J. Phys. Chem. A 2010, 114, 8923–8931
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Analysis of Bonding Patterns in the Valence Isoelectronic Series O3, S3, SO2, and OS2 in Terms of Oriented Quasi-Atomic Molecular Orbitals† Vassiliki-Alexandra Glezakou,‡ Stephen T. Elbert,§ Sotiris S. Xantheas,*,‡ and Klaus Ruedenberg| Chemical and Materials Sciences DiVision, Pacific Northwest National Laboratory, 902 Battelle BouleVard, MS K1-83, Richland, Washington 99352, Computer Science and Mathematics DiVision, Pacific Northwest National Laboratory, 902 Battelle BouleVard, MS K1-85, Richland, Washington 99352, and Ames Laboratory and Department of Chemistry, Iowa State UniVersity, Ames, Iowa 50011 ReceiVed: June 3, 2010; ReVised Manuscript ReceiVed: June 24, 2010
A novel analysis of the chemical bonding pattern in the valence isoelectronic series of triatomic molecules O3, S3, SO2, and OS2 is reported. It is based on examining the bond order matrix elements between the oriented localized molecular orbitals (OLMOs) that are localized on the three individual atoms: left (L), center (C), and right (R). The analysis indicates that there is a (L-C) and (C-R) π-bonding interaction and a (L-R) π-antibonding interaction. It supports the earlier proposed “partial biradical” interpretation of these systems, which had recently been challenged. The degree of biradical character is shown to increase from SO2 to S3 to O3 to OS2. I. Introduction Interpreting the nature of chemical bonding remains one of the challenging problems in theoretical chemistry.1,2 New approaches aimed at analyzing the physical origin of bonds continue to be developed, and it is of interest to test them on molecules with nontrivial bonding patterns. In the present study we report the results of applying a recently introduced bonding analysis3,4 to the valence isoelectronic triatomic series of molecules consisting of ozone (O3), thiozone (S3), sulfur dioxide (SO2), and cyc-disulfur monoxide (S2O).5,6 Since its discovery in 1840,7 O3 has received much attention because of its importance in atmospheric processes and as a powerful oxidizing agent. While its presence in the lower atmosphere is a harmful air pollutant, the ozone layer in the lower stratosphere (10-50 km above the Earth) absorbs over 95% of the sun’s ultraviolet light, thus preventing it from damaging life on the planet.8 Ozone has several absorption bands of interest to atmospheric chemistry: the Hartley band in the ultraviolet (200-300 nm), the Huggins band in the near-ultraviolet (320-360 nm), the Chappuis band in the visible (375-600 nm),9,10 and the Wulf band in the infrared (beyond 700 nm). All of them have been the subject of extensive experimental and theoretical studies.11–35 In recent years, the effect of man-made chemicals in destroying the ozone layer has been widely discussed.36 A surprising discovery of the closed-shell ground state of O3 and also of its valence isoelectronic analogues was the existence of a conical intersection with the lowest excited state of like symmetry (even though both states are closed shell dominated), as has been investigated in detail by
Ruedenberg and co-workers.37–43 This intersection is related to the existence of a ring structure for these molecules. In the open C2V geometry of these molecules, 14 electrons occupy 7 σ-type bond and lone-pair orbitals and form a standard structure pattern. Less obvious on the other hand is the interpretation of the bonding pattern of the 4 electrons that occupy the three π-type valence orbitals. The latter form the three symmetry-adapted molecular orbitals
π0 ) 1b2 ) {+++},
π1 ) 1a2 ) {+0-}, π2 ) 2b2 ) {-+-} (1)
where the + and - signs indicate the values of the these orbitals (above or below the molecular plane) at the positions of the left, center, and right atom respectively. All calculations (beyond SCF) show that the ground state wave function ψ is dominated by two configurations
ψ ) c1ψ1 - c2ψ2 + ...
(2)
where both coefficients are positive and ψ1 and ψ2 are closedshell determinants with the valence occupancies
ψ1 ) {σ14π20π21},
ψ2 ) {σ14π20π22}
(3)
Hayes44 pointed out that, in the case c2 ) c1, these two terms can be expressed in terms of an antisymmetrized “singlet biradical” function with the occupations {σ14π02π+π-}, where
†
Part of the “Klaus Ruedenberg Festschrift”. * To whom correspondence should be addressed. E-mail: sotiris.xantheas@ pnl.gov. ‡ Chemical & Materials Sciences Division, Pacific Northwest National Laboratory. § Computer Science and Mathematics Division, Pacific Northwest National Laboratory. | Iowa State University.
π+ ) (π1 + π2)/ √2,
π- ) (π1 - π2)/ √2
(4)
are predominantly on the right and on the left side of the molecule, respectively. Considering this case as a “perfect” biradical, Hay, Dunning, and Goddard,45,46 as well as Laidig
10.1021/jp105025d 2010 American Chemical Society Published on Web 07/28/2010
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and Schaefer,47 subsequently deduced a fractional “biradical character” for ozone from various CI-type of calculations. These authors defined the biradical character of ψ as B ) 2c22, since B ) 0 for c2 ) 0, and B ) 1 if c1 ) c2 and if ψ contains no other configurations (so that c12 + c22 ) 1). Thus, a GVB-configuration interaction (GVB-CI) yielded45 c1 ) 0.8852 and c2 ) 0.3493 (B ) 24%), and a subsequent POL-CI-1 calculation produced46 similar results of c1 ) 0.896 and c2 ) 0.319 (B ) 20%). Calculations with different reference configurations and excitation levels47 yielded a value of c2 ) 0.481 (B ) 46%) using a two-configuration SCF (TCSCF) calculation, a value of c2 ) 0.149 (B ) 4%) with a singles plus doubles CI calculation based on a single reference (HF+1+2) and a value of c2 ) 0.162 (B ) 5%) with a limited MCSCF (6,825 terms) calculation. Larger CI and MCSCF calculations47 (13 413 terms) produced values of 0.296 (B ) 18%) and 0.337 (B ) 23%) for c2, respectively. From these results, Laidig and Schaefer47 deduced an estimate for B of around 20% in agreement with the earlier POL-CI results of Dunning, Hay, and Goddard.46 These calculations have been interpreted to imply that ozone has partial “biradical character having two unpaired π-electrons on the terminal oxygen atoms weakly coupled into a singlet state”.46 Since then, ozone has often been considered as a textbook example of a biradical species, even though it has a degree of biradical character significantly less than that of a “perfect biradical”. Moreover, this partial biradical character has been considered as related to the chemical reactivity of ozone. Recently, this traditional consensus has been challenged by Kalemos and Mavridis48 through a bonding analysis based on a certain valence-bond interpretation of Lewis diagrams, which led them to conclude that ozone is a “regular ... genuine closed shell singlet” without biradical character. In view of this controversy and the importance of this molecule, we have examined the bonding in the ground state of O3 by means of a different analysis, using the aforementioned orbital-based method.3,4 This approach also addresses an issue that is a consequence of the exclusive reliance on the configurational coefficients, namely that, as Hay, Dunning, and Goddard45 already noted, the “biradical orbitals” π+ and π- of eq 4 are not really localized on the terminal oxygen atoms. We furthermore extend this analysis to the valence isoelectronic molecule series S3, SO2, and OS2. Section II describes the formal basis of the analysis and outlines the computational approach, and Section III presents the quantitative results. The conclusions are drawn in Section IV. II. Method II.1. Wave Functions. For all four molecules considered, the analysis is based on FORS wave functions, that is, CASSCF wave functions in the full optimized reactions space,49 which is the configuration space spanned by all 12 126 determinants with C2V symmetry that are generated by the 18 valence electrons from the 12 conceptual minimal valence basis orbitals. Such wave functions can also be characterized as the optimal configurational representations in the molecule-optimized minimalbasis-set space. Valence-bond wave functions would use part of this space. Another characterization is to say that the wave functions account for all nondynamic correlations. We do not expect the basic bonding pattern identified at this level to be altered by the inclusion of dynamic correlation. The wave functions are obtained by full (18 e-/12 orbitals) MCSCF calculations within C2V symmetry. The molecular
Glezakou et al. orbitals are expressed in terms of Dunning’s correlationconsistent basis set50 of triple-ζ quality, that is, cc-pVTZ. For sulfur the cc-pV(T+d)Z basis set51 was used, which includes additional tight d-type functions. All calculations were performed with the GAMESS suite of electronic structure codes,52,53 which also contains the codes for the analysis of the Ivanic-Atchity-Ruedenberg scheme3,4 described below. II.2. Geometries and Energies. The present study is only concerned with the bonding patterns at equilibrium geometries. In all four triatomic molecules, the geometries of the open forms were considered and optimized for the FORS wave functions with C2V symmetry. They were found to be:
O3:
R(O-O) ) 1.28459 Å,
θ(O-O-O) ) 116.67° E ) -224.600576 hartree
S3:
R(S-S) ) 1.94986 Å,
θ(S-S-S) ) 117.77° E ) -1192.707835 hartree
SO2: R(S-O) ) 1.43664 Å,
θ(O-S-O) ) 119.50° E ) -547.434585 hartree
OS2: R(O-S) ) 1.64707 Å,
θ(S-O-S) ) 123.87° E ) -869.975421 hartree
The geometrical parameters are similar to the ones reported in earlier studies (cf. ref 43). It should be noted that, for the first three molecules, these geometries represent the global minima. However, for OS2 the geometry considered here is the metastable local minimum with the oxygen atom in the center and C2V symmetry so that the electronic structure can be compared with that of the rest of the series. The global minimum of this molecule has the S-S-O arrangement and Cs symmetry.43 II.3. Bonding Analysis in Terms of Oriented QuasiAtomic Orbitals. Since the bonding analysis method is described in explicit detail in references 3 and 4, we give here only a brief description of the steps involved. (i) After the FORS wave function has been calculated by the above-mentioned MCSCF calculation, the 12 natural orbitals54,55 (NOs) φk are determined and the MCSCF wave function is expressed in terms of the natural-orbital-based configurations. (ii) Localized molecular orbitals (LMOs) are then determined in the full 12-dimensional space of all NOs. This localization is known3,4,49 to yield quasi-atomic orbitals, that is, molecular orbitals that, their mutual orthogonality notwithstanding, are essentially localized on indiVidual atoms in the molecule. The Edmiston-Ruedenberg localization method56,57 in the Raffenetti implementation58 is used. The localization is moreover restricted to occur only within the A′ and A′′ representations of the Cs symmetry group respectively, that is, no mixing is allowed between the nine σ-orbitals and the three π-orbitals so that the transformation from NOs to LMOs is block-diagonal. For the reasons discussed in the introduction, only the 3 × 3 block transforming the π-NOs into the π-LMOs is of major interest. The three π-LMOs will essentially turn out to be deformed atomic p-orbitals perpendicular to the molecular plane at the three atoms. (iii) The three quasi-atomic σ-LMOs on each atom are then mixed among each other by an orthogonal transformation to create oriented hybrid orbitals (OLMOs) that are either atomic
Bonding Patterns in the Series O3, S3, SO2, and OS2
J. Phys. Chem. A, Vol. 114, No. 33, 2010 8925
Figure 1. Natural orbitals (NOs) and corresponding occupation numbers for O3. Orbitals 6, 9, and 10 are the three π-type orbitals.
TABLE 1: Occupation Numbers of the σ- and π-Type NOs
lone-pair orbitals or bonding orbitals generating the σ bonds. This is accomplished by the recent method of Ivanic, Atchity, and Ruedenberg,3 which provides an automatic, unbiased and basis set independent procedure for extracting intrinsic quasiatomic hybrid orbitals that are chemically adapted. Actually, the procedure is applied to the full set of all LMOs, that is, including the π-LMOs, but the three π-LMOs turn of course out to be also the three π-OLMOs. (iv) Finally, the density matrix is expressed in terms of the 12 OLMOs:
F(x, x′) )
∑ pjkOLMOj(x) OLMOk(x′)
orbital symmetry
O3
S3
SO2
OS2
σ
1.9984 1.9977 1.9970 1.9915 1.9900 1.9550 1.9462 0.0654 0.0565 1.9614 1.7841 0.2567
1.9981 1.9977 1.9963 1.9898 1.9968 1.9702 1.9625 0.0529 0.0426 1.9979 1.8583 0.1769
1.9991 1.9991 1.9958 1.9888 1.9871 1.9763 1.9690 0.0455 0.0354 1.9772 1.9352 0.0915
1.9993 1.9991 1.9981 1.9980 1.9969 1.9628 1.9611 0.0428 0.0404 1.9718 1.6407 0.3891
π
(5)
jk
The OLMO occupations pkk and the bond orders pjk (j * k) between them provide information about the bonding patterns in the molecules. In determining the various transformations in the described sequence of operations, care has to be exercised so that the order and the signs of the generated orbitals are most appropriate for a transparent interpretation. In some cases, reordering and sign changes are expedient. III. Results and Discussion III.1. Natural and Quasi-Atomic Molecular Orbitals. Since the orbitals in all four systems have the same general structure, we display only the contours for the case of ozone. Figure 1 shows the valence natural orbitals and their occupation numbers. In this and all other figures the blue color indicates
positive lobes. The π-NOs manifestly are numbers 6, 9, and 10. The corresponding occupation numbers for the other three molecules are listed in Table 1. The localized quasi-atomic orbitals of step (ii) of Section II.3 and their occupations are displayed in Figure 2. The π-LMOs are again numbers 6, 9, and 10. The transformations from the π-NOs to the π-LMOs are listed in Table 2 for all four molecules. Although the (+++) orbital [) π0 of eq 1] is given by a similar expansion in O3 and S3, its coefficient at the center atom is markedly smaller in SO2 and markedly larger in OS2, a difference that can be ascribed to oxygen being more electronegative than sulfur. The oriented localized quasi-atomic orbitals of step (iii) in Section II are exhibited in Figure 3. Here the π-OLMOs are assembled in panel (a) and the σ-OLMOs in panel (b). The comparison of Figures 2 and 3 clearly shows that the orientation procedure generates in fact quasi-atomic bond and lone-pair orbitals. Accordingly, the following symbols are used to denote the OLMOs:
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Figure 2. Localized molecular orbitals (LMOs) and corresponding occupation numbers for O3. Orbitals 6, 9, and 10 are the three π-type orbitals.
TABLE 2: Transformations from π-NOs (Ok) to π-LMOs (λk)
L-sσl ) s-type σ lone pair on left atom, L-pσl ) p-type σ lone pair on left atom, L-σb ) σ bonding orbital on left atom. Analogous notation is used for the right (R) atom. The σ-OLMOs on the center atom are denoted by: C-σl ) σ lone pair on center atom, C-lσb )
left pointing σ bonding orbital on center atom, C-rσb ) right pointing σ bonding orbital on center atom. Finally, the π-OLMOs are denoted by: L-πb, R-πb, C-πb on the left, right and center atoms, respectively.
Bonding Patterns in the Series O3, S3, SO2, and OS2
J. Phys. Chem. A, Vol. 114, No. 33, 2010 8927 n0 e 2, n1 e 2, n2 e 2, n0 + n1 + n2 ) 4, one deduces the following extremes as regards the biradical character of this π-system. On the one hand, it is readily seen that the bond order of Eq.(8) is zero when the center-atom π-occupation of Eq.(7) is 2 and that the converse is also the case. For this type of wavefunction, the system exhibits therefore 100% biradical character. On the other hand, the value of the bond order of Eq.(8) is seen to be maximal when n0 ) 2, n1 ) 2, n2 ) 0 and γ ) 45°. This maximal bond order value is p(Cπb, Lπb) ) (1/2) ≈ 0.707. The corresponding value of the center-atom π-population of Eq.(7) is then p(Cπb, Cπb) ) 1. In this case, there is thus no contribution from the second configuration ψ2 in Eq.(1) and the principal configuration ψ1 in Eqs.(2) and (3) can be expressed in terms of a doubly occupied left bonding MO πL and a doubly occupied right bonding MO πR as follows:
ψ1 ) {σ14π02π12} ) {σ14πL2πR2}
(9)
with
Figure 3. (a) The three π-type OLMOs and (b) the nine σ-type OLMOs for O3.
III.2. Orbital Measure of the Biradical Character. The most general form of the expansions of the three π-NOs π0, π1, π2 of eq.(1) in terms of the π-OLMOs is
π0 ) {s(Lπb) + √2 c(Cπb) + s(Rπb)} / √2
(6a)
π1 ) {(Lπb) - (Rπb)} / √2
(6b)
π2 ) {-c(Lπb) + √2 s(Cπb) - c(Rπb)} / √2
(6c)
where s ) sin γ > 0 and c ) cos γ > 0. From these generally valid expressions, it follows for the population of the quasiatomic π-OLMO at the center atom:
p(Cπb, Cπb) ) n0 cos2γ + n2 sin2γ
(7)
and for the bond order between the quasiatomic π-OLMO at the center and those at the end atoms:
p(Cπb, Lπb) ) p(Cπb, Rπb) ) [(n0 - n2) /2√2] sin2γ
(8) where n0 and n2 are the occupation numbers of the NOs π0 and π2 respectively. By virtue of these relations, and the fact that
πL ) {a(Lπb) + 0.5(Cπb) - b(Rπb)}
(10a)
πR ) {-b(Lπb) + 0.5(Cπb) + a(Rπb)}
(10b)
where a ) (2 + 1)/22 ) 0.8536 and b ) (2 - 1)/22 ) 0.1464. This wavefunction has therefore no biradical character. The bonding is manifestly created by the donation of 0.5 π-electrons from the center atom towards each of the end atoms. Thus, as the π-occupation at the center changes from 2 to 1, the biradical character changes from 1 to 0. This perspective suggests considering the quantity
Borb ) (π-occupation at the center atom minus 1)
(11)
as an orbital measure of the biradical character for this system. III.3. Population and Bond Orders of Oriented QuasiAtomic Molecular Orbitals. Ozone. The occupation-bond-order matrix between the OLMOs of ozone is shown in Table 3. For each of the Oxygen atoms, which are arranged in the order left (L), center (C), right (R), the three σ-type orbitals are listed first, followed by the π-type orbital in each atom. All offdiagonal elements smaller than 0.02 are omitted. The orbital occupation numbers on the diagonal validate the designation of the quasi-atomic OLMOs as being of the lonepair and bonding type. There is no charge transfer between the σ system and the π system. There are strong bond orders (∼0.9) between the nearly singly occupied σ-bond orbitals of the center atom and those of the terminal atoms. The bond orders between the π-orbital on the center and the π-orbitals on the terminal atoms are weaker (∼0.56). These π-bonds are manifestly contingent on the π-charge transfer of about 0.28 e- from the center atom to each of the two terminal atoms. There is a counter charge transfer of about 0.08 e- in the σ-system from each end atom to the center atom, resulting in a net charge transfer of about 0.2 e- from the center atom to each terminal atom as a result of the π-bonding. There is also a substantial antibonding bond order between the π-orbitals on the two terminal atoms, although the interaction energy integral multiplying this bond order will be weaker because of the greater distance between these atoms.
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TABLE 3: Population Bond Order Matrix in Terms of the OLMOs for O3a oxygen (L) L-sσl L-sσl L-pσl L-σb L-πb C-σl C-lσb C-rσb C-πb R-sσl R-pσl R-σb R-πb
L-pσl
L-σb
1.940 -0.030
-0.030 0.982
1.996
oxygen (C) L-πb
C-σl
C-lσb
C-rσb
oxygen (R) C-πb
R-sσl
0.216 0.915 1.283
R-pσl
R-σb
-0.219
-0.219 -0.097
R-πb
-0.505
0.558 1.994
0.915
1.085 0.114
0.216
0.114 1.085
0.558 -0.219
0.216 0.915 1.436
0.558 1.996
-0.219 -0.097
0.216
1.940 -0.030
0.915
-0.505
-0.030 0.982
0.558
1.283
a
The designation L, C, R denotes the left, center, and right oxygen atoms, respectively. Meaning of orbital symbols, “σl”: sigma lone pair, “σb”: sigma bonding, “πb”: π-bonding.
TABLE 4: Population Bond Order Matrix in Terms of the OLMOs for S3a sulfur (L) L-sσl L-sσl L-pσl L-σb L-πb C-σl C-rσb C-lσb C-πb R-sσl R-pσl R-σb R-πb a
L-pσl
L-σb
sulfur (C) L-πb
C-σl
1.996 1.865 0.058
C-rσb
C-lσb
R-sσl
-0.351
0.058 1.174
0.883 1.332
R-pσl
R-σb
R-πb
-0.022 0.309
0.309 -0.130
-0.527
0.604 1.992
-0.351
0.968 0.162
0.883
0.162 0.968
0.604 -0.022 0.309
sulfur (R) C-πb
-0.351
0.883
1.340 -0.351
0.309 -0.103
0.604 1.996 1.865 0.058
0.883
-0.527
0.058 1.174
0.604
1.332
The designation L, C, R denotes the left, center and right sulfur atoms, respectively.
TABLE 5: Population Bond Order Matrix in Terms of the OLMOs for SO2a oxygen (L) L-sσl L-sσl L-pσl L-σb L-πb C-σl C-rσb C-lσb C-πb R-sσl R-pσl R-σb R-πb a
L-pσl
L-σb
sulfur (C) L-πb
C-σl
1.998 1.691 0.076
0.076 1.525
C-rσb
C-lσb
-0.583 0.029
-0.019 0.729
1.459 -0.583 -0.019
R-sσl
0.029 0.729
0.787 0.157
0.357 -0.093 -0.476
R-pσl
R-σb
-0.054 0.357
0.357 -0.093
0.157 0.787
-0.019 -0.583
0.729 0.029
1.086 -0.019 0.729
R-πb
-0.476
0.666 1.992
0.666 -0.054 0.357
oxygen (R) C-πb
0.666 1.998
-0.583 0.029
1.691 0.076 0.666
0.076 1.525 1.459
The designation L, C, R denotes the left oxygen, center sulfur, and right oxygen atoms, respectively.
Since the remaining π-population at the center atom is 1.436 for ozone, its biradical character, as defined by eq 11 in Section III.2., is ∼44%. The Isoelectronic Series O3, S3, SO2, and OS2. The population-bond-order matrices for the valence-isoelectronic molcules S3, SO2, and OS2 are displayed in Tables 4-6. In all of them, the general bonding pattern is seen to be similar to that of ozone. Since sulfur is less electronegative than oxygen, there are however interesting differences, which are notably apparent in the charge transfers between the center atom and the terminal atoms. They are exhibited in Table 7, where the first three rows list the charge decrease at the center atoms, half of which goes to each terminal atom:
It is seen that, in all molecules, there is a charge transfer from the center to the terminal atoms, which increases from OS2 to O3 to S3 to SO2 and corresponds to the relative electronegativities. In all cases, this charge transfer is dominated by the transfer in the π-system. In OS2 and O3 there is some counter transfer in the σ-system, but that is not so in S3 and SO2. The last row also lists the π-bond orders between the center atom and the end atoms. It is apparent that the bond strengths increase with increasing charge transfer to the end atoms. In view of the discussion in Section III.2., one concludes that the biradical character increases from SO2 to S2O. This trend is exhibited by the values of the orbital-based biradical assessment Borb, defined
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TABLE 6: Population Bond Order Matrix in Terms of the OLMOs for OS2a sulfur (L) L-sσl L-sσl L-pσl L-σb L-πb C-σl C-lσb C-rσb C-πb R-sσl R-pσl R-σb R-πb a
L-pσl
oxygen (C)
L-σb
L-πb
C-σl
C-lσb
1.997
C-rσb
sulfur (R) C-πb
R-sσl
-0.074
1.992 0.875
0.945
0.081
1.188 0.945
-0.074
R-σb
1.136 0.082
-0.453
-0.074
0.082 1.136
0.462
0.945 1.626
0.462 1.997
-0.074
0.081 -0.079
1.992 0.945
-0.453
R-πb
0.081 -0.079
0.462 1.997
0.081
R-pσl
0.875 0.462
1.188
The designation L, C, R denotes the left sulfur, center oxygen, and right sulfur atoms, respectively.
TABLE 7: Valence Charge Decrease at the Center Atom and π Bond Orders Molecule
SO2
S3
O3
OS2
π-orbital all σ-orbitals total valence π-bond order
0.914 0.434 1.348 0.666
0.660 0.072 0.732 0.604
0.564 -0.164 0.400 0.558
0.374 -0.269 0.105 0.462
TABLE 8: Biradical Character Assessments molecule from π-orbital occupations from CI expansion
Borb BCI
SO2
S3
O3
OS2
0.086 0.192
0.340 0.338
0.436 0.432
0.626 0.597
by eq 11 in Section III.2., at the end of the preceding subsection, which are listed in the first row of entries in Table 8. The population-based biradical characters Borb of Table 8 and the π-bond orders of Table 7 are seen to satisfy rather accurately the following relationship:
(π-bond order × √2)2 + (Borb)1.1905 ) 1
(12)
where the factor 2 represents the inverse of the maximal possible π-bond order, as was derived in Section III.2. Equation 12 exhibits the contra-gradient relationship that exists between the biradical character and the bond strength. III.4. Configuration Interaction Expansion. Table 8 lists the first terms of the configuration interaction expansions of the four molecules in terms of the natural-orbital-based determinants. Shown are all CI coefficients that are larger than 0.05. In agreement with all previous work43 all expansions are dominated by the first two terms, which were previously discussed in detail by Ruedenberg and co-workers (cf. refs 38–42) in the analysis of the difference between the global minimum “open” structure and the local minimum “ring” structures of ozone. The weights of the second configuration increase from SO2 to S3 to O3 to OS2, as is seen in Table 10 [note the minus sign in front of c2 in eq 2]. They follow exactly the pattern of decreasing charge transfer and increasing radical character established in the preceding Section III.3. As discussed in the Introduction, Hay, Dunning, and Goddard45,46 as well as Laidig and Schaefer47 based their biradical criterion on these CI coefficients by defining the biradical character as B ) 2c22. Let us, in slight generalization, use the quantity
BCI ) [2c22 /(c12 + c22)]1/2
(7)
(which always goes from 0 to 1) as a CI-based measure of the biradical character. The values of BCI obtained for the four molecules are listed in the second row of entries in Table 8. There is a manifest degree of similarity between the two measures that are obtained in so different ways. As regards the comparison of the molecules, there is thus complete agreement between the two ways of assessing the biradical character. The CI-value of (BCI)2 found for ozone (∼0.19) is in agreement with the B values found in the earlier papers.45–47 The parallelism of the two biradical measures is particularly remarkable since the transformation of the first two CI configurations into biradical form is based on the orbitals π+ and π- that, acccording to eq 4 in the introduction, are derived from the orbitals π1 and π2 and do not haVe quasi-atomic character. By contrast, the orbitals used in Section III.3 do have quasiatomic character since they were obtained by a transformation that also involves the orbital π0 of eq 1. Kalemos and Mavridis48 dismissed the presence of the second configuration in ozone as just “a usual ... GVB ... correlation”. In fact, however, an admixture of 0.29, as in ozone, is generally considered a strong nondynamic correlation in the full molecule-optimized minimal basis valence space (which encompasses the valence-bond space). Because of this “MCSCF character”, mere SCF calculations do not yield correct vibrational frequencies in ozone.59 They are however obtained by MCSCF calculations38 and by CCSD(T) calculations.60 A biradical character is of course nothing else than a specific type of a sufficiently large non-dynamic correlation in the valence space. III.5. Is Ozone a Biradical? It has been apparent since the calculations of Hay, Dunning, and Goddard,45,46 as well as Laidig and Schaefer47 that ozone is not a pure biradical as originally conjectured by Hayes,44 but a “partial biradical”. Using it as a “textbook prototype” for a singlet biradical seems therefore inapposite. On the other hand, is it simply “a regular ... genuine closed shell singlet” as suggested by Kalemos and Mavridis?48 The preceding analysis contributes the following considerations. In the SO2 molecule, the closed-shell configuration ψ1 provides 92% of the wavefunction, the central atom donates 91% of the possible maximum of 1 π-electron towards bonding, and the π-bond order between the center atom and one end atom is 94% of its possible maximum value of (1/2). The molecule is thus close to having no biradical character and our definitions assign to it a biradical character of ∼0.09.
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Glezakou et al.
TABLE 9: Wavefunction Expansion in Terms of Natural Orbitalsa O3
S3
SO2
OS2
3a1
2b1
4a1
3b1
5a1
1b2
6a1
4b1
1a2
2b2
7a1
5b1
coefficient
2 2 2 2 2 2 2 2 2 2 2
2 2 2 2 2 2 2 2 2 2 2
2 2 2 2 2 2 2 2 2 2 2
2 2 2 2 2 2 2 2 2 2 2
2 2 2 2 2 2 2 2 2 2 2
2 2 0 2 2 2 2 1 1 1 1
2 2 2 0 2 2 2 2 2 1 1
2 2 2 2 1 1 0 1 1 2 2
2 0 2 2 1 1 2 2 2 2 2
0 2 2 0 1 1 0 1 1 1 1
0 0 0 2 1 1 0 0 0 1 1
0 0 0 0 0 0 2 1 1 0 0
0.9068070 -0.2919003 -0.0734029 -0.0635505 -0.0620287 -0.0620287 -0.0590298 0.0532943 0.0532943 -0.0501662 -0.0501662
8a1
6b1
9a1
7b1
10a1
11a1
3b2
8b1
2a2
4b2
12a1
9b1
coefficient
2 2 2 2 2
2 2 2 2 2
2 2 2 2 2
2 2 2 2 2
2 2 2 2 2
2 2 2 2 2
2 2 0 2 2
2 2 2 1 1
2 0 2 1 1
0 2 2 1 1
0 0 0 1 1
0 0 0 0 0
0.9331282 -0.2291756 -0.0772509 0.0621445 0.0621445
5a1
3b1
6a1
4b1
7a1
2b2
8a1
5b1
1a2
3b2
9a1
6b1
coefficient
2
2
2
2
2
2
2
2
2
0
0
0
0.9580289
2 2 2 2 2 2 2 2 2
2 2 2 2 2 2 2 2 2
2 2 2 2 2 2 2 2 2
2 2 2 2 2 2 2 2 2
2 2 2 2 2 2 2 2 2
2 0 2 2 2 2 2 2 2
2 2 2 2 2 2 2 1 1
2 2 1 1 0 1 1 2 2
0 2 1 1 2 1 1 1 1
2 2 1 1 0 1 1 1 1
0 0 1 1 2 1 1 0 0
0 0 0 0 0 0 0 1 1
-0.1305993 -0.0696668 0.0644504 0.0644504 -0.0586815 0.0544497 0.0544497 -0.0505274 -0.0505274
5b1
6a1
6b1
7a1
8a1
2b2
9a1
7b1
2a2
3b2
10a1
8b1
coefficient
2
2
2
2
2
2
2
2
2
0
0
0
0.8792786
2 2 2 2 2 2
2 2 2 2 2 2
2 2 2 2 2 2
2 2 2 2 2 2
2 2 2 2 2 2
2 2 2 0 1 1
2 0 2 2 2 2
2 2 0 2 1 1
0 2 2 2 2 2
2 0 0 2 1 1
0 2 0 0 0 0
0 0 2 0 1 1
-0.4098239 -0.0704828 -0.0668654 -0.0595065 0.0504095 0.0504095
a Only the configurations with coefficients >0.05 are listed. The Hartree-Fock (HF) configuration is the one with the largest coefficient. The molecule lies in the (xz) plane, and the z-axis is the bisector.
TABLE 10: List of the First Two Coefficients c1 c2 (c12 + c12)
SO2
S3
O3
OS2
0.958 0.131 0.967
0.933 0.229 0.961
0.907 0.291 0.953
0.879 0.410 0.970
At the other end of the spectrum, in the OS2 molecule the closed-shell configuration ψ1 provides 77% of the wavefunction, the central atom donates 37% of the possible maximum of 1 π-electron towards bonding, and the π-bond order between the center atom and one end atom is 65% of its possible maximum value of (1/2). Clearly, this molecule is still quite a ways from being a full biradical and it still exhibits non-negligible π-bonding. Our definitions assign to it a biradical character of ∼0.63. The ozone molecule lies somewhat in the middle between these two extremes: ψ1 provides 82% of the wavefunction, the central atom donates 0.56 electrons towards bonding, and the π-bond order is 79% of the possible maximum. Our definitions assign to it a biradical character of ∼0.44. These comparisons lead to an intermediate view regarding ozone. On the one hand, it cannot be described as “a regular ... genuine closed shell singlet”, such as SO2. On the other hand,
it cannot be visualized as having single electrons in non-bonded quasi-atomic π-orbitals at the end atoms, i.e. it is far from a full-blown biradical. Manifestly, in order to assess the physical significance of the quantitative differences between the theoretically defined wavefunction characteristics of these systems, one will have to find relationships between these theoretical quantities and experimentally observed quantities. IV. Conclusions The bonding patterns in the isoelectronic series O3, S3, SO2, and OS2 have been analyzed by an examination of the population-bond-order matrices in terms of the recently formulated oriented quasi-atomic localized molecular orbitals. A systematic trend has been identified that is related to the electronegativity difference between oxygen and sulfur. The differences in the bonding patterns of the molecules are driven by the changing patterns of the four-electron-three-orbital π-bonding. From SO2 to S3 to O3 to OS2, charge transfer from the central to the terminal atoms decreases, the π-bond strength decreases, and the biradical character increases. The π-bonds are of course weaker than the σ-bonds, and there is also some antibonding effect between the π-orbitals of the terminal atoms.
Bonding Patterns in the Series O3, S3, SO2, and OS2 The assessment of the biradical character through quasiatomic orbitals is consistent with the traditional CI-based assessment.45,46 Ozone is found to be intermediate between a full biradical44 and a regular closed-shell singlet.48 It has partial biradical character in agreement with the results of Hay, Dunning and Goddard,45,46 but it has substantial π-bonding. Acknowledgment. The authors would like to thank Dr. Michael W. Schmidt of Ames Laboratory and Iowa State University and Kurt Glaesemann of PNNL for many useful discussions. This work was supported by the Chemical Sciences, Geosciences and Biosciences Division, Office of Basic Energy Sciences, US Department of Energy at Pacific Northwest National Laboratory and under Contract No. DE-AC0207CH11358 at Iowa State University through the Ames Laboratory. Battelle operates the Pacific Northwest National Laboratory for the US Department of Energy. Computer resources were provided by the Office of Science of the US Department of Energy. References and Notes (1) See, e.g. Frenking, G.; Shaik, S. J. Comput. Chem. 2007, 1, 28. (2) Ruedenberg, K.; Schmidt, M. W. J. Phys. Chem. 2009, 113, 1954. (3) Ivanic, J.; Atchity, G. J.; Ruedenberg, K. Theor. Chem. Acc. 2008, 120, 281. (4) Ivanic, J.; Ruedenberg, K. Theor. Chem. Acc. 2008, 120, 295. (5) Lo, W.-J.; Wu, Y.-J.; Lee, Y.-P. J. Phys. Chem. A 2003, 107, 6944. (6) Lo, W.-J.; Wu, Y.-J.; Lee, Y.-P. J. Chem. Phys. 2002, 117, 6655. (7) Scho¨nbein, C. F. Philos. Mag. 1840, 17, 293. (8) Seinfeld, J. H.; Pandis, S. N. Atmospheric Chemistry and Physics: From Air Pollution to Climate Change, 2nd ed.; Wiley: New York, NY, 2006. (9) Chappuis, J. C. R. Acad. Sci. Paris 1880, 91, 985. (10) Huggins, W. Proc. Roy. Soc. London, A 1890, 48, 216. (11) Anderson, S. M.; Mauersberger, K. Geophys. Res. Lett. 1992, 19, 933. (12) Angione, R. J.; Medeiros, E. J.; Roosen, R. G. Nature 1976, 261, 289. (13) Bacis, R.; Bouvier, A. J.; Flaud, J. M. Spectrochim. Acta, Part A 1998, 54, 17. (14) Baloitcha, E.; Balint-Kurti, G. G. Phys. Chem. Chem. Phys. 2005, 7, 3829. (15) Banichevich, A.; Peyerimhoff, S. D.; Beswick, J. A.; Atabek, O. J. Chem. Phys. 1992, 96, 6580. (16) Batista, V. S.; Miller, W. H. J. Chem. Phys. 1998, 108, 498. (17) Braunstein, M.; Martin, R. L.; Hay, P. J. J. Chem. Phys. 1995, 102, 3662. (18) Braunstein, M.; Pack, R. T. J. Chem. Phys. 1992, 96, 6378. (19) Burkholder, J. B.; Talukdar, R. K. Geophys. Res. Lett. 1994, 21, 581. (20) Cachorro, V. E.; Duran, P.; deFrutos, A. M. Geophys. Res. Lett. 1996, 23, 3325. (21) Chakraborty, S.; Bhattacharya, S. K. J. Chem. Phys. 2003, 118, 2164. (22) Enami, S.; Ueda, J.; Nakano, Y.; Hashimoto, S.; Kawasaki, M. J. Geophys. Res., Atmos. 2004, 109.
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