7966
J . Phys. Chem. 1992, 96,1966-1911
Computation of Accurate Electronic Molecular Polarizabilities Christophe Voisin, Alain Cartier, and Jean-Louis Rivail* Laboratoire de Chimie Thdorique, U.A. au CNRS No. 510, Universitd de Nancy 1 , Domaine Scientifique Victor-Grignard, B.P. 239, F-54506 Vandoeuvre-1;s- Nancy Cedex, France (Received: March 25, 1992)
A rule of operation to compute accurate values of dipole polarizability tensors for molecules containing the main functional groups present in peptides is presented. An extensive study of the influence of the basis set on equilibrium geometry and dipole polarizability has been performed on a series of target molecules. Methods for taking electron correlation into account are also considered. The CPHF/MPZ system associated with the adapted basis sets built is then applied to a series of 19 molecules and computed dipole polarizability tensors are found to be in good agreement with the known experimental values.
1. Introduction Knowledge of electronic polarizabilities which characterize the distortion of a molecular electron cloud by an external field is very important to describe intermolecular energies (induction term).’ In addition, the intermolecular force fields used in molecular mechanics computations are usually based on the assumption that the interaction between nonbonded atoms can be analyzed in the same way as intermolecular interactions. Therefore, a possible improvement of modern force fields is expected from the introduction of an induction term.2” Unfortunately accurate experimental data on the electronic polarizabilities of molecules are very rare, especially if one is interested in the whole tensor quantity instead of the usual mean value. It seems that modem quantum chemistry, with the help of powerful computers, is now able to become a substitute to difficult experimental determinations. The scope of this paper is first to select a convenient method which can be applied to large molecules in the sense of elaborate quantum chemistry, i.e. having more than three heavy atoms. For this purpose, we analyze the influence of the basis set and of the procedures used to include electron correlation effects in standard quantum mechanical methods which are commonly used to compute electronic polarizabilities. We then apply the method to determine the polarizability tensors for a series of molecules containing the main functional groups which are present in peptides, in order to obtain a set of reference data which shall be used in future works to define a system of polarizability increments to be used in a molecular force field. 2. Methodology
The performances of the computers allow the use of the coupled Hartree-Fock method7-I4(called CPHF). The programs used were HONDO-8,I5 which can perform CPHF at the SCF level only, and GAUSSIAN-88I6 in which electron correlation can be taken into account in a CPHF computation but is limited to the configuration interaction simple + doubles (CISD)17-’8 and to the second-order Maller-Plesset (MP2) level.l9 More elaborate methods for computing molecular polarizabilities can also be used, e.g., finite field methods in which the perturbation due to the external field is added to the Hamiltonian, or in which point charges are used to mimic the electric field. This approach offers the additional opportunity to describe the effects of an electric field gradient (by moving the charge), giving access to higher order polarizabilities. Whatever the method used for the computation, a correct description of the small distorsions of the molecular electron cloud requires an extended basis set which has to be defined carefully. 3. Selection of an Atomic Orbital Basis 3.1. Energy Optimized CCTOs. Numerous Gaussian bases are
exposed in the literature,20q21 few being used and chosen for their quality regarding the problem of polarizability computations. Two factors will be taken into account when searching for a basis set: it must be large enough to produce accurate results and at the 0022-3654/92/2096-1966$03.00/0
same time it must be limited to reduce computation time to reasonable values.22 The first step consists in choosing a good set of primitive Gaussians and to build a contracted Gaussian set (CGTO) by optimizing the contraction coefficients and, finally including additional polarization and diffuse functions adapted to the calculation of molecular polarizabilities. Despite the fact that various forms of GTO have been proposed (FSG023or GLF24), most standard ab initio calculations use primitive Gaussians centered on the atoms and include for g type functions. Primitive Gaussians are generally built up so as to minimize the H F energy in atomic computations; the first basis obtained by this method was published by Huzinaga:2s36(9s5p) and (1Os6p) for the atoms of the first series (C,N,O, ...). More recently, van DuijneveldtZ7 built up a high quality series extended to (13s8p) and to (18913~) by Partridge.28 Another approach proposed by Pop19 gives the STO-NG bases and uses the maximal overlap criterion with the corresponding Slater’s orbitals. The same author published the N-31Gmv3’series very often used in optimization or force constant calculations. The popularity of the later series is due to their short computation time, as they are highly contracted and use the same exponents { for s and p functions of the same shell = 12,; {3s = t3r..). Other bases exist, such as even tempered series32or well tempereP3 series. In our preliminary studies we observed that the number of functions needed to obtain the same accuracy as with the van Duijneveldt bases is always significantly larger. The duration of perturbation or configuration interaction calculations depends on the number of contracted functions only. So it is advantageous to use a lot of primitive Gaussians associated with a high contraction rate. Therefore, we studied several bases and the accuracy requirements associated with a low cost gave the following orientations: (i) Huzinaga’s b a ~ e s , 2(9s5p) ~ * ~ ~and (1oS6p) contracted following Dunning;3k36(ii) van Duijneveldt’s (12s7p) and (1 3s8p) contracted following David~on.~”’~ A qualitative approach of molecular polarizabilities can be achieved with more limited bases, such as the 5-31G and 6-31G bases of Ditchfield and P ~ p l e . ~ ~ * ~ ’ 3.2. Difhrpe and Polarization Orbitals. Quantitative calculation of the molecular polarizabilities requires the adjunction of additional functions to the initial basis set. In most cases a molecular property depends on the characteristics of the electronic density associated with a well-defined portion of space.40 Lowdin4I showed it was possible to build up wavefunctions which are arbitrarily close to the exact wavefunction, in the least squares sense, but whose error in the property may be arbitrarily large. Practically, we have to define an effective complete basis set, capable of yielding an atomic SCF within 1 mHartree of the Hartree-Fock limit and the molecular properties within 1%. The study of even tempered series by R ~ e d e n b e r gshowed ~ ~ , ~ ~it is easy to extrapolate the total energy to the HF limit, but the situation for properties other than the energy is more complicated. Davidson
(rb
0 1992 American Chemical Society
The Journal of Physical Chemistry, Vol. 96, No. 20, 1992 7967
Computation of Electronic Molecular Polarizabilities TABLEI: B8sbSets B1 6-31G
+
Ditchfield-Pople 6-31G30 polarization orbitals = 0.080 for hydrogen lap= 0.050 and {3d = 0.080for carbon = 0.007 and l3d = 0.100 for nitrogen fa,, = 0.090 and = 0.300 for oxygen B2 VD13 van DuijneveldtZ7with diffuse and polarization orbital^^^.^^ added. (1 3s8p3d/7s2p)/ (8s5p3d/4s2p) = B2 for C, N, 0. Hydrogen centered orbitals built up from a SadlejS8basis (13s8p3d/7s2pld)/(8~5p3d/4s2pld). B3 VD13' B4 DTZDPP D ~ n n i n g ~contracted ~.'~ in double zeta. Addition of a shell of diffuse and polarization orbitals59 (1 ls6p2d/6s2p)/ (4s3pZd/4s2p) B5 DTZP D ~ n n i n g ~ ' *contracted '~ in double zeta for C, N, and 0 and triple for hydrogen. Addition of a shell of diffuse and polarization orbitals59(9~5p2d/5sZp)/(4~3pld/3~2p) B6 Basis of R. D. AmosM (13~8p4d2f/8~3pld)/(8~6p4d2f/6~3pld)
rap rap
TABLE 11: H1O Energy, Moments, and Dipole Polarizabilities (RHF-SCF/CPHF) EHF, PI, k, 9YY 4n Ep
Nbb
hartree
exP B1 B2 B3 B4 B5 B6
31 61 73 63 40 112
-76.0288 -76.0604 -76.0618 -76.0428 -76.0333 -76.0666
D 1.85' 1.986 1.985 1.987 2.078 2.090 1.983
au 1.99 1.780 1.909 1.910 1.932 1.983 1.894
'Basis used (see Table I). bNumber of functions. *Reference 63.
au -1.85' -1.622 -1.827 -1.829 -1.858 -1.849 -1.793
au -0.09' -0,157 -0.082 -0.081 -0.074 -0.133 -0.102
€2
0.29 0.05 0.05 0.03 0.05 0.08
ax,,
aYY9
a,,,
a,
au 10.329 8.608 9.170 9.186 9.094 9.072 9.160
au 9.559 6.129 7.896 7.915 7.690 7.620 7.774
au 9.919 7.651 8.475 8.503 8.350 8.374 8.472
au 9.938 7.463 8.180 8.534 8.378 8.355 8.469
"asis
81 105
-40.2009 -40.2026
71 89
-56.2183 -56.2190
7: au
4.44 2.47 2.43 2.71 2.76 2.56
0.67 2.16 1.10 1.10 1.22 1.26 1.20
= [x(XBic - XxF')2]1/2.d y 2 = 1/2[Crs(a,, - a,,)2+ ~(CX,)~]. CReference61. 'Reference 62.
TABLE 111: CHJNHa Energy, Moments, and Dipole Polarizabilities (RHF-SCF/CPHF) B' Nbb EHF, hartree pX, D A,,: % axx= ayy,au CHI B2 B3 exP NH3 B2 B3 exP
cue
1.615 1.620 1.474e
-9.86 -10.20
a,,,au
a,au
Au: %
16.110 16.100 17.28d
16.110 16.100 17.28d
16.110 16.100 17.28d
-6.77 -6.82
12.722 12.745
13.305 13.287
12.920 12.926 14.82d
12.82 12.78
used (see Table I). bNumber of functions. cPercent of relative error versus experimental value. dReference 46. eReference 64.
and Fellerzoused a sequence of energy-optimized sets to extrapolate the dipole moment of CO to the (s,p,d) SCF limit, but their work showed the difficulty of this problem; this property is sensitive to the presence of diffuse functionsu which have almost no effect on the energy. Using a sequence of energy-optimized sets remains necessary to determine the convergence of molecular properties, but this sequence must contain additional functions which are crucial to the property of interest. A basii suitable to static dipole polarizability calculati~ns~~ must do the following: (i) It must give a good description of the modification of the electrostatic distribution by the perturbation. This can be obtained by adding polarization functions, of up to d type or more for the elements of the first series, and of p type for hydrogen.46 (ii) It must accurately reproduce the electron density of the unperturbed molecule even far from the nuclei. The energy-optimized sets found in the literature give a good representation of the density near the nuclei (core) with a high energetic contribution but have poor long-range behavior. Among the numerous papers relative to Gaussian sets for ab initio calculations, few show systematic studies on the choice of additional functions for properties e v a l u a t i ~ n . ~ ~ - ~ ' The first studies appeared in the 1970s: the polarizabilities were obtained by double perturbation (FPT),modifying some functions added to the initial basiss2,S3or to all the exponent^.^^.^^ Most of the recent works use the rules of Werner and Meyer,46 who use Hylleras' functionals6for optimizing all the functions in order to maximize the values of the polarizabilities. 3.3. Selection of Adapted Buses. From the preceding considerations, we have built up 30 sets of primitives designed to give polarizability values as accurately as possible for a series of molecules (HF, CO, H 2 0 , CH,, and NH3). In Table I, the five bases leading to the best values at the H F level are presented, complete with an extended 6-31G basis for comparison.
The results obtained for H 2 0 are given in Table I1 and it clearly appears that the results obtained with the B1 basis yields very poor results compared with the other bases. The results obtained for CH4and NH3 (experimentalgeometry) using the two bases giving the best results at the HF level for H20, are given in Table 111. The energy obtained at the SCF level for H 2 0with the B2 basis (-76.0604hartree) is very near the SCF limit (-76.067 f 0.002 hartree).6s
4. Effects of Electron Correlation 4.1. Methods. It is now well established that the effects of electronic correlation must be taken into account when predicting the quantitative values of many Three main techniques can be used: (i) many body perturbation theory (MBPT),69,70a form of perturbation theory associated with the names Rayleigh and Schriidinger (RSPT) with Ho chosen as the Hartree-Fock Hamiltonian was applied to n-electron systems by C. Maller and M. S. PlessetI9 (Maller-Plesset techniques: MP2,3,4,...); (ii) configuration interaction (CI),I7 which is conceptually (but not computationally) the simplest method for ob taining the correlation energy;69 (iii) coupled cluster method (CC),71,72 which has its origins in the so-called linked diagram theorem73which states that the exact electronic wavefunction and energy can be written as an exponential of cluster operators. This guarantees size extensivity. QCISD74is an intermediate technique between CI and CC methods, which aims at taking into account the effect of triple and quadruple excitations in a calculation of CISD type. 4.2. Uncertainty of the Correlated Model. A systematic study of the influence of the above mentioned techniques associated with the B2 basis on the computation of the dipole polarizabilities was carried out. The results concerning the water molecule are given in Table IV. Whatever the method used, the correlation energy
Voisin et al.
7968 The Journal of Physical Chemisrry, Vol. 96, No. 20, 1992 TABLE I V HzO Mwlc Polrriabllitia in Atomic Units nnd EleCtrOaiC COd&On (B2 BMb)’ wave function approximation Ear, FAD) €Po %x aYY SCF -76.060537 1.9859 0.131 9.171 7.901 -~ 0.006 10.003 9.587 -76.295596 1.8603 fc MP2 10.004 9.610 0.008 1.8631 -76.31945 1 F MP2? 9.679 8.897 0.027 1.8815 -76.297750 fc MP3 0.030 9.670 8.885 1.8844 -76.32201 7 F MP3 0.020 9.729 9.010 1.8752 -76.299446 fc MP4DQ 9.719 8.997 0.024 1.8783 -76.323599 F MP4DQ 0.003 9.876 9.266 1.8578 -76.301597 fc MP4SDQ 0.006 9.865 9.250 1.8610 F -76.325781 MP4SDQ 9.621 0.023 10.086 1.8320 fc -76.309218 MP4SDTQ 0.020 10.077 9.607 -76.333561 1.8350 F MP4SDTQ 9.61 1 8.814 1.8909 -76.289616d 0.036 fc CID -76.322017‘ 0.041 9.606 8.816 1.8953 F CID‘ -76.291 187J 0.026 9.699 8.946 1.8804 fc CISD -76.3140958 0.031 9.684 8.935 1.8856 F CISD‘ -76.308441 0.007 9.884 9.329 1.8571 fc QCISDh 9.873 9.31 1 -76.332754 0.005 1.8604 F QCISDh 8.999 0.021 9.718 -76.299537 1.8756 fc CCD 8.987 0.024 1.8787 -76.323681 9.709 F CCD 1A455 9.381 0.009 9.951 -76.3082241 fc CCD(ST4) 1.8491 9.367 0.005 9.942 F -76.332530 CCD(ST4)
a22
U
8.475 9.689 9.698 9.190 9.181 9.276 9.266 9.469 9.458 9.951 9.741 9.123 9.117 9.221 9.208 9.479 9.466 9.263 9.253 9.560 9.550
8.180 9.760 9.771 9.255 9.246 9.338 9.327 9.537 9.524 9.819 9.809 9.184 9.180 9.288 9.276 9.564 9.550 9.327 9.316 9.298 9.620
e‘,
2.47 0.39 0.39 1.16 1.18 1.00 1.04 0.69 0.71 0.25 0.30 1.29 1.30 1.11 1.13 0.65 0.67 1.04 1.06 1.27 0.55
timeb 70 3078 2242 3274 3383 3358 3468 3361 3507 3874 4456 2126 246 1 4910 6396 3664 449 1 4524 5967
-
“ t = [x;(ximIc X;~P)*]~/~. *CPU time (seconds) Gaussian 88. ?First analytical differenciation of energy followed by a second numerical differenciation(SCF-quadratic). Corrections of size consistency following P ~ p l e ’E(CIDSIZE) ~ = -76.299634 hartree. eCorrections of size consistency = -76.324259 hartree. ’Corrections of size consistency following P ~ p l e ’E(cIosIz) ~ = -76.301 735 hartree. 8Corrections following P ~ p l e ’E(CIDSIZE) ~ = -76.326391 hartrw. *Configuration interaction simple and double (CISD all single + double) of size consistency following P ~ p l e ’E(C-DsIZE) ~ followed by a correction of triple excitations (correction post CI following a perturbations scheme). ‘Experimental geometry, & = 1.81 llau., uboh = 104.45O. Unless indicated otherwise, dipole polarizabilities are computed by a double numerical differenciation of energy (SCF-DIIS).
recovered remains lower than 75% of its limit value. Nevertheless the majority of correlation associated with computed quantities, dipole moment or molecular polarizabilities, is recovered. This can be related to faster convergence of these quantities with the size of the expansion (and with the calibration of the B2 basis). The oscillation of Maller-Plesset techniques expanded over a complete space, during the passage MP2 MP3 MPBDTQ is clearly shown. In this case, the importance of simple and triple excitation (S,T) at the MP4 level is obvious. The frozen core approximation seems justified in almost all cases. The MP4DQ, MP3 calculations often give worse values than MP2. CID and CISD, very time-consuming, lead to relatively poor results when compared with perturbative methods. Finally, it appears that when a reliable description of the unperturbed function has been obtained (with diffuse and polarization functions) the CPHF method will give very homogeneous results. It underestimates the mean dipole polarizability by ca. 10-15% and gives a good estimation of the anisotropy. At the HartreeFock level, it will be preferred to the finite-field method because an analytical calculation of the derivatives does not introduce numerical errors and is faster. Nevertheless we have seen that it was necessary to take electron correlation into account. For HzO, the error on the estimation of the mean dipole polarizability computed using B2 basis is found to be -14.23% at the HF/CPHF level, -14.33% at the HF/FF level, -6.55% at the CISD/CHF level, and -1.18% for MP4/FF when using the experimental ge0metry.7~If we use a geometry corrected to include vibrational effects,46these errors fall to-13.44%, -13.42%,4.70%, and 0.8796, respectively. An underestimation of ca. 10-15% is typical of a CPHF/HF calculation as well as its correction of 1 0 4 % on introducing correlation. The vibrational corrections have very little effect (24%) and are independent of the method used. Unfortunately, the CPHF technique is still limited in its applications with multiconfiguration calculations: Gaussian 8816 program allows a CPHF determination of dipole polarizability at MP2 or CISD levels only for a full calculation (all the orbitals included in the active space). Looking at Table V the CPHF/MP2 technique seems to be a good compromise and the results recently published by Bloor” with its largest basis in a study on the influence of diffuse and polarization orbitals and of the correlation at MP2 level on the dipole polarizability of water molecule are very similar to our
- -
TABLE V HzO Polrriabiliti~(PU) (ExWriwaW Ccowtry) (W
Work)
A”! %
basis” method M ~ r p h y 6 ~exP CISD(CPHF) B2 CISD(CPHF) B2‘ B2 QCISD(FF) MP2(CPHF) B5 MP2(CPHF) B2 MP4(FF) B2 MP4(FF) B2‘ MP2(CPHF) B2?
uxr 10.32 9.68 9.97 9.87 10.08 10.00 10.08 10.38 10.29
1.11 0.82 0.67 0.69 0.38 0.30 0.18 0.17
-6.55 -4.70 -3.80 -2.38 -1.57 -1.18 0.87 0.42
d
10.0oO 9.609 9.737 0.35
-1.48
MP2(CPHF)
4..
9.55 8.94 9.02 9.31 9.19 9.61 9.61 9.72 9.72
-
u., 9.91 9.21 9.39 9.47 9.80 9.70 9.74 9.94 9.89
e/
-
“Basis (see Table I). * e = [ X : ~ ) ( U U, ,~~ P~) ~~] ~ / *A,,” = [(%lo u’cap)/uoxp] (100). Geometry corrected of vibrational deformations in its fundamental state.’* “R~8ultsobtained by J. E. Bloor” with a (7s6p6d/3s3p3d) basis.
results obtained with B2 basis set. 5. Evaluation of Moleculu Ceometriea In the choice of a suitable basis to determine molecular geometries, the problems arising will slightly differ from those encountered when computing the polarizabilities. A good compromise must be found between the size of the basis, the approximation level considered to take the correlation into account, and the algorithm used for the optimization. 5.1. Performraees of the Bmsea at tbe Hwtree-Fock Level. Many studies have been performed using STO-NG?e83 321G 4-21G,8es6 4-31G/5-31G/6-31G,’*B7 and Dunning’s seriesd-3 on m o l d e s containing elements of the first series. n e mean errors on the values of distances and angle are 0.03 A and vely, for STO-NG, 0.016 A and 2O for 3-21G/4-21G, 0.013 20* and lo for 6-31GL, and 1% and lo for DZP. At the Hartree-Fock level, it is generally found that STO-3G bases underestimate bond angles and that the 4-31G bases overestimate them. For bases larger than 6-31G. the results are better. 5.2. EIoctroa comhtion. Among all the techniques available to calculate electron correlation, only the simplest can be used to locate equilibrium geometry. Although efficient algorithms have appeared in this field, the multiple evaluation of gradients for CI, MCSCF, CCSD, ...,
resr
The Journal of Physical Chemistry, Vol. 96, No. 20, 1992 7969
Computation of Electronic Molecular Polarizabilities
TABLE VI: H,O Equilibrium Geometries No exP B1
basisa 6-3 1(PP~P) 6-31G* 6-31G* 6-31G* 6-31G* 6-31G* 6-31G* (853*/42) (853*/42) (853/42) (853/42)
B2*' B2*' B2* B2
Nbb
PGM
31 19 19 19 19 19 19 58 58 61 61
HONDO8 HONDO8 GAUSS86 GAUSS86 GAUSS86 GAUSS86 GAUSS86 GAUSS86 GAUSS86 GAUSS86 GAUSS86
method' SCF SCF SCF MP2(F) MP3(fc) CID(fc) CID(fc) MP2(F) SCF MP2(F) SCF
OPT QNd QNd B' B' FF' FPJ FP' Be B' B' QNd
Ro-h
EHF
'%oh
1.8111 1.7796 1.7903 1.7905 1.8300 1.8280 1.8253 1.8266 1.8279 1.7843 1.8272 1.7836
104.45 105.696 105.460 105.418 104.085 104.267 104.323 104.258 103.928 105.819 103.911 105.605
-76.0294 -76.0107 -76.0107 -76.0098 -76.0099 -76.0100 -76.0100 -76.0598 -76.0608 -76.0587 -76.0610
E,,
-0.18941 -0.19280 -0,009728 -0.18823h -0.25955 -0,26092
@Basis(see Table I). bNumber of basis functions. 'F = full; fc = frozen core. dQuasi-Newton algorithm (Hondo 8.0). 'Schlege19' algorithm. size consistency following P ~ p l e E~c,,srzE) :~~ = -76.204875; AE,, = -0.19484 hartree. Corrections = -76.205658; AE,,, = -0.19568 hartree. 'B2 basis (see Table I) with purely angular moment of size consistency following P ~ p l e :E(CISIZE) ~~ Gaussians. cu = - CY:XP)~]~/~. Aa% = [(amlc- aexp)/aeap](lOO).
f Flet~her-Powell~~ algorithm. 8 Corrections of
TABLE VII: Static Dipole Polarizabilities (nu) molecule
geoa
E, hartree
basisb EXP B2 EXP B2
-113.10847
HF
rE"p
-100.37256
H20
EXP B2 EXP B2 EXP B2 EXP B2 EXP B2 EXP B2 EXP B5 B5 EXP B5 EXP B5 B5 EXP B2 EXP B5 EXP B5 EXP B5 EXP B5
H2
co
CH4 NH3 NH4' H2CO CH3OH CHINH, CHINH3' CH3OCH3 CH3COOH CH3COOformamide acetamide N-methylformamide N,N-dimethylformamide N-methylacetamide
-1.159718
-76.319451 -40.371748 -56.451 115 -56.560486 -1 14.27386 -115.07477 -95.211892 -95.860992 -154.49345 -228.35382 -227.78690 -169.56788 -208.5081 1 -208.48914 -247.64672 -247.65966
we D 0.0000 -0.1 25g1 -0.2887 1.796g2 1.816 1.8Y3 1.862
0.0000 1.4784 1.525 0.0000 2.3984 2.352 1.77g5 1.746 1.29686 1.421 2.204 1.31W7 1.379 1.7v 1.479 1.3333 3.71@ 4.060 3.7281° 3.702 3.828'" 3.997 3.8W10 3.991 3.688'" 3.672
al
a2
a3
4.579h' 4.539 12.1h2 12.03 6.59h3 6.32 10.32h4 10.004 17.2gh5 16.763
4.579h1 4.539 12.lh2 12.03 5.1Oh' 4.97 9.55h4 9.610 17.2gh5 16.763
6.380h1 6.477 15.7h2 15.91 5.10h3 4.97 9.91h4 9.698 17.2gh5 16.763
13.715 9.44gh6 9.198 18.63h7 18.393 2 1.79hs 20.403
13.715 9.44gh6 9.198 12.95h7 13.047 17.8gh8 20.043
15.977 9.448h6 9.198 18.63h7 22.463 27.60h8 23.406
23.495 18.687 43.054h10 38.367
22.899 18.687 29.625h'0 30.505
26.8 12 21.620 33.337h'O 3 1.000
36.572 49.577
26.035 50.572
38.064 33.158
27.940
20.865
37.074
41.268
28.727
44.895
36.192
27.949
53.099
52.053
38.160
63.930
60.535
38.858
52.008
a
5.179h1 5.185 13.3h2 13.32 5.57h3 5.42 9.93h4 9.771 17.2gh5 16.763 14.82h5 14.409 9.44gh6 9.198 16.74h7 17.968 22.40h8 21.284 27.061h9 24.402 19.665 35.361"O 33.291 34.416"l 33.557 44.435 27.53h12 28.626 38.3h12 38.297 39.9h12 39.080 52.7h12 51.371 52.ghI3 50.467
B, d
Au: 5%
0.1 11
0.12
0.168
0.15
0.336
-2.7
0.384
-1.50
0.894
-2.99 -2.77
0.462
-2.72
3.865
7.33
4.917
-4.98 -9.82
5.310
-5.85 -3.52 3.98 -0.01 -2.05 -2.52 -4.42
"exp, experimental geometry; opt, geometry optimized as described in part 5 . bBasis (see Table I). 'Principal values of the polarizability tensor. = [ z : i ) ( a , a l c - a e )2~] 1/2~. cAu' = [(a,lc - aeX)/aeap](lOO). ,Basis of Gaussian functions of pure angular momentum (van Duijneveldt): (1 3s8p/8s) used uncontracted and with additional diffuse and polarization functions (following Clementi and Bishop) (14~9p3dlf/9s3pld).gI Taken from ref 93. g2 Reference 94. g3 Reference 61. @Reference 64. @ Reference 95. 86 Reference 96. g7 Reference 97. @Reference98. @Reference99. glOReference 100. h 1 Accurate calculated value ref 101. hZ Reference 102. h3 Reference 103. "Reference 63. h5Takenfrom ref 46. h6Reference 104. h7Reference105. h8Reference 106. h9Takenfrom ref 107. hloReference106. hllReference 108. h12Reference109. h13Reference110. 'Very close to the experimental value ref 1 1 1. deu
,
wavefunctions remain a very long process.89 Therefore the MP2 method will remain one of the most used techniques. For molecules containing more than four atoms of the first series, the use of bases including polarization orbitals and the introduction of correlation becomes rapidly prohibitive and we must reduce the active space to the valence region in the correlation calculation. This seems better than reducing the dimension of the Gaussian basis below DZP quality. This basis recovers ca. 60% of the correlation energy related to the valence region,22this proportion falling to 50% for (4-31Gl3-21G).
Gordon et a1.86consider that a 6-31G* basis is the minimum expansion at the MP2 level. found that the bond lengths computed with 6-31G*/HF are too short, and too long with 6-31G*/MP2 and that a reasonable agreement of 0.0075 A for the bond lengths and 0.2O for the angles is reached only with the TZP/MP2 combination. DeFrees et a1.% built the 6-31 lG* series especially adapted to Maller-Plesset calculations, which lead to a mean error of 0.006-0.013 A on the bond lengths at the MP4 level. 5.3. Selected Methodology for Geometry Optimization. A
Voisin et al.
7970 The Journal of Physical Chemistry, Vol. 96, No. 20, 1992
systematicstudy has been made in order to find a basis/method system for molecular geometry calculations, on a series of amino molecules similar in size to alanine. The bases used have to be better than double zeta with polarization functions. The insignificant difference between a 6-3 lG* basis and the Dunning’s DZP/TZP series directs our choice toward 6-31G* due to time considerations. Concerning electron correlation, whatever the method used, its inclusion systematically overestimates the bondlengths, while the angles are generally unchanged. However, careful analysis of the results shows that the MP3 method broadly gives a better set of results (0.005 A), but this results from compensation of errors. This fact must be taken into account before applying this system to other series of molecules. For the 6-31G*/MP3 system the frozen core approximation has no effect. So for the structural study of nitrogen-containingmolecules, we used the following method. (i) First a complete optimization is done with a 6-31G* basis at the H F level. (ii) This geometry is used as data for a single optimization cycle with 6-31G*/MP3 (frozen core). These considerationscan be illustrated by the results obtained for the water molecule with different methods summarized in Table VI.
6. Computation of Dipole Polarizabilities of a Series of Moleculea The methodology for computing molecular dipole polarizabilitia has been applied to a series of molecules containing hydrogen and atoms of the fmt series. The computed values fit the experimental ones well and the results are gathered in Table VII. The full molecular results including the atomic coordinates and the orientation of the principal axes of polarizability are given as supplementary material. Table VI1 shows that with a basis set of good quality (containing diffuse and polarization orbitals), the couple CPHF/MP2 gives homogeneous results conceming dipole polarizabilities. They are usually underestimated by less than 10% (les than 5% in most cases) and the effects of anisotropy are rather well described. This homogeneity of results leads us to consider that the discrepancy between the experimental and computed values of the polarizability in the case of formaldehyde is probably due to uncertainties on the experimental value. Dipole moments which can be considered as a test of the quality of the basis set fit well with the experimentalvalue except for methylamine and acetic acid. This has apparently no effect on the dipole polarizability computed for the latter molecule which is one of the closest to the experimental value. 7. Conclusion Sets of basis functions have been built up for computing molecular dipole polarizabilities of peptide fragments. They are described as B5 and B2, the latter being reserved for small systems in Table I. With these sets, it has been found that the best procedure to use for obtaining accurate values of the polarizabilities was the CPHF method associated with MP2 to take the electron correlation into account. When the geometries have to be op timized, the use of the B2 or B5 bases became too difficult and these bases had to be redaced. during the oDtimization steD bv 6-3 lG* at the MP3 leve~Witd these keth&ologia, good v&& of the dipole polarizabilities (mean values and anisotropy) were obtained for a set of molecules including peptide fragments. These conclusions are not general and, in particular, the sets of basis functions have perhaps to be modified when studying families of Other than -tide fragments. Nmehelm, the results obtained indicate that these rules of omration can be used for computing the polarizabilities of other peflde fragments for which the experimental values of the polarizabilities are unknown. Acknowledgment. This work is part of the “Doctorat de 1’Universitl de Nancy I” thesis of C.V. supported by a grant from the Minist€!re de la Recherche et de la Technologie. The computations have been performed on the IBM 3090-600 of the CIRCE (Orsay). We are grateful to J. M. Teuler for his computational assistance. We wish to thank the Centre National de
la Recherche Scientifique for financial support and IBM France (GS IBM-CNRS Modllisation Molkulaire) for a generous allocation of computationaltime. Helpful discussions with Drs. F. Colonna and E. Evleth are gratefully acknowledged.
Supplementary Material Auuilable: Tables of the full molecular results of the B2 basis set including the atomic coordinates and the orientation of the principal axes of polarizability ( 5 pages). Ordering information is given on any current masthead page.
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Fundamental Aspects of Lithlum Ion Transfer Xiaofeng Duan and Steve Scheiner* Department of Chemistry & Biochemistry, Southern Illinois University, Carbondale, Illinois 62901 (Received: May 4, 1992)
The transfer of the Li+ ion from one water molecule to another is studied in (H20-Li-OH2)+ by ab initio calculations, and the results are compared with analogous proton transfer in (H20-H-OH2)+. In both cases,the equilibrium geometry contains a centrally located Li or H ion, although R(0-0) is considerably longer in (H20-Li-OH2)+. Only a small stretch of the H bond is needed to yield a double-well potential, whereas the barrier does not appear in the Li potential until R(O-0) has been stretched by 1 A. The Li-transfer barrier rises much more gradually with further intermolecular stretch than in the case of the proton. Similarly,the Li barrier is less sensitive to angular deformations. Whereas the proton-transfer barrier is enlarged monotonically with each addition to the basis set, the behavior of the Li analogue relates instead to the dipole moment calculated for the water monomer. Many of these discrepancies can be understood on the basis of the highly ionic nature of the Li bond.
Introduction
In comparison to the volumes of data and interpretation that have built up over the years concerning the hydrogen bond,’-3 the analogous lithium bond remains largely unexplored. An interaction of this type was first observed in 19754using spectroscopic measurements of LiCl and LiBr combined with nitrogen bases in matrix isolation. The Li-X stretching frequency was found to shift in a direction similar to that observed in H bonds but by a much smaller amount. Another parallel was drawn to the possibility that the bridging Li, like the proton in a H bond, is capable of transferring from one group to the other. This pos-
sibility was indicated by a minimum in the curve which plots the relative frequency change vs a normalized difference in proton affinity of the two group^.^ Recent years have seen a rapid accumulation of insights into proton transfer from both experimental and theoretical perspect i v e ~ . ~It. ~has been learned, for example, that the barrier to transfer rises very quickly as the H bond is elongated and that angular deformations from linearity can produce not only increases in the transfer barrier but also strong perturbations in the relative energies of the two minima in the transfer potential?J Systematic ab initio calculations have revealed the sorts of errors likely to
0022-365419212096-7971$03.00/0 0 1992 American Chemical Society
