Electron density maps are reported for the CN−ion and the LiCN and LiNC molecules, calculated from molecular wave-functions near the Hartree-Fock limit. The electron density distribution derived from CNDO/ 2 wavefunctions does not resemble the ab initio results. The ultimate ability of a minimal basis set to represent the electron density near the Hartree-Fock limit, has been tested. The requirement of N-representability of the trial electron density has been satisfied. It is found that the molecular valence density cannot be reproduced to a satisfactory extent by a minimal set of Slater orbitals, even when the exponents of the basis orbitals are optimized

Bats, J.W.

Feil, D.

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The electron density distribution in CN−, LiCN and LiNC. The use of minimal and extended basis set SCF calculations

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Chemical Physics 26 (1977) 79-84 0 North-Holland Publish& Company THE-ELECTRON DENSITY DISTRIBUTiON IN CN-, LiCN AND LiiC. THE USE OF MINIMAL AND EXTENDED BASIS SET SCF MO CALCULATIONS J.W. BATS and b. FEIL Twente University of Technology, Chemical Physics Laboratory, Ens&de. The Netherlands Received 1 June 1977 Eiectron density maps are reported for the CN-ion and the LiCN and LiNC molecules, calculated from molecular wave- functions near the Hartree-Fock limit. The electron density distribution derived from CNDO/2 wavefunctions does not resemble the ab initio results. The ultimate ability of a minimal basis set to represent the electron density near the Hartree-Fock limit, has been tested. The requirement ofN-representability of the trial electron density has been satisfied. It is found that the molecular valence density cannot be reproduced t,l a satisfactory extent by a minimal set of Sister or- bitals, even when the exponents of the basis orbitals are optimized. 1. Introduction The electron density distribution has been the sub- ject of several quantum chemical studies [l-6]. The advances in X-ray and neutron diffraction, es- pecially in the measuring techniquesand the process- ing of the data, have resulted in an increasing accura- cy in the experimental determination of the electron density distribution. The difference electron density, i.e. the molecular electron density minus the atomic electron densities, has been observed for a large num- ber of compounds (for example refs. [7,X]). However, not much progress has been made in a quantitative analysis of the observed electron density in terms of parameters that are of direct significance in quantum chemistry. Coppens et al. [9’] have developed a meth- od in which the terms of the charge bond order ma- trix are used as parameters in a least-squares proce- dure to fit these terms to X-ray diffraction data. In order to remain significant, the number of parameters should be limited to avoid correlation. Consequently the electron density distribution has to be constructed from a limited number of basis functions. A faithful re- presentation of the electron density was obtained with only a minimal set of Slater orbitals. However, in treating the terms of the charge bond order matrix as parameters to be fmed by experiment, Coppens et al. neglected the constraints imposed on this matrix by the requirement of iv-representability, as proposed by Massa and Clinton [lO,l 11. Applying these con- straints, a minimal basis set may no longer be able to give an accurate representation of the observed elec- tron density distribution. In section 2 of this contribution the difference electron density in the CN- ion and the LiCN and LiNC molecules, is calculated from molecular and atomic wavefunctions near the Hartree-Fock limit. In section 3 the possibility is analysed to give an ade- quate description of the electron density in these molecules, with a state function based on a minimal basis set of Stater orbitals. 2. The difference electron density The theoretical electron density distribution is given by P(r) =~~p,,xJf,,(r), J! v where p,,V is a term of the ch’arge bond order matrijc and x,(r) an atomic basis function. A difference density map of the CN- ion, calcu- lated on basis of the results of Bonaccorsi et al. [12] 80 J. W. Bats, D. .&il/The electron density distribution in CN; LkNarid LNC -- :. : ‘-- : : .’ ... ’ : . . _-. _: Fig. 1. The difference density in the CN-ion, using an extended basis set. Contour interval: 0.025 e/au3. and Clementi [ 131 for the molecular and atomic wave- ty at the Hartree-Fock limit [4,5]. Siiar differ- functions respe&ively, is shown in fig. -1 _ The molec- ence density maps for LiCN and LiNC molecules, ular wavefunction is based on a double zeta Slater shown in figs. 2 and 3, are derived from the molec- basis including polarization functions and is supposed ular wavefunctions given by Bak, Clementi andKortze- to lead to a close representation of the electron densi- born [ 141 which are based on 45 contractsd gaussian I /’ I I --_ / \ /* i ; I- I ‘\ 1 i ’ ,-I- i ; I..>,/> , ; ‘\ jLi ’ ‘----_; /’ i . \ . \ Fig. 2. The difference density in LEN, using an extend* ba& set. 6ontotis as hi fGi.1.’ : ‘- ‘Y .- ‘- ’ . . : .: : . . . . . -- J W..Bats, D. FeiljThe electron density distribution in CN; LiCNand Lih’C 81 Fig. 3. The difference density in LiNC, using an extended basis set. Contours as in fig. 1. orbitals. Subtracted are again atomic densities based on Clementi’s atomic orbitals [13]. The lack of features near the Li atom in figs. 2 and 3 is a result of the very diffuse nature of the valence electron density of this atom. Comparison of figs. 1, 2 and 3 shows a striking resemblance of the electron density in the CN regions. Good correspondence is also found when compared with theoretical difference den- sity maps of HCN and NCCN [15]. This apparent agreement is contradictory to the differences in Mulliken population analyses derived from the Hartree-Fock molecular wavefunctions and reported in table 1. This should not be surprising as it has been shown by Mulliken.[l7] and Politzer and Harris [lg] that the Mulliken popuiation analysis, based on an extended basis set Hartree-Fock calcula- tion, often results in. unrealistic net atomic charges. Table 1 Net atomic charges according to Mdiken population ana- lyses Hartree-Fock CNDOJZ’) Li c. N L.iC N CN- -0.50; --0.499 a) LiCN 0.4?5. -0.39, -0.61 -&29+ -0.180b) 0.43 -0.23 -0.19 IiNC 0.817. -O.OCM -0.812 b) +?o -0.09 -0.41 a) kef. [ 161; b) Ref. 1141. C) Results after deortbogonaliza- tion.of basis functi&r- . . Charges resulting from CNDO/Z calculations, however, have been found to be consistent with the electron density distribution near the Hartree-Fock limit [ 191. They have been included in table 1. Following Giessner- Prettre and Pullman [ZO] and Shillady et al. [21] the coefficients resulting from the CNDO/Z calculations are assumed to be of the symmetric orthogonalized . basis functions. A deorthogonalization procedure has been applied b.efore performing the population ana- lyses. The resulting net charges are almost identical to values obtained for the same compounds by integra- tion over the Hartree-Fock electron densities and by estimations from core energies [22]. 3. The minimal basis set Minimal basis set quantum chemical calculations, both ab initio and semi-empirical, are generally found to give poor representations of the electron density distribution [6,9]. As example the difference density of CN- based on a CNDO/2 calculation, using stan- dard parametrization [23], is shown in fig. 4. The atomic electron density is assumed to be the spheri- cally averaged density based on the same valence or- bitals as used in the CNDO/2 calculation. Striking dif- ferences with fig. 1 can be noted in both the bondiig and the lone pair regions: the latter regions seem to be-stressed at the expense of the former. It is clear that if the miniial basis set of Slater or- 82 %W_.B~~~,.~_:F~jl/Th~eleeiron density distribut,& in W;:LiCNand_@VC.~;. :I :_ ’ ... 1 : .. ---: ,. .. I _’ _.: . . . .. i/ Fig. 4. The difference density in the CN-ion, derived from a deorthogonalized CNDO/% mole&tar wavefunction. Contours a; in iii 1. bitals is too limited for our purpose, neither ab initio nor semi-empirical method for determining the coeffi- cients of the basis functions in the molecular orbitals will give accurateresults. It can be expected, however, that an improvement in the electron distribution will be obtained if the semi-empirical parameters are de- termined on basis of a comparison with &curate elec- tron density distributions. In this case the density close to the nuclei will not play its dominant role as in the case of an energy minimization, with a benefi- cial effect on the bondingregions. In order-to assess the quality of a trial electron density distribution with respect to-its ability to.give’ ’ a proper representation of a high quality reference electron density distribution, we introduce the error functional& defined as where the superscripts (ref) and (tr) refer to the refer- ence and trial quantities. This functional is easily. cal- culated by numerical integration. Now the capability of a minimal basis set of Slater orbitals to represent the reference density can be found by mini&zmgR. In order to be physically significant, the trial electron density has to be restricted to satisfy N-representabil- ity [lO,ll]: The procedure we used- is basically different from the one suggested by Clinton et al. [l 11. The.trial electron density can be interpreted in terms of a set of orthonormal molecular orbitals. Furthermore, such a set of functions can:be regarded as the eigenvectors of a real symmetric-matrix H: We assu_med-a starting-. matrixof the right order, i.e. the Fock matrixof a-. f&O/2 calculation and varied the matrix terms one at a time, keeping the matrix symmetric. Subsequent- ly we diagonalized the matrix and calculated the trial electron density from the ~deorthogonaliied eigenvec- tors..The value of the$ror.functional R was miniY- .: mized by the.stekpest-descent procedure. By-repeat-. .- ing this procedure, the trial electron-density co& :. verged in aboutten cycles;,. ’ .:.. :..I:.. :. _.. ._ .~... Thi: relative deviation of the-trial electron density, : _.-. ., : : ;_ J. W. Bets. D. FeilfT%e elect& density dihibkion in CN; LiCNand LihT . 83 .:. :. based on a minimal set of 1s; 2s and 2p Slater func- tions, from.+e high quality density was considerable: -the minimum .irik was 12% for CN:. Restriction to a fit of the valence densityusing only 2s-and 2p basis ii&ions(asapplied in the semi-empirical methods) .made:the reMs:viorse: R =:16% for CN-, R =.9% for MN. : it is~interesting to note that no real improvement is obtained over the CNDO/2 wavefunctions which give R = 16.4% and R = 10.6%.for CN- and MN, respectively. So the CNDO/Z method results in al- most the best electron density possible with a mini- mal set of 2s,and 2p Slater orbit& with standard ex- ponents. In order to test the ultimate capability of a mini- mal set of Slater orbitals, the exponents b of these orbitals were varied as well Ransil’s work on C, and N, [24] indicates the sensitivity of the energy on the value:of $. Electron density maps of these molecules, using Ransil’s wavefunctions, however, did not show much improvement as result of the energy-optimiza- tion of the exponents when compared with the Hartree-Fock densities of Bader et al. [l] _ Full opti- mization of the trial electron density by varying both exponents and the vahres of the H-matrix was achieved by a similar procedure as described above. A fit of the valence density of CN- with a minimal set of 2s and 2p Slater orbitals, optimizing three exponents per atom(52s, 52poj <22p,) lowered the value of R to 9.7%. A similar fit of the valence density of LiCN re- duced the value of R to 6.6%. The exponents showed an overall reduction during the refmement, indicating an average increase of the distance of the electrons from the nuclei. No attention has been paid to the uniqueness of the individual exponents. The effect of the optimization can be seen in fig. 5, where the electron density contours of CN- are drawn after the fmal refinement. Atomic valence densities calculated from Clementi’s atomic orbitals have been subtracted. The massive lone pair regions of the CNDO/2 results are considerably reduced_ The quahta- tive features of the Hartree-Fock difference map.have not been reproduced to a satisfactory extent. The unsatisfactory description of the difference Fig. 5. The difference density in the CN-ion: the ultimate optimized molecular valence density with a &inul basis set mirp~s accurate atom& valence densities. Contours as in tig. 1 84 -. .~ 1 . . J. W. Bats. -D; Feil/The electron dekiiy distribution ip CN; LiCN apd L&C :- .: _ ‘: : . . :. dei&i&CNY in.the final map could be expected..as ihe-o&tied v&nce density of CN--stmshows a de- viationof about 16% of the high quality.density, or ir- absolute.tern& the. deviation is still one electron, which equals about the electron change on molecule. form&ion; [4] W;Heysq; A<‘@..& Ke+md-%.J. B&r++&&em; ..: i .~ Phys. 16 (1976) 371:. :: -.I : ..;._-._- -[S] PIE. C&i Trans:A;n.C,~st.-Ass,B (1972) ii. .’ [$I P.I(;‘$nith &id J;W. Richard&, J. PkF_ tieti::!!_ (1g67) g24;_ I: ‘I-. : .-. :-:-_’ .:_ __: . . [i] P:Goppens;.M.T.P..Internatiorial Rev..of&&~,: -Physical Chemistry, Series II; Vol. ll(l975). [8] J.W. Bats, Thesis Twente University ofTeclmolo& (1976). 4. donclusion Summarizing this work the following conclusion can be made:.quantum chemical calculations based-on a minimal basis set of Slater orbit& are inadequate to lead to electron density maps with the same qualitative features as calculations near the J&-tree-Fock level: Acknowledgement. Frequent discussions with Dr. S. Harkema contrib- uted to the final results. References [l] R.F.\V. Bader, W.H. Henneker and P.E. Cade, J. Chem. Phys. 46 (1967) 3341. [2] M.J. Hazeirigg and P. Politzer, J. Phys. Chem. 73 (1969) 1008. [3] G. de With and D. Feil, Chem. Phys. Letters 30 (1975) 279. [<I-D.S. JonesiD. Paiier’tid Pf kopp&s, @a &yst: ‘A28 (1972) 635: :-. ~-. : [lo] L.J. Massa and W.L. Clinton, T&s. Am. Cryst..Ass. 8 (1972) 149. [ 111 W;L. Clinton, C.A. Frishberg, p.j: Massa &d P.k. ‘. Oldfield, Int. J; Q&turn Chem. s’7_(19?3) 505;. [ 121 R. Bonaccorsi; C. Pekon&lo, E. Skocco and J. To&d, Chem. Phys. Letters 3 (1968) 473. [13] E. Clementi, IBM I. Res. Dev. Suppl. 9 (1965) 2. [14] B. Bak, E. Clementi and R.N. Kortzebom, J.Chem. Phys. 52 (1970) 764. [15] F.L. Hirschfeld, Actz Cryst. B2j (1971) 769. [16] R. BonaccorsI, C. Petrongolo, E. %&co ana J. iomasi; J. Chem. Phys. 48 (1968) 1500.. [17] R.S. Mulliken, J. Chem. Phys 36 (1962) 3428. [18] P. Politzer and R-R. Harris, I. Am. Chem. Sot. 9i (1970) 6451. [19] P. PoIitzer, K.C. Leung, J.D. Elliott and S.K. Peters, Tbeoret. Chim. Acta 38 (1975) 101. [20] C. Giessner-Prettre tid A. Puliman, Theor. Chim. Acta 11 (1968) 159. 1211 D.D. Shiiady, F.P. Billingsley and J.E. Bloor, Theoret. Chim. Acta 21 (1971) 1. [22] P. Poiitzer and A. Poiitzer, J. Am. Chem. Sot. 95 (1973) 5450. [23] J.A. Pople and D.L. Beveridge, Molecular orbital theory (McGraw-Hill. New York, 1970). 1241 B.J. Ransii, Rev. Mod. Phys. 32 (1960) 245. 

Actz Cryst.

Molecular orbital theory (McGraw-Hill.

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The electron density distribution in CN−, LiCN and LiNC. The use of minimal and extended basis set SCF calculations

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