Proton Power An Intuitive Approach to the Electronic Structures of Molecular Hydrides Thomas P. Fehlner University of Notre Dame. Notre Dame, IN 46556 J a m e s R. B o w e r SUNY College at Fredonia, Fredonia, NY 14063 An obvious difference between atoms and molecules is that atoms have only one positive center (i.e., a single nucleus), while molecules have more than one. The united atom model of Mulliken (1)tells us that molecules can be related to isoelectronic and isoprotonic atoms; that is, a molecule is an atom whose nucleus has been separated into two or more regions of space. For example, in Mulliken's approach the N2 molecule can be thought of as arising from a silicon atom whose 14 protons have been split into two centers of seven each, approximately 110 pm apart. This nuclear division separates the electrons of the system into two types. Core electrons are associated with only one of the separated nuclei. Such electrons are destabilized relative to their corresponding states in the united atom because of the reduced effective positive charge they experience. Alternately, electrons associated with two or more nuclei (and either more or less stable than the corresponding united atom state) are called valence electrons. Core and valence electrons share many characteristics (e.g., the Pauli principle applies to both). However, there are also significant differences between them. For example, the potential energies of valence electrons are much more sensitive to changes in internuclear distances than are core electrons. The united atom model was once considered fundamental (2),but has now fallen into disuse. In this paper we explore its applicability to a variety of chemical systems (particularly those that differ only in the spatial locations of their orotons) and demonstrate its considerable value as a teaching tool. This methodgir,es predictions that arequnlitatlrely similar to thoseof molecular orhiral IMO) theorv, hut is both more easily applied (because i t requires little orbo computation) and understood (because it is a chemically intuitive approach) than are MO methods. Since the united atom model relates the electronic structures of atoms to those of molecules, the atomic orbital (AO) model is a prerequisite to the approach advocated here (as it is to MO theory). A characteristic of our method is the facile generation of visual pictures describing the general shapes and nodal planes of molecular orbitals. A pictorial approach to MO theory is of course not new (3),and in particular the pioneering work of Hoffmann has been recognized (4). In our aopriach the MO's of hydrides are draurn as perturbations bf the atomic orbitalsof"parrnt"sysrems, making.it attractive for instructional purpoies. Another pedagogical benefit is that the predictions developed can be tested and verified by the experimental evidence of photoelectron spectroscopy (PES) (5). Ionization studies provide reliable information on electronic states for both atoms and molecules through Koopmans's theorem ( r = -1P) (6). In most texts Koopmans's theorem is applied to atoms in order to use ionization potentials in a discussion of AO's, but it is often ignored for molecules. PES is an excellent tool for the study of both core (viax-ray PES, or ESCA) and valence (via UV-PES) electronic states, and the greater 976
Journal of Chemical Education
incorporation of photoelectron spectroscopy into the undergraduate curriculum has been urged (7, 81.Unfortunatelv, spectral interpretations have traditionally been based i n MO theory, with the ionization band assignments made using group theory terminology. Therefore, understanding the spectrum of even a simple molecule may require a depth of knowledge beyond that possessed by many undergraduates. The method advocated here avoids this problem, and a secondary goal of this article is to demonstrate that the approach described here is a useful pedagogical complement to the experimental data from PES. Deflnltlon of the Method In this approach the description of molecules flows readily from the familiar hydrogenlike, A 0 description of atoms. Hydrides, for example, can be generated in a thought experiment by removing n protons from some atom Z to infinity (step a), and then bringing them back to the known (arbitrary) positions of the molecule (step b):
Both the atom and its monatomic anion are equally well described by hydrogenlike AO's. The placement of the protons around the anion perturbs the AO's in a readily understood manner. These perturbed AO's are the molecular orhitals of the hydride and can he correlated with the ionization behavior of the molecule just as the IP's of the atom correlate with its AO's. Application to H2
The general process can he applied to Hp beginning with the He atom (Fig. 1).In the first step the potential energy of the electrons increases because of the reduction in local
FlgUre 1. Energy relationshipsfw some two-proton, twc-electron systems.
Figure 3. Derivation of the molecular orbital energy levels faHF. Except for the 2s level of FF, ail energies are known and to scale.
.o.. UNITED ATOM
SEPARATED
HOH
ATOMS
Figure 2. United atom orbital correlation diagram, with levels shown for a homonuclear diatomic such as N2,which correlates with Sl. nuclear charge (as evidenced by ionization potentials, the 1s orbital energy changesfrom -24.5eV in He to -2.5 eV in the hvdrideion). Thesuhsequent placement oithe proton at the H-H equilibrium distance of 14 pm creates a perturbation of the 1s atomic orbital as the electron cloud is attracted to (oolarized bv) the oroton.' The result is a "distorted" A 0 chat we call molecular orbital. I t is entirely equivalent to the familiar a, orbital of conventional MO theory, but obtained in a much simpler way using qualitative electrostatics rather than mathematics. Experimental ionization potentials show that two protons separated by 74 pm stabilize two electrons less than do two protons in the same location (-15.4 eV for HZVS. -24.5 eV for He), but more than two protons separated by a large distance (-13.6 eV for 2 H). The additional stability in the former case can be attributed to "bonding" between the two hydrogen atoms in Hz. The nhotoelectron soectrum of H? illustrates a orooertv of he mo1ec;les that is not bossessed by- atoms-str;ctuie. snectrum consists not of a sinele line (like atomic spectra). gut aset of lines, or a band, whGhis characteristicof molec": lar spectra in aeneral. This results from the fact that the nuclei are not stationary but oscillate, thereby changing the internuclear distance. Fine structure yields information on the structure ot the cation, hut a discussion of these aspects of PES is beyond the scopeof this article (51.For hydridesof the type MH,, suffice it to say that hand width and shape permit easy differentiation between electrons associated with the central nucleus (atomiclikeand thereforesharor vs. more than one nucleus (~olecularlikeand therefore bioad) and provides additional information on the nature of the interaction.
a
--
'
Of course this actually gives the ion pair HfH-. The great polarization of this species would lead to electron redistribution to formHH. Throughout this article such redistribution to give nonpolar or polar covalent bonds will be assumed.
Figure 4. Quaiitatlveenergy changes ofthe H 2 0 MO's derived horn the 2p AO's 01 02-lor bent and linear geomebies. The energies for bent H20are estimates.
The correlation diamam in Figure 2, patterned after Mulliken's generalization'il), allowsan excited state of Hz to be easilvdiscussed. Taking the line connecting the nuclei as the z axis, one finds that the next higher ene* MO correlates with the p, A 0 of He. Again this is equivalent to the a, orbital of conventional MO theory.
..
Aoollcatlon to the First-Row Hvdrides Using the same approach, the hydrogen fluoride molecule can be "svnthesized" from the isoorotonic neon atom by the removal of one proton from the ~e nucleus to give F< followed bv its replacement (conventionally, along the z axis of a ~ a r t e i i a ncoordinate system) at the equilibr6m bond distance of 91.7 pm. The resulting energy changes are presented in Figure 3:~he placement-of the proton on the z axis polarizes the atomic 2s and 2p, orbitals, forming MO's. The p, and p, orbitals remain atomiclike because the proton lies in their yz and x z nodal planes. Thus we expect to observe three IP's in the ohotoelectrons~ectrum.one sharp at low IP (degenerate 2pZAand2p, AO's) a n d two broad athigher IP (MO's derived from the 2s and 2p, AO's). The actual spectrum confirms these predictions (56). If desired, the aualitative MO's of this model can be effectively compared tb orbital contour diagrams generated from ab initio or semiempirical molecular orbital methods (91. For example, the shapes and nodal behavior of the 20 and 3a Volume 85
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ed because of its -greater nroximitv to the two nrotons.2 The p, atomic orbital experiences no selective stabilization hecause of its node in the yz nlane, and is atomiclike. Tberefore, the prediction for linear HZOis three IP's (sharp, broad, broad), while for bent water it is four I P S (sharp, broad, broad, broad). Correlations with MO calculations (11) and contour diagrams (12) can also be made. The photoelectron spectrum, which consists four bands a t 12.6 (sharp), 14.8 (broad), 18.5 (broad), and 32.2 (hroad) eV, agrees with the predictions for the bent geometry. The MO's for NH3 in planar and pyramidal structures can be generated in similar fashion (Fie. 5). Here it is necessarv to iecognize that the electronic ;.barge distribution exnressed bv the two D AO's is cvlindricallv svmmetric relative to a plane-defined by the axes-of the AO's, e.g., p,a py2(x2 yz)lrz = 1. Thus three protons in an equilateral triangular &ray will perturb the 2p, and 2py A O ' ~of nitrogen by the same amount. The model predicts three IP's for both geometries. but for the . Dvramidal case all are hroad (as the ohoto. electron spectrum shows), whereas for the planar case the lowest IP would be sham. The MO's for CHa can be similarlv generated. Correlation of the photoelectron spectra of all o i these molecules, adapted from a diagram by Potts and Price (131,are given in Figure 6. Compare Ne vs. CH4: A symmetrical array of four protons yields the same IP pattern, but like He vs. Hz the four protons stabilize the electronic charge more effectively when they are all located in one nucleus.
+
Figure 5. Quai DllVB energies 01 the NH3 MO'Sderived fromthe 2p AO's of N'. 101 p mar an0 pyramidal geometries. Tneenerg es fw the pyrsmloa SlructLre are experimental values
+
Comparlsons Between Compounds of the First and Second Perlods
The relationship between HCI and Ar is the same as that between H F and Ne. Since the hiehest enerev orbitals of argon are less stable than those ofneon, onewould expect the same relationshins for HC1 vs. HF. The PES ionization potentials for the lowest energy (np) bands are: i.p. (eV) Ne A,
Figure 6. Correlationsot the MO's of Ne. HF, H.O. NH.. and CH,: others are broad (see ten).
sh =sharp:all
HF HCl
As predicted, there is a shift to lower energy for hydrogen chloride vs. HF. It mieht also be noted that. while the D.-"...ID. . energy differences ink^ and HC1 are equal (3.4 eV), the p, A 0 is more stabilized in H F relative to neon (16.1 vs. 21.6 eV) than is HC1 relative to argon (12.8 vs. 15.8 eV). This can he explained in terms of relative effective nuclear charges. The larger, more diffuse p, A 0 of C1- is less effectively stabilized than that of F-. When Hz0 and H2S are compared, an additional factorthat of hond angle-comes into play. The smaller angle in hydrogen sulfide causes t h e y and z coordinates of the two protons to be more nearly equal. Using the experimental hond angle (92') and distance (134 pm), the coordinates are found to be
. .
bonding MO's of H F are readily interpreted as resulting from distortions of the parent atomic orbitals due to perturbations by the displaced proton. For MHz systems both linear and nonlinear geometries are possible and can be explored easily by this method. Thus, to obtain the relative relationship for the (hypothetical) linear water molecule we visualize the removal of two protons from Ne to give 02-, followed by reprotonatation a t the coordinates (0,0,z) and (0,0,-z). The resulting energy levels are aualitativelv similar to those of HF. exceDt that the MO derived fro& the 2p, A 0 shows greate; stabilization because two protons, rather than one, are on the z axis (Fig. 4). For nonlinear HzO, it is convenient to place the oxygen at the origin of a Cartesian axis system and the two renositioned protons in the yz plane tb be consistent with Goup theory convention (10). Choosing the experimental bond length (96 pm) and hond angle (104.5'), simple geometry leads to these atom coordinates (in units of picometers):
H? 0
HI
X
Y
I
0 0 0
59 0 59
76 0 -76
The 2py and 2p, AO's are unequally polarized and have different energies (Fig. 4); the former is more strongly affect978
Journal of Chemical Education
Therefore not only are the ionization hands of HzS found a t lower energy, hut the second and third bands lie closer to one
A useful study question is to ask students at what bond angle the p, and p, orbitals would become equivalent in energy. The fact that
thev do not become eauivalent (see the examde of HISI - . indicates a fundamental limitation bf this mddel; i.e.. thereis no way to permit the mixing of 2s and 2p, with this simple approach.
another; the p,/p, energy difference is 3.7 eV for water but only 1.9 eV for HpS (14):
A comparison of phosphine and ammonia (bond angles of 92O and 107', respectively) reveals the same effect, with the energy separation between the first two ionization bands being 5.3 eV for NH3 and only 2.8 eV for PH3.3 Compounds Havlng Two Heavy Atoms N2 and C f i
Mulliken illustrated his model with a correlation of the energy levels of molecular nitrogen to those of the isoprotonic silicon atom. This is accomplished by imagining the relocation of seven protons from the Si nucleus to a distance of 110 pm along the z axis. By the same reasoning as above, this should preferentially stabilize the original 2p, and 3p, orbitals relative to the s, p,, and py orbitals. A correlation diagram for the Si Np system (adapted from ref I ) is given in Figure 2. The highest occupied energy level of Np (3ugin MO theory) correlates with the 3s atomic orbital of the parent silicon; other correlations are aJ2p,,, 2uu/3p,, and 2og/2s. Although Nz is more closely related to 2 N than to Si, the reader can nevertheless appreciate that a relationship exists between the pertinent AO's of Si and the MO's of Nz. The diagram also suggests that the atoms-in-molecules approach of MO methods gives a superior description for the Nz molecule. By removing one proton from each nucleus of Nz (giving C12-) and then re~ositioningthem on the z axis, the MO's of N; are perturbed;thereby generating the MO's of acetylene. The concentration of positive charge along z corresponds, of course, to a depletionif charge along x and y. ~ h e i e f o r the e 2p,, level is destabilized (from 16.7 eV in Nz to 11.4 eV in HCCH), while the other two valence AO's (s and Zp,) are stabilized (from 15.6 and 18.7 to 16.7 and 18.8 eV, respectively).
-
NzHZ. C2H4. and B2Hs
The concept of moving protons to generate the MO's of isoelectronic (isoprotonic) molecules can be carried further. Diazine, NpHp, has been isolated a t -180 "C. Spectroscopic evidence indicates the trans geometry, with H-N-N bond angles of about 107' (15). One can visualize the formationpf diazine from molecular oxveen "- hv. removal of one wroton from each oxygen nucleus to give Nz2-, followed by i& reintroduction in the vz wlane. The bond angles are such that the greates relative stabilization occurs wi;h MO's hwing lobal densities alone. the, axis. Thus the degeneracies ot the 2w,, and 3d,,,y, (united-atom designation;) levels are removeJ, with py and dy, becoming the more stable member of each pair. The s, p,, and p, domains are little changed from Nz2-, while the removal of the 3 d , , , degeneracy and subsequent electron pairing cause the 3d,, orbital to be unoccupied. This analysis is consistent with PES data (Fig. I ) , in which the four ionization bands at 10.0, 14.4, 15.0, and 16.9 eV correspond to MO's deriving from the 3dy,, 2p,, 2py,and 3s united atom orbitals, respectively. Next one can visualize the formation of ethylene from NpHp by the removal of an additional proton from each nucleus to form CzHp2-, followed by replacement in the yz plane. This increases the relative stabilization of the MO's having py and dy, parentage and the destabilization of those of s, p,, and p,. As can be seen in Figure 7, this leads to a 3 A discussion of the reasons behind the changes in bond angle has been given most recently by Hall. M. 6 . Inorg. Chem. 1978, 17. 2261.
Figure 7. Correlation of MO's in me series 02, N2H2,GH4, and &He. Energies for the last three from PES (-IP's). The united atom designationsare given in Figure 2. The energies for O2are calculated values (6).scaled by me factor 0.84. The 29 and 3p, energies for N,H2 are from the same source: all others are experimental values. reversal of the relative energies of 3dy, and 2p, (the latter becoming the least stable of the occupied orbitals) and of 2p, and 2s (2py becoming the more stable of the two). Diborane is now derived from ethylene by repositioning two additional protons taken from the carbon nuclei (16). Keeping the same reference axes, these protons (the B-H-B bridging hydrogens) exert their primary influence perpendicular to the molecular wlane of CIHL-that is. in the x v plane. Thus the hlO derivEd from th;'&. unitrd atom orhi;. al rnuves from -10.5 eV in C,HI to -14.7, eV in RIH.; (Ifihl. The other ionizations all oc& at slightly lowereiergies, consistent with the reduced nuclear charge of boron vs. carbon. Comparisons of orbital contour diagrams for CpH4 and B2H6are of interest in this regard (17). Cluster Systems This approach need not he restricted to simple molecules. For example, the delocalized bonding in metal, nonmetal, and mixed cluster systems has been a focus of intense study (18). The proton displacement model permits certain aspects of the bonding in these complex systems to be understood. For examole. and . . the isoelectronic clusters Fe2(CO),~ .. H.Mn,[COj,: are related by the movemen1 of lhree protons from iron nucle~into hrideme-. wositions in the wlane defined by the metal atoms (by group theory convention, the xy plane). This places these protons in existing electron domains between metals, thereby stabilizing them; i.e., wellshielded protons are brought out of the metal nucleus and buried in-the valence electron density in the bridging positions. Thus, in comparing the photoelectron spectra of these compounds, we anticipate themovement of bands associated with the metals to significantly higher I P in the hydride cluster. Consistent with this nrediction. the first band of the Fe3(C0),p spectrum (7.9 eV) 'has been cbrrelated to the third band of HaMnn(CO),., (11.4 eV). and the latter assianed to M-H-M ikization(l9). The other bands are either unchanged or move to slightly lower IP's. Similar results have been observed for O S ~ ( C Ovs. ) ~H3Re3(C0),~ ~ (20) as well as other related systems (21). To conclude, we advocate the method described here as a versatile, easily understood, qualitative approach for correlations of isoprotonic molecules. Although it does not replace MO theory, it serves to draw strong connections between AO's (familiar to students) and MO's (often confusine to students): In addition, the approach need not be restricted to simwle (diatomic) molecules as is often found for MO treatments in introductory texts. We believe it to he valuVolume 65 Number 11 November 1988
979
able for introducing the topics of molecular structure (both electronic and geometric) and photoelectron spectroscopy in the undergraduate curriculum. Literature Cited 1.
Mu1liken.R.S. Re". Mod. Phys 1932.4.1.
2. SeeCou1~on.C.A.Valence: 2nded.: Oxford University: London, 1961.pp97ff. Alsothe following quote concerning Mullik~n'smodel: "it might well be on the wall! of
chemistry huildingn. being a1mo.f worthy fooceupyaposilion heridethe Mendeleel mriodic tableso frewenflyfound thereon. Just as the latter affordsan understandillg of the structureof atoms so does the former an understandins structure of m o 1 a c u l e ~ ' ~ V aVlerk. n J. H.: Sherman. A. Rou. Mod. Phys. 1935, 7. 167. 3. See for example Davidson. R. B. J. Chem. Educ. 1977, 9,531, and referenew cited therein. 4. Hoffmann. R.Sciance 1988,211,995. 5. For summaries nf the basic principles of UV-PES. see la1 Sieqbahn, K. Science 1982. 217, 111. (bl Brundlc, C. R.: Raker, A. D.,Eds. Electron Sperfrorropy: Theory, Terhniqu~r,nndApplicolzons:Academic: London, 1977 and 1979: Vols 1and 11. lc) Rabalsis. J. W. Principles of Ultroorolaf Photoeleciron Spectroscopy: Wiley-InLeiscience: NewYork, 1977. Id) Hendrickson, D. N. InPhysiroi Meihadnin Chemirlry:Drago,R. S., Ed.:Saundaa: Phlladalphia, 1977,pp 566-58d (e) Turner. D. W.: Baker, C.: Baker. A. D.; Brundle. C. R. Molaculor Pholodrctlon Spectroscopy; Wiley: New Yark, 1970. 6. Knoprnan8,TPhyaico (Utmchl) 1934.1, 104.
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Journal of Chemical Education
Bock.H.: Molliore, P. D. J. Chem.Educ. 1974, SI.506. A useful anurce for this purpose iiDeKock, R. L.: Gray, H. B. Chemical Sfnrefure and Rondiny; BenjsminICummings: Menlu Park. CA. 1980. For aramplo,see Jorgensen, W. L.; Salem, L. The Orgnnic Chemist's Rook oiO~bitals; Academic: New YO&. 1973; p 71. Cotton. F. A. Chemirol Appiicorions of Group Theory, 2nd ed.: Wiley Intemience: New York. 1971. See for example Pitzer, R. M.: MerriBeld. D. P. J . Chem Phys. 1970, 52, 4782 and refermces therein. Rei9. p 70. Poffn,A. W.;Price, W.C.Pror.Rqv. Sor, 1972.A326, 181, For the corresponding moieculsr orbital theory interpretation see r e t 7, pp 509-511. la) Froat. D. C.: La". W. M.: McDowell. C. A.: Wesfwood. N. P. C. J. Mol. S t r u l . (THEOCHEM) 1982, 7, 283.1blFrost,D.C.;Lee,S.T.: McDowel1,C.A. Westwood, N. P.C. J. Chem. Phys. 1976.64.4719, la) Pitzer, K. S. J. Am. Chem. Soc. 1945,67, 1126; lbl Brundle, C. R.:Robin, M. B.; Bunch. H.:Pinrkv. M.: B0nd.A. J. Am. Chem. Soc. 1970.92.3663.
.....,
.. ... ... ... .
lsl Mingos,D. M.P.Chem.Soc.Reu. 1986,15,3lIb) Johnron,B.F.G..Ed. Transition MelalClusters: Wilcy: N~wYork.1980.1~1Wade. K.Adu.Inorg Cham.Rodiochsm. 1976. IS. 1. ld) Schilling, B. E. R.; Hoffmann. R. J. Am. Cham. Sac. 1979,101,3456. lel Hoflmann.R.;Lioscomb. W.N. J.Chem.Phvs.62.36.2179.iflLiwcomb.W. N.
