J. Phys. Chem. 1985,89, 2186-2192
2186
lAl state therefore cannot be regarded as a model for the surface state, However, it is no more likely that the triplet state would serve as an appropriate model, but it is tempting to speculate on the existence of two types of surface states, one outer weakly bound state corresponding to the triplet cluster state and one more strongly bound state, the two states being separated by an ap(25) J. E. Demuth, Surf. Sci., 84, 315 (1979).
preciable energy barrier. Transitions between such surface states have been observed for acetylene adsorbed on Pd(ll1) and Pt( 111) surfaces.26 The structures of the two adsorbed states closely resemble what might be predicted for the 3AIand ‘A, states of Ni(C2H2) using present results for Ni(C2H4). Registry No. Ni(C2H4),61 160-51-8. (26) J. E. Demuth, Chem. Phys. Lett., 45, 12 (1978).
Ab-Initio Prediction of Properties of COP, NH3, and CO,***NH, R. D. Amos,* N. C. Handy, P. J. Knowles, J. E. Rice, and A. J. Stone University Chemical Laboratory, Cambridge, U.K.(Received: August 8, 1984) The recent experiments by Klemperer et al. on small van der Waals molecules have demonstrated the need for a full understanding of the properties of these systems. In this paper, the results of large basis set calculations at the SCF level of accuracy are reported. The properties calculated include electric multipole moments through tenth rank, some second-order properties such as polarizabilities and magnetizabilities,and frequency-dependent second-order properties. From these, properties which describe the long-range interaction of these molecules may be calculated, such as C, and C8coefficients. A distributed multipole analysis of the charge distributions is given. The Buckingham-Fowler model to describe the interaction of CO,and NH3 is investigated and compared with SCF calculations on the complex. Both predict the T shape structure found by Klemperer.
1. Introduction Recently a series of papers has appeared which have established geometrical parameters of some van der Waals complexes. These experiments by Klemperer and his associates, and earlier work by Legon and Flygare and co-workers,’ have investigated numerous complexes involving the molecules H 2 0 , C 0 2 , NH3, H 2 C 0 , HF, H C N , and CzHzamong others. The interaction of these molecules presents a challenge to theoreticians to provide a complete understanding of their long-range interaction. The first requirement in such an investigation must be the accurate knowledge of the electromagnetic properties of each molecule, and this is the problem we are especially concerned with in this paper. We have now completed our development of a very general computer package which is specially designed for the calculation of first- and second-order properties of molecules using large basis sets, predominantly at the self-consistent field level of accuracy. Its structure and capabilities are described in section 11. Results are presented for the multipole moments, and the static polarizabilities and magnetizabilities, using standard Hartree-Fock theory. We have also calculated the dynamic counterparts of these properties, at both real and imaginary frequencies. The theory we have used for this, although fairly standard, is outlined in section 11. In section IV, a distributed multipole analysis for the oneelectron charge distribution is presented following the original ideas of Stone.2 With all the above properties calculated, it is possible to give a good description of the long-range interaction of these molecules. From the frequency-dependent properties, it is straightforward to calculate the c 6 and c8coefficients for the long-range attraction, and these are derived in section 111. Recently, Buckingham and Fowler3 presented a simple approach which used the distributed (1) F. A. Baiocchi and W. Klemperer, J. Chem. Phys., 78, 3509 (1983); G. T. Fraser, K. R. Leopold, D. D. Nelson, A. Tung, and W. Klemperer, J . Chem. Phys., 80, 3073 (1984); K. R.Leopold, G. T. Fraser, and W. Klemp e w , J . Chem. Phys., 80, 1039 (1984); G. T.Fraser, K. R.Leopold, and W. Klemperer, J . Chem. Phys., 80, 1423 (1984); K. I. Peterson and W. Klemperer, J . Chem. Phys., 80,2439 (1984); A. C. Legon, D. J. Millen, and S. C. Rogers, Proc. R.SOC.,London, Ser. A, 370, 21 3 (1980); A. C. Legon, E. J. Campbell, and W. H. Flygare, J . Chem. Phys., 76, 2267 (1982). (2) A. J. Stone, Chem. Phys. Lett., 83, 233 (1981).
0022-3654/85/2089-2186$01.50/0
multipole analysis, with the atoms embedded in hard spheres, to predict the geometry of van der Waals complexes. In section V this method is used to predict the geometry of the CO2.-NH3 complex, which turns out to be T-shaped. Of course, an obvious way to examine the long-range interaction is by an ab-initio calculation of the complex; in section V results of such SCF calculation are presented which are in good agreement with the experimental results of Klemperer’s group’ for the C-N distance. The paper therefore attempts to present all those properties of COz and N H 3 which are vital to the understanding of the interaction of these molecules. A discussion of the correlation effects on some of these properties is given in section 111. 11. Self-Consistent Field Calculations The calculations reported here use the recently developed self-consistent field program specially developed at Cambridge by Amos4for the calculation of molecular properties. The program is written for the R H F formalism and has the following capabilities. (a) Energies, energy gradients, and second derivatives may be analytically calculated by using s, p, d, and f contracted Gaussian basis sets on any center. The program is based on the Rys polynomial quadrature5 scheme and is described in detail in ref 4. There is an upper limit in practice of about 200 basis functions. (b) The one-electron properties which can currently be calculated include electric multipole moments through tenth rank and the expectation values of the angular momentum operators L and L2. (c) Static second-order properties apQ(0) may be calculated through the direct solution of the coupled-Hartree-Fwk equations6
Here L is the Hessian matrix which contains the coupling of all nonredundant orbital rotations to first order, g(p)is the perturbation (3) A. D. Buckingham and P. W. Fowler, J . Chem. Phys., 79,6426 (1983). (4) R. D. Amos, CADPAC, The Cambridge Analytic Derivatives Package, publication CCPl/84/4, SERC, Daresbury Laboratory, 1984. ( 5 ) J. Rys, M. Dupuis, and H. F. King, J . Comput. Chem., 4, 234 (1983). (6) R. M. Stevens, R. M. Pitzer, and W. N. Lipscomb, J . Chem. Phys., 38, 550 (1963).
0 1985 American Chemical Society
Ab Initio Study of C 0 2 , NH3, and C02-.NH3
The Journal of Physical Chemistry, Vol. 89, No. 11, 1985 2187
vector constructed from the one-electron property integrals (it&), and contains the first-order perturbed orbital coefficients expressed in term: of the unperturbed S C F orbitals. Notice that in the case where P is pure imaginary (e.g., L), the Hessian matrix matrix h takes a different form to the usual real perturbation case. Solution of eq 1 using the method of Pople' efficiently obtains second-order properties such as the polarizability and magnetizability, as well as the response to higher multipole perturbations, for any reasonable size of basis set. (d) Dynamic second-order properties aw(w),which are the linear response to a perturbation of the form dwfp,are calculated through the dynamic analogue of the coupled Hartree-Fock equations (Slth(2)SIh(I)- w2)X(p#) = S-lth(2)S-lg(l) (2)
(3) aPQ(w) = 2g(p).X(Q~) These equations hold for any variational method (for example, MCSCF,*q9C I with fixed orbitalsI0) but here we are concerned primarily with closed-shell SCF. In this case, S i s the unit matrix. If P is a real perturbation, then h(I)is the Hessian matrix used to obtain the polarizability, while fr(2) is that used to calculate the magnetizability. Previous applications of these equations",12 have proceeded by followed by concomplete diagonalization of the matrix b(2)h(') struction of the properties at desired frequencies and interpretation of the eigenvalues as excitation energies. However, this interpretation must be of limited value, because this single configuration approach cannot reasonably be expected to represent excited states adquately. However, we notice that for a stable Hartree-Fock solution, h(I)and h(2)are positive definite, and so with imaginary and small real frequencies problems arising from the incorrect position of the singularities of (2) are not expected to be important. is the most efficient method of obtaining Diagonalization of !d2)h(l) the properties at many frequencies for basis sets up to the size of those considered in a recent paper by Visser, Wormer, and Stam,I3but for the calculations described here where there are up to 1400 nonredundant orbital rotations, it is clearly impossible. Thus for large basis sets we solve (2) iteratively in a similar manner to the static C H F equations. This can be org?nized such that the equations are solved for all perturbations P (e.g., all tensor components) simultaneously, and when treating a sequence of frequencies, the solution from the previous calculation can be used as an initial guess. In this way the calculations reported in section 111, solving for 11 perturbations simultaneously, required an average of 9 iterations per frequency, or 43 evaluations of the action of h on a trial vector. (e) The principal application of these dynamic properties which we report here is the ab initio computation of long-range intermolecular dispersion coefficients. In the region of negligible overlap, the dispersion part of the interaction energy between two closed shell molecules A,B may be written as14-37946
u= -(4
ne,)-
x 1 I 1,11'1,'
~-11-1,~-1,-1,'-~
X
L,L, JK ,K,
where the orientation of the two molecules enters through the functions
In these expressions is a Wigner 3j symbol, oLMK(O)
,:
);
a Wigner rotation matrix element, and C,M(B,(p)a modified spherical harmonic. The molecular properties are contained in the quantities
(6) where the dispersion energy integrals ykf$f$ik2t are defined by
using a slight extension of Visser, Wormer, and Stam's notationI3 to deal with nonlinear molecules. Here ap,/ltk((if.o)is the polarizability resulting from two electric mu tipo e perturbation operators of the fprm $/k
= Ee/rt'c/k(ei,'#'i)
evaluated a t imaginary frequency iw. Thepita1 quantities in the dispersion energy are the integrals X!k${?k2k; which must be obtained for each pair of second-order If the full diagonalization of b(2)8(1) properties ( Y $ , l / l t k l l , (Y/,kZk$ki. B has been performed for both molecules A,B, then these integrals are obtained analytically from the eigenvalues and eigenvectors, together with the vectors g(p).'3However, for the size of calculation reported here, this is not open to us, and so the integrals must be computed numerically, evaluating the a(iw) for a finite set of values for iw. An attractive approach is to fit the calculated properties to a functional form which is appropriate to these quantities. The most obvious form is the truncated sum over a few states
or alternatively a truncated continued fraction representation of the property." While this is an attractive idea, it is found that the actual fitting process is rather difficult, since it is so ill-conditioned when one is considering the imaginary frequency range. An alternative is to use some standard quadrature scheme, which prescribes uniquely the values of w at which the properties are to be computed. We have found that the transformation w = iwo(l - t)/(l
+ t)
(9)
followed by Gauss-Legendre quadrature on the range -1 It I 1 gives better accuracy for a given number of integration points than the method of Koide, Meath, and Allnatt," where the transformation w = iwo tan t (7) J. A. Pople, R. Krishnan, H . B. Schlegel, and J. S. Binkley, Int. J . Quantum Chem. Symp., 13, 225 (1979). (8) P. J. Knowles, to be submitted for publication. (9) E. Dalgaard, J . Chem. Phys., 72, 816 (1980). (IO) S. Iwata, Chem. Phys. Lett., 102, 544 (1983). (11) P. Jargensen, Annu. Reu. Phys. Chem., 26, 359 (1975). (12) J. Oddershede, Adu. Quantum Chem., 11, 275 (1978). (13) F. Visser, P. E. S . Wormer, and P. Stam, J . Phys. Chem., 79, 4973 (1983).
(10)
was used. We have found in a series of small basis set calculations (14) F. London, Trans. Faraday Soc., 33, 8 (1937). (15) A. D. Buckingham, A h . Chem. Phys., 12, 107 (1967); Discuss. Faraday SOC.,40, 232 (1965). (16) A. Dalgarno and W. D. Davison, Adu. A t . Mol. Phys., 2, 1 (1966). (17) A. Koide, W. J. Meath, and A. R. Allnatt, J . Phys. Chem., 86, 1222 (1982).
2188
The Journal of Physical Chemistry, Vol. 89, No. 11, 1985
that an optimum value for wo in (9) is approximately twice the geometric mean of the first excitation energies of the interacting molecules. But for a reasonable number of integration points (28) the accuracy of the integration is rather insensitive to the precise choice of wo. Accordingly, we used the value wo = 0.3 in all our calculations, with a 12-point Gauss-Legendre integration scheme. 111. Calculation and Results ( a ) Basis Sets. The calculations on N H 3 used a 7s5p3dlf set on N and a 5s2p set on H. The 7s5p set is a contraction of a 13s8p set of van Duijneveldt.I8 The d functions were uncontracted with exponents of 1.2, 0.4, and 0.12. The largest of these was energy optimized, and the smaller ones were included to give a good description of the multipole moments and polarizabilities. The f function exponent was 0.75. The H basis set consisted of a 5s contraction of the 10s set from ref 18 and uncontracted p functions with exponents of 1.0 and 0.35. The S C F energy at the experimental equilibrium geometry RNH= 1.012 A, LHNH = 106.7’ is -56.222; the Hartree-Fock limit is estimated to be -56.225 hartree.I9 The basis set for COz contained 5s4p3dlf functions on each center. The 5s4p part of the basis is due to Dunning.2o The uncontracted d functions have exponents 0.8, 0.2, and 0.05 for C , and 1.6,0.4, and 0.1 for 0. The f function exponent was 0.25 on C and 0.5 on 0. Owing to the smaller (sp) set used this basis will not approach the Hartree-Fock limit for the energy as closely as did the N H 3 calculations, though the molecular properties should still be well described. The largest S C F calculation on C 0 2 is due to Yoshimine and McLean,*l who obtained -187.723 hartree, using a basis set of Slater functions. The energy obtained here is -187.709 hartree a t the experimental geometry [Rco= 1.161 A].22 ( b )Previous Calculations. There are no previous studies which have dealt with all the properties considered here. However, there are several calculations of the low-order multipole moments and static polarizabilities using accurate SCF calculations, and in some cases including estimates of the correlation c o n t r i b u t i ~ n . ~ The ~~’ most accurate SCF calculation of any property of NH3 is the study of the dipole polarizability and hyperpolarizability by Lazzeretti and ZanasLZ4 The most detailed calculations of the multipole moments were those by van Duijneveldt et al.,2Swho give values for the dipole, quadrupole, and octopole. Other calculations on N H 3 which should be mentioned are the studies of the dipole moment and dipole polarizability by Werner and MeyerZ6and by Diercksen and SadlejSz7 These included estimates of the correlation effects using CEPA and MBPT. The present S C F calculations are comparable in accuracy to the best of these previous results. Calculations on the properties of C 0 2 are comparatively rare, though there are a small number dealing with the quadrupole moment and the static dipole polarizability, and some of these have used correlated wave function^.^^-^^ The present S C F calculations are the largest and most detailed yet published on the properties of C 0 2 . ( c ) Multipole Moments. The simplest of the properties calculated are the multipole moments of the charge distributions. (18) F. B. van Duijneveldt, IEM Res. Rep., RJ945 (1971). (19) W. C.Ermler and C. W. Kern, J . Chem. Phys., 61, 3860 (1976). (20) T. H.Dunning, J. Chem. Phys., 55, 716 (1971). (21) M. Yoshimine and A. D. McLean, Int. J. Quanrum Chem., S1, 313 (1967). (22) G. Herzberg, “Electronic Spectra of Polyatomic Molecules”, Van Nostrand, 1966. (23) P. Lazzeretti and R. Zanasi, Chem. Phys. Lett., 39, 323 (1976). (24) P. Lazzeretti and R. Zanasi, J . Chem. Phys., 74, 5216 (1981). (25) J. G.C. M. van Duijneveldt and F. B. van Duijneveldt, J . Mol. Structure (THEOCHEM), 89, 185 (1982). (26) H.J. Werner and W. Meyer, Mol. Phys., 31, 855 (1976). (27) G.H. F.Diercksen and A. J. Sadlej, J. Chem. Phys., 75, 1253 (1981) Chem. Phys., 61, 293 (1981). (28) M. A. Morrison and P. J. Hay, J. Chem. Phys., 70, 4034 (1979). (29) W.B. England, B. J. Rosenberg, P. J. Fortune, and A. C. Wahl, J . Chem. Phys., 65, 684 (1976). (30) R. D. Amos and M. R. Battaglia, Mol. Phys., 36, 1517 (1978).
Amos et al. TABLE I: Self-Consistent Field and Msller-Plesset (Second-Order) Corrected Values for the Multipole Moments of CO, (in Atomic Units)“
Q2.2 Q4,O
Q6,O Q8,O Q10.o
SCF
SCF + MP2
-3.783 -1.223 55.79 758.0 7769
-2.739 (quadrupole) 0.822 (hexadecapole) 51.87 647.1 6429
“ See eq 11 for definitions. * Experimental value for the quadrupole moment is -3.3
* (1.1.~~
TABLE II: Self-consistent Field and MoUer-Plesset (Second-Order) Corrected Values for the Multipole Moments of NH3 (in Atomic Units) with the Origin at the Center of Mass SCF SCF + MP2
Q I .o: Q2.0 Q3.0 Q3.30 Q4,O
Q4,30
Q5.0 Q5.30 Q6.0 Q6.30 Q6,60
0.6353 -2.164 -2.462 4.311 1.203 8.968 12.44 -2.412 10.97 -34.26 34.51
0.605 (dipole) -2.200 (quadrupole) -2.509 (octopole) 4.279 1.377 (hexadecapole) 8.605 12.20 -2.672 10.28 -33.47 33.42
“Experimental value for the dipole moment is 0.57935for the ground vibrational state, leading to approximately 0.59426 at equilibrium. bExperimental value for the quadrupole moment is -2.4 f 0.3.36
In their spherical tensor form, these are the expectation values of the operators
where C,,are modified spherical harmonic^.^^,^^ It is convenient to use the real multipole moments Qlkcand Qlkrwhere
Explicit expressions for these quantities have been given by Price, Stone, and A l d e r t ~ n together ,~~ with the relationship between the multipole moments in their spherical tensor form and those in Cartesian form.33 The self-consistent field values for these moments for C 0 2 ( I < 10) and N H 3 ( I < 6) are given in Tables I and 11. The effects of electron correlation on the multipole moments are estimated from second-order Mdler-Plesset (MP2) t h e ~ r y , ~ ’ with the second-order correction to a one-electron property defined by Amos38 as ( W ) I Q p P ( ~ )+) 2( \k01Q19(2))
(1 3)
For most molecules the second-order correction accounts for the bulk of the correlation contribution, though in some cases (see especially ref 39), the higher terms are by no means negligible. (31) S.L. Price, A. J.Stone, and M. Alderton, Mol. Phys., 52,987(1984). (32) D. M. Brink and G. R. Satcher, “Angular Momentum”, Oxford University Press, London, 1968. (33) A. D. Buckingham, “Intermolecular Interactions from Diatomics to Biopolymers”, B. Pullman, Ed., Wiley, New York, 1978. (34) M. R. Battaglia, A. D. Buckingham, D. Neumark, R. K. Pierens, and J. H. Williams, Mol. Phys., 43, 1015 (1981). (35) A. L. McClellan, “Tables of Experimental Dipole Moments”, Vol. 2, Rahara, El Cerrito, CA, 1974. (36) S. G. Kukolich and D. H. Casleton, Chem. Phys. Lett., 18, 408 (1973). (37) C.M~illerand M. S. Plesset, Phys. Reu., 46, 618 (1934). (38) R. D. Amos, Chem. Phys. Lett., 73, 602 (1980);88, 89 (1982).
The Journal of Physical Chemistry, Vol. 89, No. 1 1 , 1985 2189
Ab Initio Study of C 0 2 , NH3, and C02-NH3 TABLE III: Dipole and Quadrupole Polarizabilities“of C02 as a Function of Frequency w (in Atomic Units)b at SCF Accuracy dipole
TABLE V Magnetizabilities of C02 and NH3 at SCF Accuracy with the Origin at the Center-of-Mass (in Atomic Units)
co2
quadrupole
w
a10,10
a11,1-I
0 0.06 0.07 0.08 0.09 0.1
23.700‘ 23.947 24.041 24.148 24.270 24.409
-11.943‘ -12.022 -12.051 -12.084 -12.122 -12.165
a21,2-1
a22.2-2
-193.89 -195.55 -196.17 -196.88 -197.71 -198.65
41.08 41.33 41.42 41.53 41.65 41.79
a20.20
224.11 225.25 225.66 226.15 226.70 227.32
“The relations between the spherical harmonic and Cartesian tensor definitions are alo,lo = azz,all,l-l= -a,,, = 3~2,,,2, a21,2-1= -4C,2,x,, a22,2-2 = 4CXy,. bSee text for definitions. Experimental values4 a t w = 0: alo,lo= 26.6, all,l-l= -12.9.
x”,
xi, XI2
xR X
AX
” Experimental value is 21 .36.45 0.62, ,g = 1.20.44
12.960” -12.583“ -1.276 -5.047 -6.603 57.37 -63.26 61.70 -7.629 5.221 -6.985 -13.78 -5.238
a11.1-1
a20.10
a21.1-1 a22,11
a20,20 a21.2-I
ff22.2-2 ff22.21 a30.10
a31,I-l a32.11
a33,10
QII QI-I
= (1/2)-i’2[Qiic
+ iQiisl
(l2k24%?
X
(1010) (1 11-1) (1 11-1) (1010) (111-1) (1010) (111-1) (1010) (111-1) (1010) (111-1) (1010) (1 11-1)
45.21 -26.22 15.55 510.4 -304.0 -415.5 246.1 94.0 -56.3 369.6 -217.3 -230.5 137.0
EZ,Z,,
= -(1/2)-1’2[~x
+ by1
= (1/2)-1/2[Q1~c - iQ11s1= ( 1 / 2 ) - i / 2 [ ~ ,- b y ]
and therefore
are written in terms of the Cartesian A, C, and E tensors.33 For NH3, there have been previous calculations for the dipole-dipole polarizabilities, and we are in agreement with those, but these are the first calculations on the higher polarizabilities. The same is true for COz. The SCF static polarizabilities of C 0 2 are within about 10% of the experimental values,40and most of the error can be attributed to correlation effects. The frequency dependence in the optical region is also in agreement with experimental values, showing an increase in magnitude of 1% over the range w = 0.0 to w = 0.1. Similar agreement can be seen for NH3, but note that the experimental values are obtained at optical frequencies for the ground vibrational state.41 The magnetizabilities of C 0 2 and NH3 are given in Table V. There are previous SCF calculations of comparable quality in the case of NH3,42,43 though the present results are probably the most reliable. There do not appear to be any earlier calculations of similar accuracy for C 0 2 . Those experimental values which are a ~ a i l a b l e ~ ~ , ~ ~ indicate that the paramagnetic terms have been obtained with an accuracy of 5-lo%, which is all that can be expected from an SCF calculation. The diamagnetic terms will be considerably more accurate.30 (e) Dispersion Energy Integrals. The integrals X!,f$f?&,’defined in eq 7 are given in Tables VI-VI11 for C02.-C02, NH3-NH3, and C02-NH3. The tables include values for those integrals which are necessary R-’, and R8in the for the evaluation of the coefficients of R6, dispersion energy formula. The isotropic C, and Cscoefficients are given by the following formula for dissimilar molecules: c - 2 xiiii 1111 - 2 x 1 1 1 1 + 4x1111 6
+ ayyl These negative values are a consequence of the phase conventions in the definition of spherical harmonics32and have no physical significance. This notation is entirely consistent with the usual definition of CY through al1,i-l
= -(1/2)[a,,
= Ca/k./’VF/’k* where F/tKis the applied field and Qlk is the induced moment. Values for the dipole-dipole, dipole-quadrupole, quadrupolequadrupole, and dipolmtopole polarizabilities are given for C 0 2 and N H 3 in Tables 111 and IV. Also in the tables, the (Y/Ikl/l,kl, Q/k
(39) G. H. F. Diercksen, V. Kellb, and A. J. Sadlej, J. Chem. Phys., 79, 2918 (1983).
xtx =
2
(1010) (1010) (1 11-1) (2020) (2020) (2 12-1 ) (212-1) (222-2) (222-2) (3010) (3010) (31 1-1) (31 1-1)
-(2/ 3 1/2)Ax,xz -(2/ 3 1/2)Ax,xx 3 CZZ,, -4CxzJ2 4CXYJY -4cxx,xz
The results for C 0 2 should be the best SCF values published to date; they are in agreement with previous values where publ i ~ h e d . ~ ~ For - ~ ONH3, the results are in agreement with those in ref 23-27. In NH3, MP2 for the dipole moment yields a value closer to the experimental dipole moment, picking up 60% of the correlation effect. For C 0 2 , the MP2 correction has overshot the experimental value: the hexadecapole is very small and its sign is uncertain. Indications are that the higher terms in the Maller-Plesset series are important for C 0 2 . For NH3 several basis sets were tried, and the higher moments were stable to 5%. (d)Polarizability and Magnetizabilities. The formulae for the frequency-dependent polarizability a/l!,l(k,,(u) have been given earlier, eq 3, and correpond to the change in the multipole moment ( Qllkl)due to the external field Qlllklt, in the static case. If one is using the real multipole moments (12), then the corresponding dipole polarizabilities are all positive. However, it is convenient to express the polarizabilities in terms of the spherical harmonic operators (1 l ) , as this simplifies the averaging of the dispersion energy integrals (see section IIIe). If this is done then some polarizabilities are negative, for example
1
(likili’ki)
Az,22
“Experimental values a t optical frequencies for the ground vibrational state:41 a l o , , o = 16.28, a l l , l - = l -14.33.
Experimental values are
c02...c02
a22
-(3 / 2)1/2Ex,xzZ -(3/5)1/2E,, -(4/5)”2EzJxx
-4.31 0.59b -4.66 1.15b -3.65 -0.21
TABLE VI: Dispersion Coefficient IntegralsX)l~l’!~’k’, (Eq 7) for
TABLE IV: Static Polarizabilities of NH3 at the SCF Level (in Atomic Units) a10,.10
“3
-25.71 20.02“ -5.51 0 -5.63 0.18
- /3[
0000 - 2x001-1
cs= [ - y 2001-1 11 - xi11-100 22 ~X;!;;-I -
*211 1-100
-
1-100
1-11-11
+ ~ 2 1 +1 2x1122 + 4x2211 0000
001-1
- 4%!;!-1
ow0
1-11-1
1122 + - 4x1-12-2
+ W 2 1 1 2-200
2XAA:!2]
+
(1 4)
and are shown in Table IX. (40) M. P. Bogaard, A. D. Buckingham, R. K. Pierens, and A. H. White, J . Chem. SOC.,Faraday Trans. 1 , 75, 3108 (1978). (41) N. J. Bridge, A. D. Buckingham, Proc. R. SOC.London, Ser. A, 295, 334 (1966). (42) R. Holler and H. Lischka, Mol. Phys., 41, 1017 (1980). (43) P. Lazzeretti, R. Zanasi, and B. Cadioli, J. Chem. Phys., 67, 382 (1977). (44) S.G. Kukolich and W. H. Flygare, Mol. Phys., 17, 127 (1969). (45) J. W. Cederberg, C. M. Anderson, and N. F. Ramsey, Phys. Rev. A, 136, 960 (1964).
2190 The Journal of Physical Chemistry, Vol. 89, No. 11, 1985 TABLE VII: Dispersion Coefficient Integrals xk,kl'!&ir (Q 7) for NHy.NH3
(1010) (1 11-1) (2010) (2010) (21 1-1) (211-1) (221 1) (221 1) (2020) (2020) (212-1) (212-1) (222-2) (222-2) (2221) (2221) (3010) (3010) (31 1-1) (31 1-1) (3211) (3211) (3310) (3310) (2010) (2010) (2010) (211-1) (21 1-1) (221 1)
(1 11-1) (1 11-1) (1010) (111-1) (1010) (101 1) (1010) (111-1) (1010) (111-1) (1010) (1 11-1) (1010) (111-1) (1010) (1 11-1) (1010) (111-1) (1010) (11 1-1) (1010) (111-1) (1010) (1 11-1) (2010) (21 1-1) (2211) (211-1) (221 1) (221 1)
-12.55 13.03 -0.84 0.87 -4.36 4.51 -5.35 5.52 57.9 -60.1 -63.4 65.8 67.3 -69.9 -6.32 6.53 3.17 -3.23 -2.99 2.99 -10.1 10.4 -4.00 4.12 0.094 0.335 0.41 1 1.60 1.98 2.49
TABLE VIII: Dispersion Coefficient Integrals Xklkl'!i'i for 2 1 NH*.*.CH, I
(ioioj (1 11-1) (1 11-1) (2010) (2010) (21 1-1) (21 1-1) (21 11) (21 11) (1010) (1010) (1010) (1010) (1010) (111-1) (111-1) (1 11-1) (111-1) (111-1) (2020) (212-1) (2221)
(1 1 1 4 ) (1010) (111-1) (1010) (1 11-1) (1010) (1 11-1) (1010) (111-1) (2020) (212-1) (222-2) (3010) (31 1-1) (2020) (2 12- 1) (222-2) (3010) (31 1-1) (1010) (1010) (1010)
(222-2)
(1010)
(3010) (311-1) (321 1) (3310) (2020) (212-1) (2221) (222-2) (3010) (31 1-1) (321 1) (3310)
(1010) (1010) (1010) (1010) (111-1) (111-1) (1 11-1) (111-1) (111-1) (111-1) (111-1) (111-1)
-13.53 -24.28 14.08 -0.83 1.62 -8.42 4.77 -10.32 5.75 263.1 -214.4 48.48 190.8 -1 18.9 -274.1 223.1 -50.5 1 -198.4 123.7 112.0 -122.7 -12.19 130.3 6.07 -5.64 -19.39 -7.71 -65.4 71.6 6.83 -76.7 -2.98 2.66 10.67 4.28
Because this is the first time that these integrals have been calculated, we cannot make any comments; indeed we believe this
Amos et al. TABLE I X Isotropic C,and C8 Values for C02--C02, NH3-NH, and C02--NH3(in Atomic Units) CO2...C0* N H yNH, COy..NH,
C 6
C*
141.58 76.28" 103.6
7100 1960 3800
'Experimental value is 89.08.51 is the first time they have been evaluated for nonlinear molecules.
Iv. Multipole Moment and Distributed Multipole Anaysis (a) Carbon Dioxide. The multipole moments of COz are given in the second column of Table X, up to rank 10. The quadrupole moment is quite large and negative, the hexadecapole moment small, and the higher even moments increase to large positive values. Since the contribution of moment Q,to the electrostatic potential at a distance R is proportional to R("'),the contributions of the higher moments Q4,Qs,... are small at moderate distances ( R 3 5 bohr) and C02behaves very much like a pure quadrupole at such distances. At shorter distances, such as occur in as solid, the higher moments cannot be ignored, and have a dramatic effect on the proper tie^.^^ It is then helpful to use a distributed multipole expansionZ to describe the charge distribution. The technique depends on the well-known property that the product of two spherical Gaussian functions on different centers is another Gaussian at an intermediate center. The product of Cartesian Gaussian atomic functions with angular momentum 1, and l2 (Le., containing factors xilJzk of total degree i + j + k equal to I , and 12, respectively) is similarly a sum of Cartesian Gaussians of angular momenta up to 1, + 1,. The associated charge distribution has a multipole expansion about the center of the product Gaussian which terminates at rank lI + 12. This finite expansion can be represented by an infinite multipole expansion about the center of the molecule, and this is a convenient way to calculate highorder moments. However, it is possible to designate several sites in a molecule (e&, the nuclei) and to represent each overlap distribution by a multipole expansion about the nearest site. The resulting many-site expansion yields a much more rapidly convergent expansion of the electrostatic potential, because the multipole expansion at each site describes only the charge distribution in its immediate vicinity. (The radius of convergence of a multipole expansion is determined by the distance to the farthest point of the charge distribution represented by the expansion; beyond this point, the multipole expansion formally converges, though convergence may be slow out to significantly greater distances.) The third and fourth columns of Tables X give the contributions for carbon and one of the oxygen atoms to a distributed multipole expansion of C 0 2 . The contributions for the other oxygen atoms are obtained by changing the sign of the odd moments. It can be seen that the contributions from oxygen are all small, while those on carbon increase more slowly than before. A comparable degree of convergence in the electrostatic potential is now obtained at distances of about 4 bohr or less. The distributed multipole analysis can also give a picture of the charge distribution which relates quite closely to conventional ideas of chemical bonding.2 For example, the DMA for C 0 2 exhibits inward-pointing dipoles on the oxygen atoms, associated with lone-pair electron density beyond, and these are responsible for most of the overall negative quadrupole moment of the molecule. However, the chemical interpretation is very largely obscured when diffuse high-angular momentum basis functions are used; for example, a diffusef,,, function on C can represent some of the electron density in the region of the oxygen atoms, but the electron density associated with this function is attributed in the DMA to carbon. Such effects are particularly noticeable in the five-center DMA (46) A. J. Stone and R. J. A. Tough, Chem. Phys. Le??.,110, 123 (1984). (47) C. S. Murthy, S. F. OShea, and I. R. McDonald, Mol. Phys., 50, 531 (1983).
Ab Initio Study of C 0 2 ,NH3, and C02-NH3
The Journal of Physical Chemistry, Vol. 89, No. 11, 1985 2191
TABLE X Distributed Multipole Expansions for Carbon Dioxide SCF with MP2 Corrections (atomic units) 3
moment
1 site, C,
0.0 bohr
Qo
site
C, 0.0 bohr
0, 2.194 bohr
C, 0.0 bohr
-0.1632
0.08 16 -0.5 134 0.2257 0.6601 -1.555 2.55 -3.43 4.16 -4.20 2.84 0.60
-0.8797
QI
Qz
-2.7387
0.5301
0.822
18.956
51.87
75.90
647
216.9
6429
482.6
Q3 Q4 Q5 Q6 Q7 Q8 Q9
QIO
TABLE XI: Distributed Multipole Expansion for Ammonia SCF with MP2 Corrections" site position X
Y z
1 site, center of mass
4
N 0.0 0.0
0.0 0.0 0.0
-0.1278
0.605
-0.543 0.208
-2.200 -2.509 4.279 1.377 8.605
-0.913 -0.420 -0.520 -0.436 0.811
5 site bond center, 1.097 bohr 1.8689 2.0865 0.7945 1.6039 -0.6 178 2.4010 -1.2302 4.4937 -1.8673 4.8420 -1.9909
-5.7720 -8.2473 -9.2319 0.1592 0.0741
3-21G
QIO
(PA
Q11c ( ~ x ) Q20 e30 Q33c Q40 Q43c
-1.4290 0.1045 0.3163 -0.2067 0.0593 0.1292 -0.1 163 0.0978 -0.0782 0.0585 -0.0416
STRUCTURE
site H 1.7717 0.0 0.5922
I
moment QOO
0, bohr
2.194
0.182 0.0022 0.108
H
Hydrogen contributions above dipole have been transferred to nitrogen. given in the last three columns of Table X. Here an additional site is included a t the center of each bond. As before, the contributions are shown for the site at position z 3 0; the contributions for the corresponding sites with z < 0 are obtained by changing the sign of the odd moments. This DMA yields an electrostatic potential which can be expected to converge reasonably well at distances down to 2 bohr, but it will be seen that it contains, as a result of the presence of moderately diffuse high angular momentum functions, such counter-intuitive features as positive charges at the bond centers. (b) Ammonia. Carbon dioxide is a linear molecule and so has only multipole moments Qlmwith m = 0 (Qio being abbreviated to Q, above). The symmetry also ensures that all moments at the carbon atom with odd I vanish, though they survive at the oxygen atom which has a lower local symmetry. In ammonia, there are many more nonzero moments, especially at the hydrogen atoms, which are a t sites of only C, local symmetry. To avoid a proliferation of moment contributions to the DMA, it is convenient to allow the hydrogen atoms to accumulate only charge and high dipole contributions; all higher order terms are represented by expansion about the N atom. A DMA constructed in this way is given in Table XI, together with the overall moments relative to the center of mass. Once again it is evident that the DMA provides a more rapidly convergent description. It must be emphasized that the whole technique of distributed multipole analysis, like the use of the multipole expansion itself, is based on the premiss that the spatial extent of the charge distribution can be ignored. When charge distributions overlap, any multipole expansion, even if it converges, will give the wrong answer for the electrostatic energy. Nevertheless, multipole expansions have proved extremely useful as a way of describing long-range interactions, and it seems likely that distributed multipole expansions have an important contribution to make when the distances are s h ~ r t e r . ~ . ~ ' V. Discussion In this paper we have reported values for many of the properties which affect the long-range interaction of CO2.-CO2, NH3-.NH3,
2'677
6-31G'
STRUCTURE
176.6
0-c-0
1 ' 1 4 4 1 1"44
2-950
H
(A
BOND LENGTHS BOND ANGLES ("1 Figure 1.
and C02-NH3. We have also investigated the van der Waals molecule CO2.-NH3 by a number of methods. Figure 1 shows the results of two S C F calculations on the equilibrium geometry of C02-NH3. The basis sets are 3-21G and 6-31G*.49 It is interesting that both calculations obtain the T-shaped geometry, and that although the C-N bond length is very poor for 3-21G, for 6-31G* it is in good agreement with the experimental value of 2.99 A obtained by Klemperer et al.' Our calculations on this and related systems will be described more fully elsewhere.48 The fact that these SCF calculations reproduce the experimental geometry for C02-NH3 suggests that the dispersion energy contribution to this interaction energy is not sensitive to geometry changes. The electrostatic contribution dominates the interaction (48) J. E. Rice,A. D. Buckingham, N. C. Handy, and V. Dykgraaf, to be submitted for publication. (49) P. C. Hariharan and J. A. Pople, Theor. Chim. Acta 28,213 (1973); J. S.Binkley, J. A. Pople, and W. J. Hehre, J . Am. Chem. SOC.,102, 939 (1980).
J . Phys. Chem. 1985, 89, 2192-2194
2192
energy at the equilibrium geometry; a small basis set (6-31G) perturbation c a l c ~ l a t i o nshows ~ ~ the electrostatic energy to be -0.0102 hartree, and the exchange-repulsion energy to be +0.0040 hartree., while polarization and charge transfer together contribute only -0.0010 hartree. Accordingly, the Buckingham-Fowler model,3 in which structures are predicted on the basis of electrostatic interactions as described using distributed multipoles, together with a hardsphere repulsive potential, should give a good account of the geometry. The structure predicted by this model is indeed the T-shaped structure found by Klemperer,] although a hydrogenbonded structure was also found at much higher energy. The intermolecular distances found by such a model have no predictive significance, being merely sums of hard-sphere radii, and in the absence of a suitable van der Waals radius for carbon we used (50) I. C. Hayes and A. J. Stone, Mol. Phys., 53, 83 (1984). (51) G.D. Zeiss and W. J. Meath, Mol. Phys., 33, 1155 (1977).
a value which gives the observed C-N distance when combined with the standard Pauling radius for nitrogen. For the electrostatic description we used the five-center DMA for COz given in Table X and the four-center DMA for N H 3 given in Table XI. In this paper we have attempted to show the very great wealth of information which can be obtained on the properties of molecules. Here we have concentrated on the ab initio calculation of those properties which affect the long-range interaction of molecules, several of which have not been calculated before, but are very important if that interaction is to be understood. These calculations have been performed at the S C F level with a good basis, and this is probably sufficient for the time being, especially when there is not much experimental information. Now that we have realized that these calculations are possible and meaningful, we aim to perform further investigations on the series of molecules mentioned in the opening paragraph of this paper. Registry No. C 0 2 , 124-38-9; NH,, 7664-41-7.
Theoretical Study of the Structure and Spectroscopic Characteristics of Protonated Carbon Dioxide Michael J. Frisch,* Henry F. Schaefer 111, Department of Chemistry, University of California, Berkeley, California 94720
and J. Stephen Binkley Theoretical Division 8341, Sandia National Laboratories, Livermore, California 94550 (Received: August 8, 1984)
Protonated carbon dioxide has been examined theoretically by using geometries optimized at the MP2/6-3 1G(d) level and energies computed at the MP4/6-3 11++G(d,p) level. It is concluded that the C, 0-protonated complex is the only observable form of C02H+when it is produced by association of H+ with C 0 2and under most other conditions. The enthalpy of protonation of CO, is found to be 130.7 kcal mol-' at 298 K. Rotational constants are predicted to be 773.74, 10.79, and 10.65 GHz for C02H+and 43 1.18, 10.17, and 9.94 GHz for C02D+. Stretching vibrational frequencies are predicted to be 1292, 2330, and 3348 cm-I for C02H+ and 1270, 2316, and 2485 cm-I for C02D+. The 0-H (or 0-D)stretching mode is expected to produce the most intense fundamental transition in both the infrared and Raman spectra, and the 2330 (2316) cm-I C-O stretch is found to be the only other intense mode.
Introduction Protonated carbon dioxide, C02H+,is of interest in combustion chemistry and may be found in interstellar space. A recent ion cyclotron resonance experiment has observed what is tentatively proposed to be C 0 2 H +and further suggests that two forms of this complex are sufficiently stable to be observable.' Direct spectroscopic observation of this species has been hampered in part by the lack of reliable predictions of its vibrational frequencies and rotational constants. We have therefore examined the C02H+ singlet potential energy surface theoretically in order to determine (1) the stable structures (local minima on the potential energy surface) that exist for this system, (2) the energetics of the complex, including the heat of formation of C 0 2 H + ,and the isomerization and activation energies for interconversion of the isomers (if more than one isomer exists), and (3) the vibrational and rotational frequencies of the stable structure(s). Methods The ground-state structure of carbon dioxide and the stationary points on the C 0 2 H +potential energy surface corresponding to (1) P. C. Burgers, A. A. Mommers, and J. L. Holmes,J. Am. Chem. Soc.,
105, 5976 (1983).
0022-3654/85/2089-2192$01.50/0
protonation at an oxygen (l),protonation at the carbon (3), and the intervening 1,2-hydrogen shift transition structure (2) were
L
2
3
optimized by use of second-order Merller-Plesset perturbation theory2 (MP2) and the polarized split-valence 6-3 1G(d) basis set3 Analytic energy derivatives4 and conjugate gradient optimization methods5 were used in the searches for the stationary points. The closed-shell singlet wave functions for the structures 2 and 3 were found to have instabilities with respect to relaxing the spin restrictions (allowing the wave function to have the unrestricted Mlaller-Plesset (UMP2) form).6 The lower energy unrestricted (2) C. Molller and M. S. Plesset, Phys. Reu., 46,618 (1934). (3) P. C. Hariharan and J. A. PoDle. Theor. Chim. Acta. 28.213 (1973). (4) J. A. Pople, R. Krishnan, and'H.'B. Schlegel, Int. J . Quantum Chem., Quantum Chem. Symp., No. 13, 325 (1979). ( 5 ) H. B. Schlegel, J. Compur. Chem., 8, 214 (1982). (6) R. Seeger and J. A. Pople, J . Chem. Phys., 66, 3045 (1977).
0 1985 American Chemical Society
