ARTICLE pubs.acs.org/JPCA
Are There Atomic Orbitals in a Molecule? I. Mayer,* I. Bako, and A. Stirling Chemical Research Center, Hungarian Academy of Sciences, H-1525 Budapest, P.O. Box 17, Hungary ABSTRACT: Effective atomic orbitals (AOs) have been calculated by the method of the “fuzzy atoms” analysis by using the numerical molecular orbitals (MOs) obtained from plane-wave DFT calculation, i.e., without introducing any atom-centered functions. The results show that in the case of nonhypervalent atoms there are as many effective AOs with non-negligible occupation numbers, as many orbitals are in the classical minimal basis set of the given atom. This means that, for nonhypervalent systems, it is possible to present the MOs as sums of effective atomic orbitals that resemble very much the atomic minimal basis orbitals of the individual atoms (or their hybrids). For hypervalent atoms some additional orbitals basically of d-type are also of some importance; they are necessary to describe the backdonation to these positive atoms. It appears that the d-type orbitals play a similar role also for strongly positive carbon atoms. The method employed here is also useful to decide whether the use of polarization functions of a given type is a matter of conceptual importance or has only a numerical effect.
1. INTRODUCTION A qualitative discussion of the electronic structure of a molecule is traditionally done in terms of molecular orbitals (MOs) formed as a linear combination of the atomic orbitals (AOs) of the constituting atoms (the LCAO principle). The first quantum chemical calculations used directly the orbitals occupied in the free atoms, but it became evident very soon that the atomic orbitals undergo deformations (e.g., orbital shrinking and polarization) during the bond formation. Later on, the original minimal basis set calculations have been replaced by those using extended basis sets—mostly of Gaussian orbitals—but most chemists continue thinking in terms of minimal basis atomic orbitals, or their hybrids, at least if nonhypervalent systems are concerned. One of us has showed that this picture chemists are using can actually be justified by performing a special a posteriori “Hilbert-space” analysis of the wave functions:13 usually one can construct as many orthogonal linear combinations of the basis orbitals centered on the given atom and having a nonnegligible contribution to the molecular wave function, as is the number of orbitals in the respective classical minimal basis set. These orbitals can be said to form an effective minimal basis. Depending on the system, the effective minimal basis either consists of orbitals of nearly pure s, p, etc. character or represents some hybrids of them. To obtain these results, one has to perform a localization procedure for each atom separately, by using the MagnascoPerico localization criterion,4 in which the net Mulliken populations on the selected atom, corresponding to the localized orbitals, are requested to be maximal or, at least, stationary. Similar results were obtained also in the “fuzzy atom” analysis,5 when the localization criterion was selected as to make stationary the net atomic population6 in a threedimensional (3D) “fuzzy” atomic domain. (The “fuzzy atoms” analysis divides the 3D physical space into atomic regions, r 2011 American Chemical Society
but the regions assigned to the individual atoms have no sharp boundaries and exhibit a continuous transition from one to another.) Although the populations on the “fuzzy atoms” do not directly correspond to those on the individual atomic basis functions, the results could heavily be influenced by the employment of the atom-centered basis sets. One might argue that such basis sets inherently bias the localization procedures toward the formation of atomic-type orbitals. In other words, “one is very much tempted to think that the LCAO approach somehow 'induces’ rather than 'describes' the 'chemistry'.”7 It is, therefore, an important conceptual question whether the existence of effective minimal basis sets featuring s, p, d-type orbitals or their hybrids is a “law of nature” or an artifact of the computation. To address this issue, we have performed the respective analysis of wave functions obtained in plane wave calculations, without introducing any atom-centered basis sets in the course of the calculations.
2. THEORY The formalism of defining effective AOs in the “fuzzy atoms” framework has been developed in the recent paper,5 so here we repeat only the most important points. (At the abstract theoretical level it makes no difference whether the MOs considered were calculated by using an atom-centered basis or a plane-wave basis.) In the “fuzzy atoms” analysis one introduces a set of nonnegative atomic weight functions wA(rB) for each atom A and Special Issue: Richard F. W. Bader Festschrift Received: April 20, 2011 Revised: June 6, 2011 Published: June 23, 2011 12733
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every point B r of the 3D space, satisfying the requirement:
∑A wA ðrBÞ ¼ 1
components as
everywhere. These weight functions are requested to be ∼1 “inside” the atom (near the nucleus) and vanish gradually with the distance from the nucleus. In the present work we have used the simple Becke weight functions.8 (For a detailed discussion see the Appendix in ref 6.) It is a polynomial function, having the property that each wA(rB) is exactly equal one at the “own” nucleus and is zero at the other nuclei. In accord with the definition (1) of the weight functions, any orbital ψ(rB) can be presented as a sum of the components ψðrBÞ ¼
∑A wA ðrBÞ ψðrBÞ
ð2Þ
which may be attributed to the different atoms A. Then the integral Z ð3Þ M ¼ wA 2 ðrBÞjψðrBÞj2 dv defines the weight of the intraatomic part of ψ(rB), assuming that it is normalized to unity. Now, starting from the set of the occupied canonic MOs ψj(rB), and subjecting them to a proper unitary transformation, one can find the set of localized orbitals ji ðrBÞ ¼
occ
∑ Uji ψj ðrBÞ j¼1
ð4Þ
which have maximal, or at least stationary, weights of their intraatomic parts wA(rB) ji(rB). It is easy to see that matrix U should diagonalize matrix Q with the elements Z wA 2 ðrBÞ ψi ðrBÞ ψj ðrBÞ dv ð5Þ Q ij ¼ (The canonic orbitals are assumed orthonormalized, Æψi|ψjæ = δij.) That means U† Q U ¼ diagfMi g
ð6Þ
The diagonal values Mi may be called “degrees of localization” (they measure the extent to which the molecular orbital ji is localized on the given atom) or simply “occupation numbers” (see below). By virtue of the unitary character of the transformation (4), and the definition of the quantities Mi as eigenvalues, the renormalized functions 1 jAi ðrBÞ ¼ pffiffiffiffiffi wA ðrBÞ ji ðrBÞ Mi
r 0Þ ¼ ρσ ðrB, B
ð1Þ
ð7Þ
form an orthonormalized set of effective atomic orbitals. The sum of the corresponding eigenvalues Mi gives the net atomic population for each spin of the atom A, as defined in ref 6. The net atomic population is, of course, invariant of the unitary transformation, but the transformation influences how many orbitals give a considerable contribution to it. By using the decomposition (1), the first-order density matrix for spin σ, ρσ(rB,rB0 ), can obviously be presented as the sum of
¼
∑ ρσAB ðrB, Br 0 Þ
A, B
∑ wA ðrBÞρσ ðrB, Br 0 ÞwBðrB0 Þ
ð8Þ
A, B
It is easy to show that the functions jAi (rB) are the eigenvectors of r 0 ), and thus represent a rather the intraatomic part ρσAA(rB, B straightforward generalization of McWeeny’s classical “natural hybrid orbitals”,9 with the occupation numbers being equal to Mi for each spin. (The analogous relation for the analysis performed in the Hilbert space of the atomic basis orbitals has been discussed in ref 3.) The procedure described above may also be considered as L€owdin’s canonic orthogonalization of the intraatomic components wA(rB) ψi(rB) of the MOs. This also means that the present orthogonalization scheme is the only one leading to an orthogonal set of effective AOs, defined as intraatomic parts of the localized MOs.
3. METHOD OF CALCULATIONS The plane-wave calculations of the valence electron-structure have been performed by using the known CPMD code10 within the KohnSham DFT formalism,11 employing the BLYP exchangecorrelation functional12 and replacing the core electrons with an effective potential. The electronic orbitals were expanded in a plane-wave basis set up to a kinetic energy cutoff of 70 Ry. Test calculations with a higher cutoff (150 Ry) have shown no qualitative differences in the derived effective orbitals and their occupancies.The interaction between the valence and core electrons has been accounted for using norm-conserving pseudopotentials generated with the TroullierMartins scheme.13 The KleinmanBylander construction14 has been applied to the evaluation of the nonlocal part of the pseudopotentials. The molecules have been placed in a cubic box with an edge length of 15.0 Å. Their structures have been fully optimized, and the corresponding KohnSham orbitals have been calculated on a grid in the Gaussian “cube file” format. The orbitals represented in this way have been the subject of the subsequent wave function analysis. When we have processed the “cube files” of molecular orbitals, we have essentially performed an analysis of the numerical solutions of the KohnSham equations, and it is only a matter of technical relevance that the orbitals originate from a plane wave calculation. The Becke weight functions wA(rB) have been calculated in every grid point of the cube file by using the routine of the program described in ref 6. The numerical integrations have been performed simply by performing summations over the grid points, attributing equal weights to every point. (The individual MOs have been normalized in the same manner.) The “Hilbert-space analysis”1,3 shown in Figure 10 has been performed by a version of the program EFF-AO.15 4. SAMPLE CALCULATIONS We have performed several calculations of different molecules and ions. Tables 15 display the occupation numbers (eigenvalues Mi) obtained for the different atoms in several molecules and ions. It can be seen that for every atom there are as many “strongly occupied” effective AOs (having considerable Mi values) as the number of classical valence orbitals on that atom—one for 12734
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Table 1. Largest Occupation Numbers (Degrees of Localization) Mi for Different Atoms in the Molecules CO, H2O, SF6, C2H2, C2H4, and C2H6 CO orbital
C
H2O
O
O
SF6
H
S
C2H2
F
C2H4
H
C
H
Table 4. Largest Occupation Numbers (Degrees of Localization) Mi for the Heavy Atoms in the Alanine Molecule orbital
C1
C2
C3
O1
O2
N
C2H6
1
0.54
0.50
0.51
0.89
0.89
0.81
H
2 3
0.42 0.40
0.39 0.35
0.46 0.36
0.61 0.53
0.69 0.53
0.67 0.48
C
1
0.97 0.98 0.94 0.39 0.39 0.98 0.58 0.33 0.56 0.38 0.53 0.34
4
0.34
0.32
0.27
0.24
0.37
0.39
2
0.33 0.67 0.82 0.04 0.20 0.91 0.41 0.01 0.40 0.01 0.39 0.01
5
0.01
0.01
0.22 0.67 0.46 0.02 0.20 0.91 0.41
0.40
0.38
0.025
0.11
3
0.008
0.008
6
0.22 0.48 0.43
0.20 0.67 0.34
0.36
0.34
0.014
0.09
4
7
0.04
5 6
0.004 0.007 0.004 0.007
0.06 0.02 0.01 0.06 0.02 0.01
0.01 0.01
0.01 0.01
8
0.02
7
0.03
8
0.03
9
0.03
Table 5. Largest Occupation Numbers (Degrees of Localization) Mi for Different Atoms in the Ions OH, NO3, ClO4, and SO42
Table 2. Largest Occupation Numbers (Degrees of Localization) Mi for Different Atoms in the Dimethyl Ether and Dimethyl Sulfide Molecules dimethyl ether orbital
C
O
C
S
H1
NO3
ClO4
SO42
orbital
O
H
N
O
Cl
O
S
O
1
0.98
0.37
0.62
0.98
0.79
0.91
0.62
0.96
2
0.89
0.05
0.52
0.82
0.59
0.65
0.40
0.75
3
0.89
0.05
0.43
0.68
0.59
0.65
0.40
0.75
H2,3
4
0.51
0.01
0.43
0.34
0.59
0.25
0.40
0.42
0.05
0.006
0.23
0.002
0.14
0.02
dimethyl sulfide H2,3
H1
OH
1
0.56
0.92
0.32
0.37
0.53
0.97
0.33
0.37
5
2
0.44
0.83
0.008
0.006
0.40
0.91
0.006
0.008
6
0.05
0.23
0.14
7 8
0.04 0.04
0.23 0.19
0.14 0.12
9
0.02
0.19
0.12
10
0.01
0.07
0.04
3
0.41
0.58
0.38
0.61
4
0.25
0.48
0.24
0.50
5
0.02
0.01
0.007
0.004
Table 3. Largest Occupation Numbers (Degrees of Localization) Mi for Different Atoms in the Acetone and Thioacetone Molecules acetone orbital C1,3
thioacetone
C2
O
H1
H2,3
C1,3
C2
0.36
S
H1
H2,3 0.36
1
0.53
0.50
0.98
0.37
0.54
0.51 0.99
0.36
2 3
0.41 0.39
0.36 0.30
0.84 0.60
0.008 0.008 0.40 0.39
0.35 0.92 0.27 0.64
0.008 0.009
4
0.35
0.29
0.49
0.33
0.25 0.52
5
0.005 0.035 0.011
6
0.025 0.007
Figure 1. Strongly occupied oxygen orbitals in the CO molecule.
0.005 0.01 0.003
H, four for C, N, O, S, etc. (We note again that in the present plane wave calculations the core electrons have been compressed to the effective potentials.) The Mi values are close to 1 for lone pair orbitals and scatter around 1/2 for the bonding ones. The “weakly occupied” AOs have significantly lower, in most cases negligible, Mi values.† Hypervalent atoms represent intermediate cases; they have also some small but non-negligible Mi values, in full analogy to that already observed in the framework of the Hilbert-space analysis.1 These orbitals are necessary to describe the “back-donation” to the positive hypervalent atoms. (An analogical effect is observed, although to a lesser extent, for the (obviously positive) central carbon atom of the acetone molecule and the carboxylic carbon in alanine.)
Figure 2. Strongly occupied oxygen orbitals in the water molecule.
Figures 19 show the shapes of the effective AOs for some of the molecules considered. It can be seen that the “strongly occupied” orbitals in all systems resemble the conventional s and p atomic orbitals, or their hybrids, with the limitation that the orbitals of a given atom do not significantly extend to the spatial domains of the other atoms. (This implies significant deviations from the conventional orbital shapes if not too large orbital cutoff values are used on the figures.) The “weakly occupied” orbitals, where they are not negligible at all, have basically the shapes resembling conventional d-orbitals. That sheds some new light to the importance of using them as “polarization functions” in the SCF calculations. Inspecting these orbitals, we should recall once again that the calculations have been performed by using plane 12735
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Figure 7. Weakly occupied orbitals of the sulfur atom in the SF6 molecule.
Figure 3. Strongly occupied hydrogen orbital (above) and the first weakly occupied hydrogen orbital (below) in the water molecule.
Figure 8. Strongly occupied orbitals of the nitrogen atom in the NO3 ion.
Figure 9. Weakly occupied orbitals of the nitrogen atom in the NO3 ion. Figure 4. Strongly occupied orbitals of the central carbon atom in the acetone molecule.
Figure 5. First two weakly occupied orbitals of the central carbon atom in the acetone molecule.
Figure 6. Strongly occupied orbitals of the sulfur atom in the SF6 molecule.
wave basis, and no atom-centered functions were employed in the calculations. (The role of d-orbitals could be further clarified by performing also an explicit decomposition of the “strongly occupied” effective AOs into pure s, p, d, etc. components.) The oxygen orbitals of the CO molecule show a classical example of the sp hybridization. The first, lone pair, atomic
orbital is more s-like, while the second, bonding, orbital is more p-like. This ordering can easily be explained by the energy order of the s and p atomic orbitals. Then follow two orthogonal p-orbitals, as expected. The situation for the occupied oxygen orbitals of the water molecule (Figure 2) is quite similar to that in the CO, except that the bonding sp-like orbital has not the second but the fourth largest occupation number. On this figure it appears more spectacularly that the AOs are sharply “cut” where the domain of another atom is started. Of course, the π-type molecular orbital in the water is not strictly confined to the spatial domain of oxygen atom; however, it has only a very small tail in the hydrogen domain, leading to a small (ca. 0.04) occupation number of the hydrogen’s first p-type orbital, shown in Figure 3. The strongly occupied orbitals on the central carbon atom of the acetone molecule (Figure 4) are similar to those of the oxygen orbitals considered previously. The most peculiar feature is the basically s-type hybrid having the largest occupation number because of the characteristic hollow on the side directed to the oxygen atom. This is not an artifact of the inverse Fouriertransform or other numerical effects: an analogous hollow on the oxygen side can be observed on the respective HartreeFock orbital obtained in a Hilbert-space analysis1,3 of a cc-pVTZ basis set calculation, shown in Figure 10. Very interesting are the weakly (but not quite negligibly) occupied orbitals of that carbon atom (Figure 5): they have wellexpressed d-orbital character. This result questions the common belief that back-donation to d-type orbitals has an importance 12736
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’ ADDITIONAL NOTE † In the case of water molecule and of the OH ion, there are only four valence molecular orbitals, so there are no “weakly occupied” AOs on the oxygen atoms. The same holds for the carbon in methane molecule. ’ REFERENCES
Figure 10. First strongly occupied valence orbital of the central carbon atom in the acetone molecule, calculated by the Hilbert-space analysis from the results of a cc-pVTZ HartreeFock calculation.
only started from the third row of the periodic system (silicon, etc.) as there are no free atomic orbitals of 2d-type. This result also indicates that plane wave calculations with the present type of analysis can serve, in general, to decide independently whether orbitals of the given type are of conceptual importance for systems in question. The sulfur orbitals shown on Figures 6 and 7 corroborate our picture about the behavior of hypervalent atoms. The most occupied orbitals represent an sp-type set (with the significant cuts where the domains of the fluorine orbitals start) while a set of d-type orbitals (four of them are shown) has smaller, but in no way negligible, occupation numbers, permitting a classical backdonation to the positive sulfur. Figures 8 and 9 show the nitrogen effective AOs in the NO3 ion. The formally pentavalent nitrogen shows all the characteristics of a hypervalent atom: there is an sp set of strongly occupied orbitals, whereas the weakly (but not quite negligibly) occupied orbitals are basically of d-character, although a diffuse π-type p-orbital also appears there. Finally we mention that quite analogous calculations could also be performed by using Bader’s topological definition of the atoms in molecules16 instead of “fuzzy atoms”; the respective equations have been derived already in ref 2.
(1) Mayer, I. Chem. Phys. Lett. 1995, 242, 499. (2) Mayer, I. Can. J. Chem. 1996, 74, 939. (3) Mayer, I. J. Phys. Chem. 1996, 100, 6249. (4) Magnasco, V.; Perico, A. J. Chem. Phys. 1967, 47, 971. (5) Mayer, I.; Salvador, P. J. Chem. Phys. 2009, 130, 234106. (6) Mayer, I.; Salvador, P. Chem. Phys. Lett. 2004, 383, 368. (7) Salvador, P. personal communication to I.M., 2011. (8) Becke, A. D. J. Chem. Phys. 1988, 88, 2547. (9) McWeeny, R. Rev. Mod. Phys. 1960, 32, 335. (10) CPMD, version 3.11.1, http://www.cpmd.org; IBM Corp.: Armonk, NY, 19902006; MPI f€ur Festk€orperforschung Stuttgart: Stuttgart, 19972001. (11) Kohn, W.; Sham, L. J. Phys. Rev. A 1965, 140, 1133. (12) Becke, A. Phys. Rev. A 1988, 38, 3098. Lee, C.; Yang, W.; Parr, R. Phys. Rev. B 1988, 37, 785. (13) Troullier, N.; Martins, J. L. Phys. Rev. B 1991, 43, 1993. (14) Kleinman, L.; Bylander, D. M. Phys. Rev. Lett. 1982, 48, 1425. (15) Mayer, I. Program EFF-AO, Budapest, 2008. May be downloaded from the site http://occam.chemres.hu/programs. (16) Bader, R. F. W. Atoms in Molecules: A Quantum Theory; Oxford University Press: Oxford, U.K., 1990.
5. CONCLUSIONS Effective atomic orbitals have been calculated by the method of the “fuzzy atoms” analysis by using the numerical molecular orbitals obtained from plane-wave DFT calculation, i.e., without introducing any atom-centered functions. The results show that in the case of nonhypervalent atoms there are as many effective AOs with non-negligible occupation numbers as there are orbitals in the classical minimal basis set of the given atom. This means that, for nonhypervalent systems, it is possible to present the MOs as sums of effective atomic orbitals that resemble very much the atomic minimal basis orbitals of the individual atoms (or their hybrids). For hypervalent atoms some additional orbitals basically of d-type are also of some importance; they are necessary to describe the back-donation to these positive atoms. It appears that the d-type orbitals play a similar role also for strongly positive carbon atoms. The method employed here is also useful to decide whether the use of polarization functions of a given type is a matter of conceptual importance or has only a numerical effect. ’ ACKNOWLEDGMENT We acknowledge the partial financial support of the Hungarian Scientific Research Fund (grant OTKA 71816). 12737
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