emphasized by pictorial representation in fig l_ It is first clear that all basis sets lead to some overestimation of the dipole moment of both NH, and OH,_ However, the .deggee of exaggeration varies a great deal among the various basis sets_ Beginning with the smallest basis sets on the left-hand side, the three double-valence basis sets all lead to substantial overestimates aI--
62 -l-Able 2
rzzdahtaldecvic
moments *’ of NH,
6-3lG’
63LG’-
and H,O &31GC+
6-311G’
&311G’*
6-3lG-: 0P.W
NH,.
P CD) @, 0%
n .T (=) Qzz (;ru)
Hz0 P (0) @A
szE (au}
_ I.946 r4clb
1559
1.830
1893
L55-l
2515
2768
3324 -2199
3364 -2230
3361 -2229
z?zi 2339 -0_052 0324 l-737
llgs uO7 -0-063 0956 I_783
LlTI 722s4 -0_108 0361 1.795
6-3lG” ~(zpa
+-311r.. Gy2d)
Js&B.. :
1524:
3350 -z27
1.772 2612 3329 -2210
=OT 26% 3.406 -2304
2700 3575 -2415
xss6 2410 -o_Os7 0881 lxxs
li73 2319 -0_103 O-912 l-734
I_904 2436 -0_060 1.010 1.844
l-924 2441 -0_100 1.036 1.914
-*_*b’ ‘3255C’
2167 2553 -o_Oa5 0-96s ls40
_.
X85” 2636” -0.134 =’
=’ Only uniquecxm~ponears shown.Remain& nott-zerocompot~ct~tsm;iy be obtained from consideration of qmmctry and sum NICS E,#_=E‘$&=O‘) i&f_ [3Ib c) R&_ [32f ” Rcf_ [33b =’ R& [34b
though there is a sIi& trend toward smaller vaIues as the basis set is en@@ by addition of poIarization functions on H atoms Extension of the basis set from double to triple valence yields no substantial improvements in the dipoIe moments. That is. comparison of 6-31G’ with 631lG* and 6-31G** with 6-3llG** geneally show only smaI1 changes An excepticn is the 6-3llG* vabre of the dipole moment of water which is particuIarly poor_
AU the above basis sets contain at most a single set of polarization functions on each atom Dramatic improvements in the dipole moments of both mokcules resuh from inclusion of second sets of polarization functions_ both 6-31G**(lp,2d) and 6-31G**(ZpZd) produce dipole moments in e~~ceilent agreement with the experimentaI values despite the fact that the valence shell is doubly rather than triply split_ It is interesting that the- fairly large 6-311+ G**(2d) basis set leads to a rather
-
:I. _
Z htajka.
’
S. ZCcheiner / P&k-a*iofi
poor
estimate of the -dipole -moment.- of. water, ~identical :to- the. 6311G’it _and 6-31GY. values_ The poor performan ce of the tliple+ziknce basis sets may Abe attributed -perhaps- to’ the fact: .~ that the --exponents of. these basis sets .were evaluated- via post-Hart&-Fock treatments; An important conchsion from fig. 1 is that addition of a second set of diffuse. d functions to. the standard 6-31G** basis set Ieads to dipole moments in &xl &cord with experiment :. Table 2 ako contains the calculated quadrupole (@and o&pole (Q) moments of NH; and OH,In. genera& all calculated components of the quadrupole moments ire somewhat -smaller in _ magnitude than experimental vaIues.- However, these underestimates are least severe for basis sets including two sets of d functions. As noted for the dipole moments, the triply split 6-31l.G type basis sets produce results only slightly better than 6-31G types;. more appreciable improvements arise from addition of second sets of polarization functions. The best ovemh agreement with experiment is associated with the 6_31G**(2p,2d) basis set_ AIthough there are no experimental data available
E
--
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._.
.:
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binding,
-..:- ..:..,.- :_63-.1 -I y _,;__-. _r‘:_:.
-for .the~.octupOle ~moments;r the ciikilatdi iGIG& 1’ are tabula:ed for purposes of interrialdmI@ri&%.: The basis set .-dependence of .th&e,&me& 1&L’. fairly. small; nevertheless; .a general -trend t&a-d’ : higher values with Iaro,er basis sets does seem to be;: p&em_ i .~ 1: : .;. 1 . . . .y’. .“:__. .,-
_ : 3-2 &km&n
energies
I
_-:
-, I( _ -_
.:_
I -
a?
t
-.-
The en&& 1of interaction. of Iii bith : NH, and OHz are presented in fig_ 2 Along with theSCF data are included -interaction _-energies corn- : puted at the correlated lMP2 and MP3 level+Note that the data in fig 2 are %ranSed such. that the Iaro,est interaction energies (a negatives quan&y) are near the top of the figure. The similarities in shape of the~curves in figs. 1 and 2 reemph&ize the strong dependence of the total interaction energy u&on -the dipole moment of the base_ pe 6-311+ G**(2d) results are an exception_) The general trend in the data is toward overestimates for the singly polarized basis sets (cspccially 6311G*) and much smaller values when two sets of d functions are included.
1
Y
-.
in &on
. .
nearly
z
fmchtas
r, -1
--MP2 ---_-- Mp 3
NH3--ti* -KY.
SCF
_ <*_
H20--ti+
\I-u,
.
basis Fig 2 Total interactiona&k% uklitiodly
the MP3 com&mt-
l&P2 rc&
..I-
set’-
include&‘SCF
-3lG&2d).
(2d)
~. : coi@&at:as
Note that larga negative values arc plkd
:
csll+G-
well & the MP2 contriiu~on near~thcitop of the figure
l@Z_d&a
coon@ii
.
A second fundamental distinction between t&se two types of- basis sets involves the effects of eicctron correlation npon the interaction energies_~ For the basis sets with a single set of d functions, tire correlated complexation energies are generally bighcr than the SCF vahtcs while the reverse is true for doubly polarized sets_ The effects of electron correlation may be anaiyzed in more depth using fig_ 3 which explicitly illustrates the secondand third-order components of the interaction energy_ (Note the return to normal sign convention; neogtive values are at the bottom of fig 3.) Comparison of the MP2 and MP3 curves sho>s that the second-order contributions to the interaction energy are of larger magnitude *&art third-order terms; hence the total correlation effect tends to parallel the MP2 curve Moreover, the MP2 and MP3 components are aiways of opposite sign- Of greatest importance is the behavior of the total Ml’2 f MP3 correlation contribution to the interaction energy_ lhere is a clear trend in both systems t0rrai-d less negative values as the basis set is improved_ Hou-ever, it is only when a second set of d functions is added to the basis set that the axreiation contribution to the interaction energy
beoornes poz5itivq’i.e. repulsive_ The presence of p. functions on hydrogen appears @qxxtant for.the : caldation -of positive correlation contributions; : indeed their absencein 6-31G* and 6-311G? leads to latge negative values. The H p functions, particuIarly the second set, make smaller contriiu--tions to the dipole moments of-the bases (see fig--. 1). In the case of H,O-Li’ it is possible to cornpare our results to the extensive calculations of Diercksen et al_ [12] involving~ a CI procedure including all single and double ticitations and a. flexible basis set_ These workers found the corrclation contribution to the interaction energy to be a positive value of l-16 kcal/moL-not much different than the resuhs calculated here with the 631G**(lp,2d) and 6-31G**(2p,2d) basis sets_ ft therefore appears that for systems of this type, a minimum of two sets of d functions are necessary for proper repulsive- behavior of the correlation contribution to the interaction. This conclusion is not confined to systems containing Li* but also to interactions with a proton, as Pople and co-workers [35-371 found incorrect attractive contributions of electron correlation with the 6-31G* and 6-31G**
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sets to yieId better dipoIejnoments,~lo$r~ iker&._~~~ tion energ@, and s&er_ basis Set Isupe&%iiion~ 2 errpm; i Particular ‘attention. sho$d be:paid'-' to:.+& -i fact ~thattjte maguitude-of tkM~;BSSEik~.&&5 !. comparable- to the’correlation:.contuti~~~_~~~.~e interaction en& (fig- 3) ReduCtioti of-the super+ L position eiiors is therefore. espq%IIy~erueiaI for.& ._ -. mei+ni@fuI examination- of the1k-rekition ef&s-The smal& irroti upper limits obtained *tt;. the doubly polarized basis ~sets are therefer& of es& cial importance, At ail levds of.calculation, the BSSE computed -: ~. for H,O--I_i* is greater than the analogous $xtrk .tity for H,N-Li*_ Sin&e the two systems are ko“electronic, the jarger va!ues m H&--Li* tiy be ascribed to the smaller number of atomic centers and associate. orbita& Due to its positive charge, there is little tendency for the electron;deficient Li+ cation to-use the orbitals of the base. Indeed, the BSSE for -this direction -of charge flow k-q calculated to be negligible .in ali cases. Instead;
t-$0--- Li+
0.0.
,
.
.
I 1.0
_:
>y
Mf&--~_--bn&&g~..
,~
basis sets.. Thus; &ter&q ..rbasis -sets :of the :lattertype ‘am fre&entIy: sufficient for ..reaso;lable de-. seriptions of H+ and Lit.bindiug at the $CF Ieve&partieular cue must be .taken in: the Selection of basis set for post-SCF- treatments. -The interaction emzrgies reported --up to. this point have not been corrected for the- effects of basis set supe~siticn. ,The WE, evaluated by the counterpoise method. at SCF and correlated levels, is ilhtstrated. in fig. 4. for ,each. basis set, Wliile:this estimate of BSSE represents an upper .mt to the true error, it is nonetheless instructive to consider its dependenee‘upon basis:set_~.It is notable ~that-the magnitude, of the MP2 BSSE is not unlike that at the SCF level; the third-order BSSE is c&siderabIy smaller and is positive in sign_-The SCF and -My BSSE curves are very nearly parallel to one another. Moreover; the similarities to the dipole moments in fig J and to the totaI interaction energies in fig 2 are rather. striking_ There is thus a clear trend for superior basis
:
_
* *
MP3
MP3
moved the qualitative discrepancies between basis sets with one and two &ts of d functions at the Ml2 level, i.e. all yield-positive values, a distinction at the M.P3 Icvel retnains. For both systems in fig_ 5, the doubly polarized basis sets produce negative third-order correIation contriiutions while positive vahtes arise from the smaller sets_ In order to obtain some insights into this problem, Iet us cousider the simpler case of the interaction of the bw Hz0 and HjN with a point c&& l%e~e systems are quite similar to base-C* except that there is no possibility of any superposition error. The interaction in either case will be dominated by eIectrostatie forces, espezially point-d+&= Due
effectively ah of the BSSE arises from electron donation from the base to Li*_ As p&ted out above a principal inadequacy of the smdI basis sets is the incorrect sign of the axreIation contribution to the interaction energy_ However. by subtracting the BSSE in fig 4 from this contribution, it is possibie to get a much more appropriate treatment of correlation_ This behavior may be seen in fig 5 which contains the correIation contributions, corrected for BSSE For both systems, this correction procedure leads to positive values for dmost aI1basis sets, in qualitative agreement with the abovementioned resufts of Diercksen et al_ The general basis set dependence of the data in figs. 3 and 5 are not very different; instead. there is an overall vertical translation of 2ach curve with some flattening of the curves resuhing from BSSE corrections_ The high vahtes on the right-hand side of fig_ 5 indicate that the counterpoise method is probably overcorrecting, which is not unexpected with reggd to the ahove discussion concerning primaty and secondary effats_ One should therefore be cautious about subtracting the full BSSE computed by the counterpoise approach_ WhiIe the counterpoise corrections have re-
to the lack of any electrons on the point charge, the correlation energy will be exclusively of intramoIecular type within the base molecule ln the case of LP, there is the possibility of an intermolecuiar component as well, but this quantity shouId be quite smaIl as a result of the very small polarizability of Li+ (O-027 A3) [38]_ In fact, an empirical estimate of the dispersion energy in the base-Li* system by the London formula is less than -0-4 kcal/mol [39]_ The similarity of the systems containing the point charge with those containing Li* is apparent from the data in table 3
MP2 20
~20---Li* ,_--“‘A’
n
,_“,:, D
---cY
-I
(lp2d)
basis
set
(2&!dl
C2d)
which compares the correlation contributions to the in&action en&&s at the MP2 and Ml?3 levels In order to analyze the reasons for the different behavior of third-order interaction energy between the different types of basis sets, calculations were carried out for the base-point charge systems with the 6-31G** and 6-31G**(2p,2d) basis sets. For any distance of the point charge from the base, the MP2 contribution to the interaction energy is positive with either basis set This result is consistent with the data in fig. 5 which have been corrected for BSSE and with the fiidiugs of Diercksen and Sadlej [40] that second-order MBPT produces a decrease in the dipole moment of both water and ammonia, and hence a reduction in the electrostatic interaction with a positive chart3e_ These authors found also that the third-order correction to the dipole moment has the opposite effect and increases the magnitude of this property_ Therefore, one would expect the MP3 contribution to the interaction energg to be a negative one This is in fact what is found with the 6-31G**(2p,2d) basis set; however, the equivalent basis set lacking the additional set of polarization functions (631G**) leads to the opposite result of a positive MP3 contribution- The opposite sign of MP3_contributions for the singly and doubly polarized basis sets in figs_ 3 and 5 may therefore be ascribed to the incompleteness of the smaller bases and their inability-to properly describe third-order changes in subsystem properties_ Similar results were obtained by Zeiss et al_ [41] who also found incorrect -third-order changes in. the- dipole mo ment with a $malI basis set conta$ir&a_ single set ofpolarizationftmctions_ --. ‘.I‘. -We note fmally that- Wells and Wilson have very recently published a paper [42]. which con- f!!~some of our findings here_ BSSE was studied
in a sequence of even-tempered basis sets on both the SCF and correlated levels, the latter via .diagrarnmatic many-body perturbation theoj;. These authors found- that in the case of Ne, the magnitude of BSSE- at correlated levels -can ‘be numeri_cally greater than the SCF BSSE It therefore appears that our conclusions conkming base-Lie interactions may be extended to a wide range ofother systems as well. 3-3. Li + and H + affinitie The foregoing analysis has demonstrated that the doubly polarized 6_31G**(2p,2d) basis set provides a sound framework for an accurate and well-balanced treatment of the LiC-complexation reactions_ In an effort to determine the best possible theoretical vaIue for the interaction energy, the distance of the lithium cation from the base moIecule was optimized at the SCF- and correlated
levels t&g
this b&is set_ These optimized
dis-
tances are presented in the fiit row of table 4 where it may be seen that a significant lengthening of the bond length (0.03 A) occurs as a result of inclusion of second-order effects, followed by a small contraction of 0_006 A at’ the MP3 level. Both of these changes are fully consistent-with the MBPT changes in dipole moment discussed above and the alternating repulsive and attractive nature of the interaction_ The interaction energies,. com: puted at. thenSCF level, for -each intermolecular distance are contained in the second row; followed by total interaction energies up to the MP2 and MP3 levels in the ensuing rows_ Due. to fairly small differences. in -R(X-Ii’) from one‘ level to. the next, the interaction energies are rather insensitive to the choice of. method for’ optimiza_tion of -this distance_ Thus, the interaction energies &.zhputed : attthe UP3 level .are quite similar re$rdless~ uf
Z LnKajkaS Schher/
6s Table 4 Lithium cuiozt int&w
tnergiQ (XmkcaI@ol)
Pohvizationfbnctionsin taxianbin&g
using the 6-31Gg*(2p,2d) basis set
H,N-5’ SCF R(x-ri’) (A) - AE, - ~‘%fZ -A&, W(29SK)”
LOO2 40_60 3967 39.62 3832
H@-Li+ MP2
MP3
SCF
MP2
MP3
ZQ>7 4052 39-n 39-72 33.42
2031 4054 3952 39_72 38.42 (39-l 22) =’
I.868 35.90 34.46 346S 33.68
1.900 35.81 3450 34-69 33-69 (34-O&2) =’
1.894 35.84 3450 34.69 33-69
a)
W
LS42 =’ 3x7 3492 =’
352 342
=’ Rcf_ 1121;[54L/31~ Lq5.q b’ Rcf_ [rrl; 6-311f G”(M)_ c’ SCF+CL d: Jii.4 coxnnurcd
b?raddingAZPE (240 for HJLfi*:LlO
for H,O-Li+
whether the R(X-Li*) distance used comes from a SCF or MP3 op timization The discrepanci~ between rMP2 and MP3 are even smaller_ F+erimental valua with which the theoretical data may be compared are generally lithium cation affinities measured at 298 K_ To bring the two types of property into coincidence, the cakxlated interaction energges must f-t be corrected for the difference in zero-point vibrational energies betweep the isolated bases and the complexes, dZPE Del Bene et al. [27) have previously obtained a vah,re of AZPE for H,O-Li’ of 210 kcal/mol by a vibrational anaIysis of the 6-31G* force constants and we have adopted this value here; raising the tem_perature to 298 K adds an additional O-4 kcal/moL However, there is no currently available data of comparabl accuracy for the H,N-Li* system_ We have aecordin& made use of 3-21G data by the following procedure Del Bene et aL provided both 3-21G and 6-31G’ data for HP-Li+ which allows the former to-be seaIed to the htter_ Applying the same sealing constant to the 3-21G data for H,N-Li’, we arrive at a value of &ZPE of 240 kcal/moL Further corrections include A( pY) of -0-6 kcal/mol to convert from AE to QN and $RT for loss of translational degrees of freedom as a result of complexation, The f& estimates-of Li* affinities are provided in the last row of table 4 where the agreement with experimental VaIUes (in parentheses) may be seen to be quite good_ It is interesting that the 6-311+ ,**@I) basis set leads to an exceUent Li* affinity
). a rhcmri amwZionof0.4.aud
-$RTtoAE_
despite its rather poor estimate of the dipole moment of H,O_ For pruposes of comparisnn, we have performed calcuiations also of the interactions of the bases with a proton. The geometries of the protonateci species were taken from previous FOG0 optimizations. The interaction energies and proton affinities are presented in table 5 which includes previous caIculations and experimental data as well_ As above, the interaction energies have been adjusted by inclusion of zeropoint vibrational energges and thermal considerations to yield a theeretical estimate of the proton affinity at 298 K_ Comparison with the experimentaI values reiterates the high accuracy of the doubly polarized 6-31Gf*(2p,2d) basis set for treatment of cation binding, and especially for binding H* and Li+ to NH, and H,O_ One important distinction between Hf and Li+ binding concerns the relative contriiutions of various orders of MP theory_ In general. Ii+ affinities are substantially Iess sensitive to correlation effects at all orders_ For example, MP2 lowers the SCF proton aftinities of NH, and H,O by 3.6 kcaI/mcl as compared to a drop of only = 1 kcal/mol in the Lit afftities_ Whereas the-third-order contribution to the proton afftities is = 2 kcal/&roI, the Li’ affinities are changed by less than 02 With regard-to fourth-order.corrections, Frisch et aI._[35] have found changes in the H+ afftities of NH, and Ha0 of less than 0.5 kcal/moL Based upon the aforementioned tren& we expect changes
--bexF -_bE&
216.96 .-
-i%E,
AZPE PA(OK) PAWK) PA (298K),
21336 21493
9.1h’ 205.83 207.13 207533
216.2
175_~8
1743
171.6
2132 o
1722a. 17421
~1728’)
in.iP.
83’
53
16591 16721
1675
9.1 204.1 205.6
-
165_3+1.8 n
83.‘. 1635 164+
_
* Au altrics in kcal~moL a’ FOG0 gwmenics=’ From I-& (441;sro[542/21] basisset a’ From ref. [IZ]; -541/31]_ =I From rcf_[t7]: 6-311-i-G**(2d). a) R&44_ 0 SCF+CL * AEm 3 R&_ 27_ j) Rcf_45
in the Li+ binding energies of substantially less than this amount to be introduced by MP4 calcu-
ductions in the BSSE at both SCF and correlated levels. By correcting the correlation contribution
MiOllS_
for BSSE, it is possibie to improve the resuhs such that all basis sets yield a positive value for the corrected correlation contribution However; there remains a discrepancy in that the corrected thirdand second-order contributions are of the same
4_ Condusions There are a number of problems encountered with polarized basis sets of double or triplevaIence type (e.g., 6-31G” and 6-311G*‘) which may be greatly ameliorated or even eliminated by addition of a second set bf polarization functions_ The additional d functions on first-row atoms are particularly helpful with auxiliary H p functions playing a somewhat less important role_The singly polarized sets lead to exaggeration of the dipole moments of the, bases and consequent overestii mates of the cation binding energies_The contributions of electron correlation to the interaction are improperly described by these basis sets which yieId attractive cc~ntriiutions; in contt%st~to the repulsive.character known from’previous calculations- It is essential to reduce basisset superposition errors as they are not negligiile; even with larger basissets In facL. the. BSSE. at the corre: Iated level is compaiable in magnitude to both the SCF BSSE and the total correlation ch*Jibution to the inteiktion en&gym Inclusion of second sets of polarization functions leads to sixbstantial re-
sign (positive) with the singly polarized sets, while previous MBFT calculati&s confirm *e’find&g with the doubly polarized sets that these two terms
should have opposite sign. This conclusion is further verified by comparison with the resuhs where
Ii* is replaced by a point charge, thereby eliminating the possibility of basis set superposition or intermolecuku correIation_ Ftiyafter combining t&e CalcuIatedinkaction ener&es with appropriate ground vibrational and thermal effects, the estimates of the I’ (and H+) affinities,
computed with the doubly po@iz$d basis set (and neglecting superposition errors), are found to be in excellent agreement with experiment..
Acknodx&ement
.-
This work was supported f%anciaI.Iy by the National Institutes of Health (GM29391 and AMO1059) and the Research Cot-p_Computer time was provided by the SIU Computer Center_
Ref-
[19] JS Binkky and JA (1975)B_ 1201 J-GCM-
P_ Ho&a and R Zduadn& Weak into intaactions ia m and bidogy (E?s&er. Amstadam. lpsat [2) H-Rza+z&~W_l_Orrilk_Thonras&~~ kr~voL1(Wiky_NcuuYo&_1980~ [3] PA KotImaq ix IModczE thexukd chemist. VOL 4_ Applicatioasot~nnarurr~.~H_F_ zscbefcr HI (PkxuIm Press, New York. 19Ti) pp_ 109-15z [4] JA Pople. Faraday Discusions Ghan Sot 73 (1982) 7_ [5j K_ Mot-okand K KitaIm?_in: Mo!a&r intcmctions. VoL 1. uk K Bamjczak and W_L Orvilk-Thomas (W&y. Ncwz York. 1980) pp_ 2x-66_ [61 irt Urban and P_ Hobza. Theoret_ Chin Acta 36 (1975) 2B7.215; NS_ Oskmd and DL rMeniGdd_ Ghan Phys Letters 39 (1976) 612; HB&kidG_Ckh&nkiTheom~_~44 (i9-m 399: ‘W_ Koros, lhrorct Chirn Acfa 51(1979) 219; J-M_ Lcderoq. M_ AU.a-ena and Y_ Boutcikr. J_ than Ph>x 7s 0983) 4606; P_HobzaandR&kzdnikIntan_J_QuntumChem_23 (1983) 325; B&X_W&s and S Wilson. Chem_ Ph>x_Letters lOl(l9S3) 429_ m P_ Ho&a and R Zduadnk Chau Ph>s Letters S2 (1981) 473_ [S] S_F_ Bo)3 and F_ &mar&. MoL Ph>x_ 19 (1970) 553_ [9~A_Johason.P_KolI1mm~SRorticnbcrgThwrec_ Chixn Acm 29 (1973) 167_ [lOI K Morokuma and K_ Kicawa. in- Chcucca appEutio= . dcstmsrsic potc~tiaik cds_ D_ ofatomkti~ PIUS. New York. 1981) TrLthbr and P_ PoIifzer (PIpp_ 215-24z and A-J_ &ikj_ Theores_ Chixn &xa 61 (1x1 G_ Kz * (1982) I_ [I21 GH_ Diecbn. W_P_ Kncmcr and RO_ Roos. Theoret_ claim Acca 36 (1975) 249_ [13I P_D_ Dacrc Ckm PJqs Lcctas 50 (1977) 147: MoL Phyx 37 (1979) 1529_ [I]
M_D_ N-u and N-It‘ K escner.ckQLPh~~~91: (1983) 19s [la 2 Latajka and S. Schcker. Chem_ Wyr Letters 105 (1984) 435 [lS] C M@lkr and MS PZcsscc.F’hys Rev_ 46 (1934) 615. [lq
Pop16 Inran -.
I Quantum Ghan 9 -.
._
xan Dvxijjddt-T
kc Rijdt_ and* F-B_ yiin Duijncwkis J. MoL Struct 89 0982) 185. (211 V- staaumkr. alem Phys 7 0975) 17; K Lkhka. J_ Am Chem Sot 96 (1974) 4761_ pl JS Binkky, RA Whiteside, R K&~&MD. R Se& DJ. DeF-KaSdkgeLSTo$l.LRKabrpd~-4 Popk Ge4USSL4N SO. QCPE No_ 4% 0981)_ 1231M_R Petand RA_ Poirier. MONSTERGAUSS. Departma~t of Chemistry. Univusity of Toronto. Toronto, [24] z?k -* andJAPop~lheorcc_ChirnActa28 (1973) 213. [25] RHe Nobes, W-R Ropwdl and L Radom, J_ Compur alcm 3 (l932) Ml_ [261 B- Kkhoan. JS Binkky. JS setga and JA Pople. J_ Cbcms Phys 72 (1980) 650_ [27j JE Dd Fbu. =_ Mettee. hU_ Frisch B-T_ Luke and JA PO& I Ph,x 5 87 (1983) 3279_ [2Sl H Huber and I Vogs Chcm_ Ph>s 64 0982) 399_ [29] H Huber and 2. Latajka, J. Compur C&an 4 (1933) 252_ [3@] P- Schuster. W_ Jakubetz and W. Marks. Topics Curr_ Chcm 60 (1975) l_ [31] EL Cohen and AL Poynrcr. J. MoL SW_ 53 (1974) 131_ 1321S-G- Kukdich. Ghan Phy= Lcttcrs 5 (1970) 401; 12 (1971) 216_ 1331SA. Ckqh Y_ Beers. G-P_ Klein and Ls Rothman. J_ Ghan Phys_ 59 (1973) 2254. [34] J. Vcrhocwn and A Dymanus, J. Ghan Phyx 52 (1970) 3222 [35l MI Frisch, JE Dd Bauc. K Fb&awzhari and JA Pop!c ulan Phys Leftas 83 (1981) 240. [36] J-E DcI Ben% MJ_ F&i&, R -X&X& and JA Pcpk I Phyr Ghan S6 0982) 1529_ [3fl J-E Dd Bmg M_J. Frinh. K lbgba and JA PopIe I P&x_ Chem_ 87 (1983) 73_ [38] JJ. Sakmann and CK_ Jrirgtnsca Hdv. Chin Acta 51 (1968) 1276_ 1391KG- Spears. J. C’hem Ph>x_57 0972) 1850_ [401 G-HI=_ Diacksaa and AX Sadkj. J_ Ghan Phys_ 75 (1981) 1253; GwHeF_ Dia-cksaz V. Kdlb and A_J_ Sadkj. J. chao_ Phys 79 (1983) 2918. [41] G-D. Zeiss, WJt Scar_ N_ scrmki D_P_ Chong and SR Langhoff. MOL Phys. 37 0979) 1543_ (421 B-He Wdfs and S Wilson, Mol. Phyr 50 (1983) 1295. [43) RI_ Woodin and JJ_ Bauchamp.J_AmChan.Soc100 (1978) 5OL [44lRAkdCS,KSlXlIotk.M_REUCZl~.DADiXoZI and DS Maqnick, J. Phyx Ghan 84 (1980) 2840_ [45] Dx- Bohme &d G-L Mackay. J. An Ghan Sot 103 (1981) 2173_
Ab initio mokcular orbital calculations have contributed a great deal to our understanding of molcctdar interactions [lJ]_ These calculations have been successful in characteriziug the nature of the forces involved and in predicting geometries of various complexes prior to their experimental determinatiou. A traditionally popular means of studying these interactions is via the supermolec&e approach wherein the entire compkx is treated. as a single @rge system. .Within this framework, the interaction energy is computed indirectly as a small difference between very large quantities and is hence extremely susceptible to small errors in the subsystems properties such as electric moments and polarizabilities. Despite the.fact that doublezeta basis sets with a single set Of- polarization functions a&z..well. known. to substantially overt on lute fromblstituteof Chemistry. university OFW~w,~ XI-383 wro&~,PohldR&arch’
= Raipimt of NIIi wJ8Ls-o. ..
dipore moments, recent years have witnessed a large number of ab- initio. studies. of molecular interactions using basis sets of this type+ resulting in exaggeration of the electrostatic attraction and associated inaccuracies in the total iuteraction energy [3-S]_ = -The necessity of using basis sets of fkte size leads to a second problem as well. In order ‘to improve the description of its electronic:structure, each molecule may take advantage of the vacant orbitak of the other subsystem and thereby lower its computed energy. ‘Ilkrefore, in addition to-the genuine ~s~tab,ilityof the_ complex relative- to the isolated subsystems, use of an incomplete basis set results in au. additional .and artificial component knoti as the basis set-superposition error (BSSE) (for a selected .J.istof examples see ref.’ [6J)_ In the --case of weak complexes. the magnitude of- the. BSSE tibtained with a- poor basis set may be comparable to the actual interaction energy itself,. making it difficult ~to extract useful informatiou from_ the ~caI&h.kionS [A_-. An .eff&tive-.means .ef corrkcting for_ the BSSE is by use of the-counter~poiseprocedtire where the spurious lowering of .the estimate
Introductiorl
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-.
;_
DcveIopmkt
:._ -,-y -..
A&n5
-_-_I.__ -_-
0301-0104/85/$03.~0 SElsevier Science Publishers B.V. :. .(North-Holland Physics Publishing Division)
-.
.:
: :
,
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_-
energy of each mokuIe by the presence of the orbit& of the other is explicitly c&x&ted and subtracted from the previousIy computed interaction energy IS]_ One drawback of the counterpoise procedure is that it furnishes OIIIY-~.U upper limit to the BSSE Attempts to scale the counterpoise correction [9] or to include onIy the vacant MOs of the other subsystem flO] have met with moderate success and appear to generally Iead to underestimates of the true BSSE In addition to the primy artificial Iowering of the energy of each subsystem mentioned above, utihzation of orbitaIs of one _moIecuIeby the other can lead to a secondary effect which, as pointed out by KarIsUiim and SadIej [ll], may improve the caIcuIated electronic properties of each moIecuIe and thereby result in a more a ccurate description of the moIecuIar interaction One should therefore not regard the effeets of basis set superposition as purely an error whose influence is to be totaIIy discardedBasis set superposition errors are not limited to the SCF IeveI but may occur in post-Hartree-Fock treatments as well [X2-17)_ Indced. the error in the cot-reIation energy may be Iarger than the SCF BSSE Despite this fact. the infIuence of basis set superposition upon interaction energies at the correlated IeveI has received very IittIe attention to date_ One difficuhy in evaluation of BSSE at correIatcd IeveIs is that the counterpoise procedure is valid onIy for those CI techniques which are size consistent_ Thus, it would not be appropriate to evaluate the BSSE within the framework of the often used Iimited CI method including only singIe and doubIe excitations_ Many-b&y perturbation theory (MBIT) with M#Ier-PIesset partitioning of the hamihonian [18,19] offers an attractive opportunity to study moIecuIar interactions at correlated levels_ The procedure has been demonstrated to provide accurate results in a fraction of the time required by comparable CI techniques [4]_ Moreover9 its size consistency aIIows the evahxation of BSSE within the framework of the counterpoise procedure Previous calculations have underscored the importance of ehmination of the spurious effects of the BSSE for a ccurate resuhs. For example, inch&on of counterpoise corrections at the MP2 and MP3
IeveIs Ied Newton and ICestner 1161to much superior agreement with experimental data for the water dimer_ Our own recent caIcuIations with the 631G** basis set found rather substantial superposition errors for the complexation reactions of H+ and Li* with a number of bases [lfl_ In fac& the BSSE at correlated levels was comparable in magnitude to the actual contribution of electron correlation to the total interaction energy_ The cakzulations reported in the present paper were carried out with a number of goals in mind. Aithough singly polarized double-zeta basis sets ~eneraIIy lead to .exa~erated dipole moments. several workers have recently suggested that addition of a second set of d functions on each fmt-row atom can yield marked improvements in both the electric moments and polarizabihties and therefore much better estimates of interaction energies [20,21]_ We have accordingly undertaken a systematic study of the effects of basis set size upor. the electric properties and interaction energges of a number of complexes. The caIcuIations are carried out at levels ranging from SCF to third-order M$lIer-PIesset theory to analyze the contributions of eIectron correlation to the interactions_ We have also carefully investigated the effects of basis set superposition error with each basis set at correIated as weII as SCF levels. The systems chosen for study are the complexes formed between the Lie cation and the bases NH3 and OH,_ The smaII sixes of these systems allow us to bring to bear fairly large basis sets and their simplicity aIIows a straightforward interpretation of the data For example, use of an electron-deficient Li’ cation leads to the assumption, substantiated below, that the BSSE may be wholly attributed to utihxation of the Li orbitah by the eiectrons of the base with no contribution from the reverse situation- Another objective of the present work is accurate determination of the proton and lithium cation affinities of ammonia and water, incIuding the effects of zero-point vibrational energies and BSSE at both SCF and correlated levels_ ZMechods Ab initio calculations were carried out using the GAUSSIAN 80 package of computer codes [22].
_- :.
.EIectron correlation was studied using .M&erPlesset perturbation theory [19] to second (MP2)_ -and third (MP3) orders, keeping the ‘cores of the fir+row atoms frozen_ Quadrupole and octupole moments were evaluated by the molecular properties pa&age of the MON!$IERGAUSS program WIInteraction energies were computed as the difference in total energy between each complex and the reference subunits at infinite separation_ The counterpoise method [S] was used to evaluate the basis set superposition error-which was computed as the difference in energy between the isolated subunits on one hand and the subunit energies, each calculated within the basis set and geometry of the entire compIex, on the other. A number of different basis sets were 2ppIied to these systems The smallest, 6-31G* is of split-valence type and contains polarization functions of d type on fmt-row atoms [24]; 6-31G** is identical but includes p functions on H atoms as well. The 6-31G’ * basis set-proposed by Nobes et aI_ [25] is simiIar to 6-31G** except that the exponents used for the polarization functions are somewhat different: the p functions on H are more diffuse while the d functions on N and 0 are somewhat more contracted_ Other basis sets used were of triple-vaIence type with polarization functions on first-row atoms (6311G*) and on ah atoms (6-311G**) [26]. To examine the effects of different numbers of sets of polarization functions, the 6-31G** set was modifted as follows_ 6-31G**(lp,2d) contains a second set of d functions on first-row atoms (3 = 032); the sir&e set of p functions on H has been made more diffuse (5 = 015) [20]_ Ah atoms
Table 1 Gcofncuia optimized by FOG0
NH, NH? Li+-NH3 OH2 OH; Li+-OH2
Ct., _-
h
C,, C 2W C3u C,
=’ Refs. [2&29].
-
method *
Lo21
1095
-
1.016 0.958 O-979 0963
106.4 1042 1135 106_9
1.955 1.795
-L.
functions in the’ 6-31G**(2p,2d) basis set. &I which the pre&$I basis has been supplemented by a contracted (S = 1-l) set of p functions on H_ A last b_asis set, &ed only for the H20-LiC system, is denoted as6-311. -t- G**(2d) [273_ It differs -from $i7311G** onIy in that the single set of d functionCon_ oxygen. with exponent l-292 has been repIaced by 2 pair of sets with exponents 2.584 and 0.646; in addition, a set of diffuse s and p functions has been added ‘to first-row atoms The “doubly pohuize8’ terni is used here to refer to the latter three basis sets where first-row atoms contain two sets of d funccontain
two
sets of poIarization
tiOIlS_ The systems studied in this paper are the complexes of NH, and OH, with the cations H* and Li’_ The geometries of the relevant systems were taken from the previous optimizations by the FOG0 method [28,29] and are listed in table L
3. Results 3. I. Monomer properties
it is well known that molecular interactions are frequently strongly influenced by electrostatic considerations This will be particularly true for the interactions of a lithium cation with the two bases [30]_ Since the charo,e distribution of Li* is sphericaI, the components of the electrostatic energy in order of decreasing magnitude will be ion-dipole. ion-quadrupole, and ion-octupole. It is therefore extremely important that any basis set used for systems of this type accura tely represent the electric moments of the bases Moments calculated with each of the basis sets are presented in table 2 along with experimental data where availableBecause of the particular importance of the dipole moments, the basis set dependence of this property has been emphasized by pictorial representation in fig l_ It is first clear that all basis sets lead to some overestimation of the dipole moment of both NH, and OH,_ However, the .deggee of exaggeration varies a great deal among the various basis sets_ Beginning with the smallest basis sets on the left-hand side, the three double-valence basis sets all lead to substantial overestimates aI--
62 -l-Able 2
rzzdahtaldecvic
moments *’ of NH,
6-3lG’
63LG’-
and H,O &31GC+
6-311G’
&311G’*
6-3lG-: 0P.W
NH,.
P CD) @, 0%
n .T (=) Qzz (;ru)
Hz0 P (0) @A
szE (au}
_ I.946 r4clb
1559
1.830
1893
L55-l
2515
2768
3324 -2199
3364 -2230
3361 -2229
z?zi 2339 -0_052 0324 l-737
llgs uO7 -0-063 0956 I_783
LlTI 722s4 -0_108 0361 1.795
6-3lG” ~(zpa
+-311r.. Gy2d)
Js&B.. :
1524:
3350 -z27
1.772 2612 3329 -2210
=OT 26% 3.406 -2304
2700 3575 -2415
xss6 2410 -o_Os7 0881 lxxs
li73 2319 -0_103 O-912 l-734
I_904 2436 -0_060 1.010 1.844
l-924 2441 -0_100 1.036 1.914
-*_*b’ ‘3255C’
2167 2553 -o_Oa5 0-96s ls40
_.
X85” 2636” -0.134 =’
=’ Only uniquecxm~ponears shown.Remain& nott-zerocompot~ct~tsm;iy be obtained from consideration of qmmctry and sum NICS E,#_=E‘$&=O‘) i&f_ [3Ib c) R&_ [32f ” Rcf_ [33b =’ R& [34b
though there is a sIi& trend toward smaller vaIues as the basis set is en@@ by addition of poIarization functions on H atoms Extension of the basis set from double to triple valence yields no substantial improvements in the dipoIe moments. That is. comparison of 6-31G’ with 631lG* and 6-31G** with 6-3llG** geneally show only smaI1 changes An excepticn is the 6-3llG* vabre of the dipole moment of water which is particuIarly poor_
AU the above basis sets contain at most a single set of polarization functions on each atom Dramatic improvements in the dipole moments of both mokcules resuh from inclusion of second sets of polarization functions_ both 6-31G**(lp,2d) and 6-31G**(ZpZd) produce dipole moments in e~~ceilent agreement with the experimentaI values despite the fact that the valence shell is doubly rather than triply split_ It is interesting that the- fairly large 6-311+ G**(2d) basis set leads to a rather
-
:I. _
Z htajka.
’
S. ZCcheiner / P&k-a*iofi
poor
estimate of the -dipole -moment.- of. water, ~identical :to- the. 6311G’it _and 6-31GY. values_ The poor performan ce of the tliple+ziknce basis sets may Abe attributed -perhaps- to’ the fact: .~ that the --exponents of. these basis sets .were evaluated- via post-Hart&-Fock treatments; An important conchsion from fig. 1 is that addition of a second set of diffuse. d functions to. the standard 6-31G** basis set Ieads to dipole moments in &xl &cord with experiment :. Table 2 ako contains the calculated quadrupole (@and o&pole (Q) moments of NH; and OH,In. genera& all calculated components of the quadrupole moments ire somewhat -smaller in _ magnitude than experimental vaIues.- However, these underestimates are least severe for basis sets including two sets of d functions. As noted for the dipole moments, the triply split 6-31l.G type basis sets produce results only slightly better than 6-31G types;. more appreciable improvements arise from addition of second sets of polarization functions. The best ovemh agreement with experiment is associated with the 6_31G**(2p,2d) basis set_ AIthough there are no experimental data available
E
--
. .
._.
.:
:.
binding,
-..:- ..:..,.- :_63-.1 -I y _,;__-. _r‘:_:.
-for .the~.octupOle ~moments;r the ciikilatdi iGIG& 1’ are tabula:ed for purposes of interrialdmI@ri&%.: The basis set .-dependence of .th&e,&me& 1&L’. fairly. small; nevertheless; .a general -trend t&a-d’ : higher values with Iaro,er basis sets does seem to be;: p&em_ i .~ 1: : .;. 1 . . . .y’. .“:__. .,-
_ : 3-2 &km&n
energies
I
_-:
-, I( _ -_
.:_
I -
a?
t
-.-
The en&& 1of interaction. of Iii bith : NH, and OHz are presented in fig_ 2 Along with theSCF data are included -interaction _-energies corn- : puted at the correlated lMP2 and MP3 level+Note that the data in fig 2 are %ranSed such. that the Iaro,est interaction energies (a negatives quan&y) are near the top of the figure. The similarities in shape of the~curves in figs. 1 and 2 reemph&ize the strong dependence of the total interaction energy u&on -the dipole moment of the base_ pe 6-311+ G**(2d) results are an exception_) The general trend in the data is toward overestimates for the singly polarized basis sets (cspccially 6311G*) and much smaller values when two sets of d functions are included.
1
Y
-.
in &on
. .
nearly
z
fmchtas
r, -1
--MP2 ---_-- Mp 3
NH3--ti* -KY.
SCF
_ <*_
H20--ti+
\I-u,
.
basis Fig 2 Total interactiona&k% uklitiodly
the MP3 com&mt-
l&P2 rc&
..I-
set’-
include&‘SCF
-3lG&2d).
(2d)
~. : coi@&at:as
Note that larga negative values arc plkd
:
csll+G-
well & the MP2 contriiu~on near~thcitop of the figure
l@Z_d&a
coon@ii
.
A second fundamental distinction between t&se two types of- basis sets involves the effects of eicctron correlation npon the interaction energies_~ For the basis sets with a single set of d functions, tire correlated complexation energies are generally bighcr than the SCF vahtcs while the reverse is true for doubly polarized sets_ The effects of electron correlation may be anaiyzed in more depth using fig_ 3 which explicitly illustrates the secondand third-order components of the interaction energy_ (Note the return to normal sign convention; neogtive values are at the bottom of fig 3.) Comparison of the MP2 and MP3 curves sho>s that the second-order contributions to the interaction energy are of larger magnitude *&art third-order terms; hence the total correlation effect tends to parallel the MP2 curve Moreover, the MP2 and MP3 components are aiways of opposite sign- Of greatest importance is the behavior of the total Ml’2 f MP3 correlation contribution to the interaction energy_ lhere is a clear trend in both systems t0rrai-d less negative values as the basis set is improved_ Hou-ever, it is only when a second set of d functions is added to the basis set that the axreiation contribution to the interaction energy
beoornes poz5itivq’i.e. repulsive_ The presence of p. functions on hydrogen appears @qxxtant for.the : caldation -of positive correlation contributions; : indeed their absencein 6-31G* and 6-311G? leads to latge negative values. The H p functions, particuIarly the second set, make smaller contriiu--tions to the dipole moments of-the bases (see fig--. 1). In the case of H,O-Li’ it is possible to cornpare our results to the extensive calculations of Diercksen et al_ [12] involving~ a CI procedure including all single and double ticitations and a. flexible basis set_ These workers found the corrclation contribution to the interaction energy to be a positive value of l-16 kcal/moL-not much different than the resuhs calculated here with the 631G**(lp,2d) and 6-31G**(2p,2d) basis sets_ ft therefore appears that for systems of this type, a minimum of two sets of d functions are necessary for proper repulsive- behavior of the correlation contribution to the interaction. This conclusion is not confined to systems containing Li* but also to interactions with a proton, as Pople and co-workers [35-371 found incorrect attractive contributions of electron correlation with the 6-31G* and 6-31G**
,-;
-:
~. :-
~I:_ -..
.,
_
z..
:. .
& ~--
isnajk4.s sihe&r/-Pdon-an‘ -7
_‘_
_~
__
. .
I PI--
o---Q-----__
~~---~~~_~~_~~~~
tJH3-ii
=.
: _
..
:
-
_.::-
I
.;._:;
.z
I;
i
~-
;
.-z ^ f.
~. ._y:, _ :;;.yq
7
-.-
;
:‘_:_;
.
_.
.-
?.
.
_
_-
_-
.--..
:
__
s:: ::_:;;-:: ..F 5
:-...
-;--__zI
sets to yieId better dipoIejnoments,~lo$r~ iker&._~~~ tion energ@, and s&er_ basis Set Isupe&%iiion~ 2 errpm; i Particular ‘attention. sho$d be:paid'-' to:.+& -i fact ~thattjte maguitude-of tkM~;BSSEik~.&&5 !. comparable- to the’correlation:.contuti~~~_~~~.~e interaction en& (fig- 3) ReduCtioti of-the super+ L position eiiors is therefore. espq%IIy~erueiaI for.& ._ -. mei+ni@fuI examination- of the1k-rekition ef&s-The smal& irroti upper limits obtained *tt;. the doubly polarized basis ~sets are therefer& of es& cial importance, At ail levds of.calculation, the BSSE computed -: ~. for H,O--I_i* is greater than the analogous $xtrk .tity for H,N-Li*_ Sin&e the two systems are ko“electronic, the jarger va!ues m H&--Li* tiy be ascribed to the smaller number of atomic centers and associate. orbita& Due to its positive charge, there is little tendency for the electron;deficient Li+ cation to-use the orbitals of the base. Indeed, the BSSE for -this direction -of charge flow k-q calculated to be negligible .in ali cases. Instead;
t-$0--- Li+
0.0.
,
.
.
I 1.0
_:
>y
Mf&--~_--bn&&g~..
,~
basis sets.. Thus; &ter&q ..rbasis -sets :of the :lattertype ‘am fre&entIy: sufficient for ..reaso;lable de-. seriptions of H+ and Lit.bindiug at the $CF Ieve&partieular cue must be .taken in: the Selection of basis set for post-SCF- treatments. -The interaction emzrgies reported --up to. this point have not been corrected for the- effects of basis set supe~siticn. ,The WE, evaluated by the counterpoise method. at SCF and correlated levels, is ilhtstrated. in fig. 4. for ,each. basis set, Wliile:this estimate of BSSE represents an upper .mt to the true error, it is nonetheless instructive to consider its dependenee‘upon basis:set_~.It is notable ~that-the magnitude, of the MP2 BSSE is not unlike that at the SCF level; the third-order BSSE is c&siderabIy smaller and is positive in sign_-The SCF and -My BSSE curves are very nearly parallel to one another. Moreover; the similarities to the dipole moments in fig J and to the totaI interaction energies in fig 2 are rather. striking_ There is thus a clear trend for superior basis
:
_
* *
MP3
MP3
moved the qualitative discrepancies between basis sets with one and two &ts of d functions at the Ml2 level, i.e. all yield-positive values, a distinction at the M.P3 Icvel retnains. For both systems in fig_ 5, the doubly polarized basis sets produce negative third-order correIation contriiutions while positive vahtes arise from the smaller sets_ In order to obtain some insights into this problem, Iet us cousider the simpler case of the interaction of the bw Hz0 and HjN with a point c&& l%e~e systems are quite similar to base-C* except that there is no possibility of any superposition error. The interaction in either case will be dominated by eIectrostatie forces, espezially point-d+&= Due
effectively ah of the BSSE arises from electron donation from the base to Li*_ As p&ted out above a principal inadequacy of the smdI basis sets is the incorrect sign of the axreIation contribution to the interaction energy_ However. by subtracting the BSSE in fig 4 from this contribution, it is possibie to get a much more appropriate treatment of correlation_ This behavior may be seen in fig 5 which contains the correIation contributions, corrected for BSSE For both systems, this correction procedure leads to positive values for dmost aI1basis sets, in qualitative agreement with the abovementioned resufts of Diercksen et al_ The general basis set dependence of the data in figs. 3 and 5 are not very different; instead. there is an overall vertical translation of 2ach curve with some flattening of the curves resuhing from BSSE corrections_ The high vahtes on the right-hand side of fig_ 5 indicate that the counterpoise method is probably overcorrecting, which is not unexpected with reggd to the ahove discussion concerning primaty and secondary effats_ One should therefore be cautious about subtracting the full BSSE computed by the counterpoise approach_ WhiIe the counterpoise corrections have re-
to the lack of any electrons on the point charge, the correlation energy will be exclusively of intramoIecular type within the base molecule ln the case of LP, there is the possibility of an intermolecuiar component as well, but this quantity shouId be quite smaIl as a result of the very small polarizability of Li+ (O-027 A3) [38]_ In fact, an empirical estimate of the dispersion energy in the base-Li* system by the London formula is less than -0-4 kcal/mol [39]_ The similarity of the systems containing the point charge with those containing Li* is apparent from the data in table 3
MP2 20
~20---Li* ,_--“‘A’
n
,_“,:, D
---cY
-I
(lp2d)
basis
set
(2&!dl
C2d)
which compares the correlation contributions to the in&action en&&s at the MP2 and Ml?3 levels In order to analyze the reasons for the different behavior of third-order interaction energy between the different types of basis sets, calculations were carried out for the base-point charge systems with the 6-31G** and 6-31G**(2p,2d) basis sets. For any distance of the point charge from the base, the MP2 contribution to the interaction energy is positive with either basis set This result is consistent with the data in fig. 5 which have been corrected for BSSE and with the fiidiugs of Diercksen and Sadlej [40] that second-order MBPT produces a decrease in the dipole moment of both water and ammonia, and hence a reduction in the electrostatic interaction with a positive chart3e_ These authors found also that the third-order correction to the dipole moment has the opposite effect and increases the magnitude of this property_ Therefore, one would expect the MP3 contribution to the interaction energg to be a negative one This is in fact what is found with the 6-31G**(2p,2d) basis set; however, the equivalent basis set lacking the additional set of polarization functions (631G**) leads to the opposite result of a positive MP3 contribution- The opposite sign of MP3_contributions for the singly and doubly polarized basis sets in figs_ 3 and 5 may therefore be ascribed to the incompleteness of the smaller bases and their inability-to properly describe third-order changes in subsystem properties_ Similar results were obtained by Zeiss et al_ [41] who also found incorrect -third-order changes in. the- dipole mo ment with a $malI basis set conta$ir&a_ single set ofpolarizationftmctions_ --. ‘.I‘. -We note fmally that- Wells and Wilson have very recently published a paper [42]. which con- f!!~some of our findings here_ BSSE was studied
in a sequence of even-tempered basis sets on both the SCF and correlated levels, the latter via .diagrarnmatic many-body perturbation theoj;. These authors found- that in the case of Ne, the magnitude of BSSE- at correlated levels -can ‘be numeri_cally greater than the SCF BSSE It therefore appears that our conclusions conkming base-Lie interactions may be extended to a wide range ofother systems as well. 3-3. Li + and H + affinitie The foregoing analysis has demonstrated that the doubly polarized 6_31G**(2p,2d) basis set provides a sound framework for an accurate and well-balanced treatment of the LiC-complexation reactions_ In an effort to determine the best possible theoretical vaIue for the interaction energy, the distance of the lithium cation from the base moIecule was optimized at the SCF- and correlated
levels t&g
this b&is set_ These optimized
dis-
tances are presented in the fiit row of table 4 where it may be seen that a significant lengthening of the bond length (0.03 A) occurs as a result of inclusion of second-order effects, followed by a small contraction of 0_006 A at’ the MP3 level. Both of these changes are fully consistent-with the MBPT changes in dipole moment discussed above and the alternating repulsive and attractive nature of the interaction_ The interaction energies,. com: puted at. thenSCF level, for -each intermolecular distance are contained in the second row; followed by total interaction energies up to the MP2 and MP3 levels in the ensuing rows_ Due. to fairly small differences. in -R(X-Ii’) from one‘ level to. the next, the interaction energies are rather insensitive to the choice of. method for’ optimiza_tion of -this distance_ Thus, the interaction energies &.zhputed : attthe UP3 level .are quite similar re$rdless~ uf
Z LnKajkaS Schher/
6s Table 4 Lithium cuiozt int&w
tnergiQ (XmkcaI@ol)
Pohvizationfbnctionsin taxianbin&g
using the 6-31Gg*(2p,2d) basis set
H,N-5’ SCF R(x-ri’) (A) - AE, - ~‘%fZ -A&, W(29SK)”
LOO2 40_60 3967 39.62 3832
H@-Li+ MP2
MP3
SCF
MP2
MP3
ZQ>7 4052 39-n 39-72 33.42
2031 4054 3952 39_72 38.42 (39-l 22) =’
I.868 35.90 34.46 346S 33.68
1.900 35.81 3450 34-69 33-69 (34-O&2) =’
1.894 35.84 3450 34.69 33-69
a)
W
LS42 =’ 3x7 3492 =’
352 342
=’ Rcf_ 1121;[54L/31~ Lq5.q b’ Rcf_ [rrl; 6-311f G”(M)_ c’ SCF+CL d: Jii.4 coxnnurcd
b?raddingAZPE (240 for HJLfi*:LlO
for H,O-Li+
whether the R(X-Li*) distance used comes from a SCF or MP3 op timization The discrepanci~ between rMP2 and MP3 are even smaller_ F+erimental valua with which the theoretical data may be compared are generally lithium cation affinities measured at 298 K_ To bring the two types of property into coincidence, the cakxlated interaction energges must f-t be corrected for the difference in zero-point vibrational energies betweep the isolated bases and the complexes, dZPE Del Bene et al. [27) have previously obtained a vah,re of AZPE for H,O-Li’ of 210 kcal/mol by a vibrational anaIysis of the 6-31G* force constants and we have adopted this value here; raising the tem_perature to 298 K adds an additional O-4 kcal/moL However, there is no currently available data of comparabl accuracy for the H,N-Li* system_ We have aecordin& made use of 3-21G data by the following procedure Del Bene et aL provided both 3-21G and 6-31G’ data for HP-Li+ which allows the former to-be seaIed to the htter_ Applying the same sealing constant to the 3-21G data for H,N-Li’, we arrive at a value of &ZPE of 240 kcal/moL Further corrections include A( pY) of -0-6 kcal/mol to convert from AE to QN and $RT for loss of translational degrees of freedom as a result of complexation, The f& estimates-of Li* affinities are provided in the last row of table 4 where the agreement with experimental VaIUes (in parentheses) may be seen to be quite good_ It is interesting that the 6-311+ ,**@I) basis set leads to an exceUent Li* affinity
). a rhcmri amwZionof0.4.aud
-$RTtoAE_
despite its rather poor estimate of the dipole moment of H,O_ For pruposes of comparisnn, we have performed calcuiations also of the interactions of the bases with a proton. The geometries of the protonateci species were taken from previous FOG0 optimizations. The interaction energies and proton affinities are presented in table 5 which includes previous caIculations and experimental data as well_ As above, the interaction energies have been adjusted by inclusion of zeropoint vibrational energges and thermal considerations to yield a theeretical estimate of the proton affinity at 298 K_ Comparison with the experimentaI values reiterates the high accuracy of the doubly polarized 6-31Gf*(2p,2d) basis set for treatment of cation binding, and especially for binding H* and Li+ to NH, and H,O_ One important distinction between Hf and Li+ binding concerns the relative contriiutions of various orders of MP theory_ In general. Ii+ affinities are substantially Iess sensitive to correlation effects at all orders_ For example, MP2 lowers the SCF proton aftinities of NH, and H,O by 3.6 kcaI/mcl as compared to a drop of only = 1 kcal/mol in the Lit afftities_ Whereas the-third-order contribution to the proton afftities is = 2 kcal/&roI, the Li’ affinities are changed by less than 02 With regard-to fourth-order.corrections, Frisch et aI._[35] have found changes in the H+ afftities of NH, and Ha0 of less than 0.5 kcal/moL Based upon the aforementioned tren& we expect changes
--bexF -_bE&
216.96 .-
-i%E,
AZPE PA(OK) PAWK) PA (298K),
21336 21493
9.1h’ 205.83 207.13 207533
216.2
175_~8
1743
171.6
2132 o
1722a. 17421
~1728’)
in.iP.
83’
53
16591 16721
1675
9.1 204.1 205.6
-
165_3+1.8 n
83.‘. 1635 164+
_
* Au altrics in kcal~moL a’ FOG0 gwmenics=’ From I-& (441;sro[542/21] basisset a’ From ref. [IZ]; -541/31]_ =I From rcf_[t7]: 6-311-i-G**(2d). a) R&44_ 0 SCF+CL * AEm 3 R&_ 27_ j) Rcf_45
in the Li+ binding energies of substantially less than this amount to be introduced by MP4 calcu-
ductions in the BSSE at both SCF and correlated levels. By correcting the correlation contribution
MiOllS_
for BSSE, it is possibie to improve the resuhs such that all basis sets yield a positive value for the corrected correlation contribution However; there remains a discrepancy in that the corrected thirdand second-order contributions are of the same
4_ Condusions There are a number of problems encountered with polarized basis sets of double or triplevaIence type (e.g., 6-31G” and 6-311G*‘) which may be greatly ameliorated or even eliminated by addition of a second set bf polarization functions_ The additional d functions on first-row atoms are particularly helpful with auxiliary H p functions playing a somewhat less important role_The singly polarized sets lead to exaggeration of the dipole moments of the, bases and consequent overestii mates of the cation binding energies_The contributions of electron correlation to the interaction are improperly described by these basis sets which yieId attractive cc~ntriiutions; in contt%st~to the repulsive.character known from’previous calculations- It is essential to reduce basisset superposition errors as they are not negligiile; even with larger basissets In facL. the. BSSE. at the corre: Iated level is compaiable in magnitude to both the SCF BSSE and the total correlation ch*Jibution to the inteiktion en&gym Inclusion of second sets of polarization functions leads to sixbstantial re-
sign (positive) with the singly polarized sets, while previous MBFT calculati&s confirm *e’find&g with the doubly polarized sets that these two terms
should have opposite sign. This conclusion is further verified by comparison with the resuhs where
Ii* is replaced by a point charge, thereby eliminating the possibility of basis set superposition or intermolecuku correIation_ Ftiyafter combining t&e CalcuIatedinkaction ener&es with appropriate ground vibrational and thermal effects, the estimates of the I’ (and H+) affinities,
computed with the doubly po@iz$d basis set (and neglecting superposition errors), are found to be in excellent agreement with experiment..
Acknodx&ement
.-
This work was supported f%anciaI.Iy by the National Institutes of Health (GM29391 and AMO1059) and the Research Cot-p_Computer time was provided by the SIU Computer Center_
Ref-
[19] JS Binkky and JA (1975)B_ 1201 J-GCM-
P_ Ho&a and R Zduadn& Weak into intaactions ia m and bidogy (E?s&er. Amstadam. lpsat [2) H-Rza+z&~W_l_Orrilk_Thonras&~~ kr~voL1(Wiky_NcuuYo&_1980~ [3] PA KotImaq ix IModczE thexukd chemist. VOL 4_ Applicatioasot~nnarurr~.~H_F_ zscbefcr HI (PkxuIm Press, New York. 19Ti) pp_ 109-15z [4] JA Pople. Faraday Discusions Ghan Sot 73 (1982) 7_ [5j K_ Mot-okand K KitaIm?_in: Mo!a&r intcmctions. VoL 1. uk K Bamjczak and W_L Orvilk-Thomas (W&y. Ncwz York. 1980) pp_ 2x-66_ [61 irt Urban and P_ Hobza. Theoret_ Chin Acta 36 (1975) 2B7.215; NS_ Oskmd and DL rMeniGdd_ Ghan Phys Letters 39 (1976) 612; HB&kidG_Ckh&nkiTheom~_~44 (i9-m 399: ‘W_ Koros, lhrorct Chirn Acfa 51(1979) 219; J-M_ Lcderoq. M_ AU.a-ena and Y_ Boutcikr. J_ than Ph>x 7s 0983) 4606; P_HobzaandR&kzdnikIntan_J_QuntumChem_23 (1983) 325; B&X_W&s and S Wilson. Chem_ Ph>x_Letters lOl(l9S3) 429_ m P_ Ho&a and R Zduadnk Chau Ph>s Letters S2 (1981) 473_ [S] S_F_ Bo)3 and F_ &mar&. MoL Ph>x_ 19 (1970) 553_ [9~A_Johason.P_KolI1mm~SRorticnbcrgThwrec_ Chixn Acm 29 (1973) 167_ [lOI K Morokuma and K_ Kicawa. in- Chcucca appEutio= . dcstmsrsic potc~tiaik cds_ D_ ofatomkti~ PIUS. New York. 1981) TrLthbr and P_ PoIifzer (PIpp_ 215-24z and A-J_ &ikj_ Theores_ Chixn &xa 61 (1x1 G_ Kz * (1982) I_ [I21 GH_ Diecbn. W_P_ Kncmcr and RO_ Roos. Theoret_ claim Acca 36 (1975) 249_ [13I P_D_ Dacrc Ckm PJqs Lcctas 50 (1977) 147: MoL Phyx 37 (1979) 1529_ [I]
M_D_ N-u and N-It‘ K escner.ckQLPh~~~91: (1983) 19s [la 2 Latajka and S. Schcker. Chem_ Wyr Letters 105 (1984) 435 [lS] C M@lkr and MS PZcsscc.F’hys Rev_ 46 (1934) 615. [lq
Pop16 Inran -.
I Quantum Ghan 9 -.
._
xan Dvxijjddt-T
kc Rijdt_ and* F-B_ yiin Duijncwkis J. MoL Struct 89 0982) 185. (211 V- staaumkr. alem Phys 7 0975) 17; K Lkhka. J_ Am Chem Sot 96 (1974) 4761_ pl JS Binkky, RA Whiteside, R K&~&MD. R Se& DJ. DeF-KaSdkgeLSTo$l.LRKabrpd~-4 Popk Ge4USSL4N SO. QCPE No_ 4% 0981)_ 1231M_R Petand RA_ Poirier. MONSTERGAUSS. Departma~t of Chemistry. Univusity of Toronto. Toronto, [24] z?k -* andJAPop~lheorcc_ChirnActa28 (1973) 213. [25] RHe Nobes, W-R Ropwdl and L Radom, J_ Compur alcm 3 (l932) Ml_ [261 B- Kkhoan. JS Binkky. JS setga and JA Pople. J_ Cbcms Phys 72 (1980) 650_ [27j JE Dd Fbu. =_ Mettee. hU_ Frisch B-T_ Luke and JA PO& I Ph,x 5 87 (1983) 3279_ [2Sl H Huber and I Vogs Chcm_ Ph>s 64 0982) 399_ [29] H Huber and 2. Latajka, J. Compur C&an 4 (1933) 252_ [3@] P- Schuster. W_ Jakubetz and W. Marks. Topics Curr_ Chcm 60 (1975) l_ [31] EL Cohen and AL Poynrcr. J. MoL SW_ 53 (1974) 131_ 1321S-G- Kukdich. Ghan Phy= Lcttcrs 5 (1970) 401; 12 (1971) 216_ 1331SA. Ckqh Y_ Beers. G-P_ Klein and Ls Rothman. J_ Ghan Phys_ 59 (1973) 2254. [34] J. Vcrhocwn and A Dymanus, J. Ghan Phyx 52 (1970) 3222 [35l MI Frisch, JE Dd Bauc. K Fb&awzhari and JA Pop!c ulan Phys Leftas 83 (1981) 240. [36] J-E DcI Ben% MJ_ F&i&, R -X&X& and JA Pcpk I Phyr Ghan S6 0982) 1529_ [3fl J-E Dd Bmg M_J. Frinh. K lbgba and JA PopIe I P&x_ Chem_ 87 (1983) 73_ [38] JJ. Sakmann and CK_ Jrirgtnsca Hdv. Chin Acta 51 (1968) 1276_ 1391KG- Spears. J. C’hem Ph>x_57 0972) 1850_ [401 G-HI=_ Diacksaa and AX Sadkj. J_ Ghan Phys_ 75 (1981) 1253; GwHeF_ Dia-cksaz V. Kdlb and A_J_ Sadkj. J. chao_ Phys 79 (1983) 2918. [41] G-D. Zeiss, WJt Scar_ N_ scrmki D_P_ Chong and SR Langhoff. MOL Phys. 37 0979) 1543_ (421 B-He Wdfs and S Wilson, Mol. Phyr 50 (1983) 1295. [43) RI_ Woodin and JJ_ Bauchamp.J_AmChan.Soc100 (1978) 5OL [44lRAkdCS,KSlXlIotk.M_REUCZl~.DADiXoZI and DS Maqnick, J. Phyx Ghan 84 (1980) 2840_ [45] Dx- Bohme &d G-L Mackay. J. An Ghan Sot 103 (1981) 2173_