R. M. Gavin, Jr. Hoverford College Hoverford, Pennsylvania 19041
Simplified Molecular Orbital Approach to h0rgtlniC Stereochemistry
O n e trend in honding theory has been to devise simple models which account for the observed geometries of molecules. Models of molecularatructure based on electron repulsion (1, 2) or on hybridization (5) of atomic orbitals have received particular attention. These approaches have been attractive for a numher of reasons, chief among which is that they provide insight into chemical honding yet require a minimum of computational effort. The Huckel method provided the organic chemist with an uncomplicated, yet informative, scheme for deriving electron densities and bond orders of planar aelectron systems. Unfortunately molecular orbital (MO) techniques for predicting molecular geometries and relative bond lengths in other types of molecules have not possessed the simplicity of the original Huckel scheme. The trend with molecular orbital models has been in the direction of increasing complexity rather than toward simplicity. Much effort has been expended in the direction of developing a truly selfconsistent-field molecular orbital approach to chemical bonding. Although considerable information of chemical interest is available, potentially, from these calculations, once an approach reaches the complexity of the SCF model, the average chemist is unlikely to possess the time or the resources necessary for the calculations. The purpose of the present paper is to outline the simplest of the Huckel-type molecular orhital models for inorganic molecules and to explore the information on molecular geometry implicit in this model. Attention will he on sigma bonding in nontransition metal but extensions of the method to other typesof bonding can easily he made. ~ ~predicted from this modified Hiickel molecular orbital (HMO) approach will he compared to those predicted by the valence-shell electronpair repulsion (VSEPR) model of Gillespie. The similarities between the two approaches will be discussed along with an evaluation of the limitations of the HMO model. In addition, the HMO approach will be compared to the molecular orhital approaches of Walsh ( 4 ) ~of Hoffmann (6)1 and of Allen (6).
are constructed from a linear combination of basis orbitals
where the cti are called expansion coefficients. The allowed energy levels for this one-electron system may be deduced from the integral form for the Schroedinger equation
where H is the Hamiltonian operator, and
Since the energy E is a function of the expansion coefficients CC,minimization of E with respect to variation of each expansion coefficient will yield the desired energy states. The series of equations
results from taking the partial derivative of E with respect to the N coefficients G and setting the e w e s sions eqn. ~ equal to ~ zero. Values ~ of E satisfying t ~ (1) may i be from the determinant
IH - ESI
=
o
(2)
where H represents the matrix of energy elements H,, and represents the of overlap integrals For an N-orbital system, there are energy roots (eigenvalues), and sets of coefficients (eigenvectors), ck,, which satisfy eqn, (2). A wavefunction #, constructed from the kth eigenvector will have energy
'..
-a.
LCAO MO Theory
The linear-combination-of-atomic-orbitals molecularorbital (LCAO 110) technique is well known and has been described in a numher of textbooks (7-9). Electrons move in orbitals which are delocalized over the entire molecule rather than localized about a single atom or between two atoms. Wavefunctions for these electrons are obtained by assuming that when an electron is close to a given atom its behavior can he approximated by an atomic orbital 6, centered on that atom. A set of one-electron molecular wavefunctions #n
For a many-electron system, it is assumed that the energy of the molecule may be obtained from the solutions of the one-electron problem. In accordance with the Pauli exclusion principle, the energy eigenstates rk are filled pairwise starting with the lowest energy level. The expression for the energy is taken to be the sum over occupied energy levels of the energy eigenvalues, or Earo
=
Cnrtr k
where n, is the number of electrons (O,1, or 2) in the kth energy level. Volume 46, Number 7, July 1 969
/
41 3
~
The electron density in the molecule is derived from the eigenvectors for each occupied orbital. Since #k*#r is related to the probability distribution for the electrons in the kth eigenstate N
Slbl'h dr
=
Eco2 i
+
. .
the components of each can be used to estimate the electron density. The relative bond order between orbitals i andj, pi$, and the relative population in orbital i, p i t , are defined by the relations (10)
.
. .
..
.....,.....,.,,..,.... ,.
Table 1. A Comparison of Approximations Adopted in Hickel Molecular Orbital Calculations on Inorganic Molecules and on Planar Pi-Electron Systems. Inorganic Molexles
N ~ E c i ~ c e S i i if9
...,
Pi-Electron Syatuns
-
., -
-
Hamindan ~ a t r i x H . . - vslence state ieniration Hii m potential Ii i and j are oeotral atom and If i end j represent oarbon abma limnd siams orbitals, bonded to eaoh other resyectively H -1 .. - B Hg K(Hii Hj;)Sij All other terms All other terms Hjj = O . O Hij = 0 0 Overlap Matrix Sii = 1 . 0 s i i = 1.0 s,=o.o,i+j Sg-0.0,iZj Secular Equation IH EII = 0 IH - EIl = 0 Bond Lenzths All lipand-centrsl atom bond All carbon-osrbon bond lengths equal lengths equal for s mven t v ~ of e licand
-
+
-
where nk is the number of electrons in the kth molecular orbital, and S,, is the overlap integral between the ith andjth orbitals. The total population for each atom is the sum of populations for all orbitals on that atom. The total bond order between two atoms is the sum of the relative bond orders between all pairs of orbitals on the two atoms. Approximations Adopted
The integrals Hu are not calculated in semiempirical molecular orbital approaches. Rather, approximations are adopted for these integrals and the approximations are evaluated with respect to their ability to predict experimental results. The basic assumption to be tested in the present model is that stereochemistry in sigm*bonded molecules is determined to a large extent by interactions between electrons pairs in the valence shell of the central atom. I n line with this assumption, the following approximations will be adopted in the LCAO J40 framework 1) Only valence shells- and p-arbitals on the central atom and
one o-symmetry vahmce orbital on each ligand will be considered. 2) Only #-type bonded interactions between ligand orbitals and central atom orbitals will beincludedin tbeHamiltonian matrix; all ligand-ligand interaction energies will be set to zero. 3) Diagonal elements H i ; of the Hamiltonian matrix will be approximated by valence state ionization potentials (VSIP)
from the tables of Mulliken, et al., (IS). The constant K in the Wolfsberg-Helmholz formula was taken to be 0.875 (5). Energies for different geometries were obtained by changing the orientations of the ligands in the coordination sphere of the central atom. Sample Calculation-AB2
Molecules
The simplicity of the proposed HMO scheme may be demonstrated by investigating molecules of the type AB,, a central atom A with two ligands B. We need to consider ones orbital and three p orbitals on atom A and one u-symmetry orbital on each B atom. An LCAO one-electron wavefunction for the molecule may be written as
+ = c&s) + eA(p.) + C&P.) +
+
c ~ P . ) c&od
+ caB(4
where A(p,) represents a p, atomic orbital on atom A, B(uJ represents a u-symmetry orbital on ligand 1, etc. A sketch of the orbitals is given in Figure 1. If the
(11).
4) ~ b Wolfsberg-Helmnalz k parametrization (18)
+
H .v- - K(Hii Hjj)So where K is a constant, will be assumed for all nonzero offdiagonal elements. 5 ) The zero differential overlap approximation will be adopted in the solution of eqn. ( Z ) , i.e., the eigenvalues and eigenvectors will be obtained from the Hiiekel-type secular equation IH - EIl = 0 where I represents the identity matrix. 6) All ligands of a given type will be placed at the same distance from the central atom. Relative bond lengths in the molecule will be inferred from calculated bond orders.
These approximations adopted for sigma-bonded nontransition metal compounds are very similar to the approximations of the simple Hiickel scheme for planar pi-electron systems. For purposes of comparison, assumptions of the two models are listed in Table 1. The calculations to be reported were performed on a digital computer. Input for each molecule consisted of VSIP from the work of Jaffe (11) and overlap integrals 414
/
Journal of Chemical Education
Figure 1 .
Geometry for AB8 molecules in a b e b coAgurotion
approximations listed in Table 1 are adopted, the secular equation becomes
where H,, H,, and HBrepresent the VSIP of the A(s), A(p), and B(c) orbitals, respectively. The HI6, Hie, HS1,and He, elements are all equal and may be expressed as H.. =
K(H.
+ HB)S,D
where S,. is the overlap integral for the A(s) and B(u) orbitals. The other off-diagonal elements are functions ., as of the B-A-B bond angle, 2a. If we define HO the energy for the integral
where the axes of A(p) and B(u) are coincident, the offdiagonal terms become
General expressions for the dependence of off-diagonal elements on bond angles are given in the Appendix. The elements Hz,, H,, H,,, and He, are all zero since the v. orbital is orthogonal to both B(u) orbitals in the ?/z plane. The roots of the secular equation, eqn. ( 5 ) ,may be ohtained by solving a sixth order polynomial in E. This is usually accomplished by use of a standard eigenvalueeigenvector routine on a digital computer (14). I n the case of AB1 molecules, however, it is possible to use the symmetry of the molecule to reduce considerably the complexity of the computations. I n this case, eqn. (5) can be factored into three equivalent equations of third, of second, and of first order in E by use of a basis set for ligand orbitals which reflect the symmetry of the molecule. For any choice of bond angle 2a an AB2molecule possesses at least C2. symmetry (15). Let us designate the Cz axis as the z axis and the yz plane as the plane of the molecule (see Fig. 1). The appropriate symmetry orbitals and symbols for the respective irreducible representations may be written as follows A@),
at
(6)
AS+= (B(WL) B ( d ) / 4 %
at
(7)
A(p.1,
ax
(8)
+
A(Pv), bz 8- = ( B ( d - ~ ( d ) / d Z b1 ,
(10)
MP.),
(11)
bl
Figure 2. Schematic diogrom showing lhs dependence of lhe six eigenrtotes for on ABz molecule on the bond angle ond showing mugh sketcher of the molecular orbitolr.
shown in Figure 2. If there are four valence electrons in the orbitals of interest, only the la1 and lbz levels are occupied in the ground state. From Figure 2 it can be seen that a bond angle of 180' would be expected in this four electron case because of'the stability of the l b z level for the linear geometry. Molecules with six valence shell electrons and molecules with eight valence shell electrons will have the same energy dependence on bond angle because the energy of the bl level (the A(p,) orbital) is the same for a11 or. For the six or eightelectron systems either a linear or a bent geometry will he more stable depending on the angular dependence of the sum el,, rz.,. Bond angles from 120' to 180' were obtained for the minimum energy depending on the exact VSIP and bond lengths assumed. The water molecule, for example, (eight valence shell electrons) had a linear geometry for some choices of parameters and a bent geometry for others. Iflolecnles with five electron pairs in the valence shell of the central atom, such as XeFZ,KrF,, IC1,-, and Ia-, were linear because of the stability of the highest occupied orbital (3al) a t the 180' bond angle. This result was obtained for all five electron pair cases tested and was not sensitive to the choice of parameters. If we adopt the notation of Gillespie (1) (A for central atom, X for ligand, E for lone pair) we may summarize
+
(9)
Since orbitals belonging to different irreducible repre sentations are orthoeonal. the use of the svmmetrv orbitals of eqns. (6)-(71) reduces eqn. (5) to the form " 4ZH.c Hs_- E 4 2 H w o cos a 0 0 0
0 &H,2 cos a H, - E 0 0 0
There are, therefore, three al eigenvalues corresponding to mixing of the A(s), A(p,), and S+ orbitals, two bz eigenvalues from mixing of the A(pJ and S- orbitals, and one bl eigenvalue corresponding to the energy of the A(p,) orbital. Calculations were performed on a number of ABz molecules to determine the relative spacings and angular dependence of the eigenvalues. The results varied somewhat from molecule to molecule and with different bond lengths for a given molecule but the qualitative appearance of the eigenstates can be represented as
0 0
0
0 Hp - E 4 2 H , 4 sin a 0
0 0 4 2 H w 0 sin a HB - E 0
0 0 0 0 0 H,
=
0
-E
the HMO predictions for ABzmolecules as follows 1 ) AXn molecules will be linear 2) AXJ3 and AX2& molecules will be bent or linear depending an the sum e., m, 3 ) AXIEomolecules will be linear
+
A review of the literature on molecular structures of AB2 molecules reveals the following systematic results for a large percentage of the cases 1 ) AX, molecules are generally linear 2 ) AXIE and AX2E2molecules are generally bent 3) AX2Esmolecules are linear Volume 46, Number 7, July 1969
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41 5
Correct predictions of linearity or nonlinearity for AB2 molecules may be obtained, therefore, in a large number of instances from HMO calculations. Care should be exercised, however, when dealing- with AX2E and AX& molecules. The influence of ligand electronegativity on bonding in rare gas molecules may be demonstrated by considering two diierent xenon halides. Shown in Figure 3 are correlation diagrams for XeF2 and for a hypothetical XeB2in which the ligand B has a VSIP of 9.0 eV. It is
Figure 3.
Correlation diagrams for two different types of xenon dihalides.
apparent from Figure 3 that the stability of the mole cule, relative to the isolated atoms, is much greater for XeF2than for XeB2. We might expect that stable AB2 rare gas molecules are possible only when the ligand is considerably more electronegative than the rare gas atom. This might help rationalize why XeF2and KrF2 are stable molecules whereas NeF2 and Xe12apparently are not stahle. I t appears from the model that XeCI, should he stable with respect to atomization but no such compound has been reported. Pentacoordinate Phosphorous Compounds
An iuvestigatidn of the series of compounds PFs, PF4(CH3), and PF3(CH& provides an interesting demonstration of the information obtainable from the HMO model. For these molecules we need to consider nine basis orbitals, four from the phorphorous and one from each of the five ligands. The VSIP adopted for the calculations were 17.5 eV for P(3s), 11.0 eV for P(3p), 17.4 eV for F(2s), and 11.3 eV for CHs(a) Overlap intepals were computed for a P-Fo bond length of 1.6 A and for a P-C hond length of 1.8 A. The trigonal hipyramid was found to be the most stahle configuration for all three molecules. A correlation diagram showing the energy levels for PF5is given in Figure 4. I n the mono- and di-substituted compounds, equatorial positions for the methyl groups were favored over axial positions, and the molecules were stabilized when the fluorines were bent away from the methyl groups. For PFI a square pyramid structure of Gosymmetry was found to he only about 0.5 kcal/mole less stable than the trigonal bipyramid (Dabsymmetry). These results are in accord with experimental results. Electron diffraction studies of the three molecules confirm trigonal bipyramid structures (16) and indicate that in the methyl-substituted compounds the methyl groups occupy equatorial sites. There is also evidence based on nmr data (17) which implies that these compounds are undergoing rapid internal rotations between equivalent conformations. One suggested path for the internal rotation is through the C4, symmetry square pyramid structure (18). The results imply a very low barrier for this path. Additional information on the structures may he derived from an analysis of the eigenvectors. Bond 416
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Journal o f Chemical Education
P
PF,
5F
Figure 4. Correlation diagram for the PFs molecule in a trigonol bipyromid geometry. The energy of the d orbitoh on phosphorous are included along with the other atomic and molecular energy levels.
orders calculated utilizing eqn. (3) are compared to experimental hond lengths in Table 2. If we assume the usual relationship between hond order and bond length, i.e., long bonds have lower bond orders than short bonds, we see that the relative bond lengths are correctly predicted for each compound. The axial bonds have the lowest hond orders and the longest bond lengths. I n addition, the axial-equatorial hond order difference increases in the series PFs, PFa (CHa), P F r (CH&, as does the axial-equatorial hond length difference. The F-P-F-axis was constrained to he linear in the calculations as a simplifying assumption. If the axis had been allowed to bend slightly, as in the case of the actual molecules, agreement between calculated and experimental distance splits would have been improved. Calculations on other pentacoordinate phosphorous compounds with mixed ligands indicated that ligands with the largest VSIP preferentially occupy axial positions. I n PF4H and PF,H2 the hydrogens occupy equatorial positions, and in PF4Cl and PFCL the chlorine atoms occupy equatorial sites. The principal features of stereochemicistry and honding in pentacoordinate phosphorous compounds can be rationalized by a HMO scheme which utilizes only s and Table 2. Experimental Bond Lengths. and Calculated Bond Orders for PF3(CHs)a PFI(CHJ, and PFs. The Most Stable Geometry for All Three Molecules is a Trigonol Bipyramid -PFI(CHI)Bond Bond Lenath Order
----PFr-----Bond Len.th
-PP.(CHa)Bond Bond Leneth Order
Bond Order
p orbitals. It is probable that inclusion of cl orbitals in an expanded basis set calculation will provide significant improvements in the total energy of the system but not change the predictions relative to stereochemistry. For calculations of the type outlined in this paper, nonetheless, d orbitals are too high in energy in comparison to the other orbitals, and the overlap integrals too small, to contribute appreciably to stereochemistry.
Eigenvectorr for Orbital
101
+ = 80'
pairs of electrons in. the coordination sphere of the central atom. I n the case of five coordination the lone pairs should occupy equatorial sites in a trigonal bipyramid. According to this model, AXzEBand AX,E2 molecules should be linear and "T-shaped," respectively. Even though the H M O scheme deals with delocalized orbitals, the same geometries are predicted from this model. I n the VSEPR model the deformation from a perfect "T" for the AX3E2 molecules is attributed to repulsions from lone pairs of electrons. The H M O approach attributes the stabilization to a decrease in the antiboding character of the 3al orbital, i.e., an orbital with a large contribution from the lone pairs on the C1 atom. Case 11.
1 bs
201
3a1
I bt
0.00 0.60 0.00 0.00 0.56 0.00 0.56
-0.21 0.00 0.52 0.00 -0.27 0.73 -0.27
-0.44 0.00 0,52 0.00 0.50 -0.13 0.50
0.00 0.00 0,00 1.0 0.00 0.00 0.00
Hexocoordinofion
The two types of hexacoordinate compounds whichwe shall be concerned with may be designated AX4E2 and AX,E. h Let us consider XeF4 as a prototype F* for AXIEZ compounds. Two geome.no h Colculded Experimental tries, square planar and tetrahedral, bond order bond length are prevalent for molecules with four 0 CI-F, 0.202 1.698 lieadds in the coordination s ~ h e r e . In CI-FS 0.218 1.598 tce case of XeF4 the square planar Figure 5. Chonges in the energy eigenvdues for CIFa as a functionof bond angle deformation. S ~ N C ~ UisT more ~ stable by several ekcThe eigewecton and calculated bond orders are given for a ten degree deformation fmm the tron volts per molecule. Shown in Figidealized geometry. ure 6(a) are the symmetry orbitals and a correlation diaeram for the DAB ... (square planar) geometry. A bond Xenon Fluorides and lnterhologen Compounds length of 1.95 and VSIP of 27.0 eV for xenon 58, of 12.0 eV for xenon 5p, and of 17.4 eV for fluorine 2 p The H M O approach provides numerous insights into were adopted in the calculations. Similar correlation the honding and stereochemistry of the xenon fluorides and interhalogen compounds. To illustrate the simidiagrams result for other A X E 2molecules such as CIFIand IF4-. larity between the H M O results and the VSEPR preTwo different deformations from the square planar dictions let us divide the molecules into three groups. structure were investigated. They may be designated If we define coordination numher as the number of as e, and e , and are shown in Figure 6(h). These dehonding pairs and lone pairs of electrons in the valence formations are closely related to two normal modes of shell of the central atom, we may classify the compounds vibration for the square planar geometry. It was found as penta-, hexa-, and hepta-coordinate. that the e, structure was less stable than the square Core I. Penfocoordinafion planar whereas the e , structure was more stable due to Two different types of pentacoordinate molecules are found, AX2E3 and AX3E2. The AXzE3 compounds SYMMETRY O R B I T A L S (XeFz, 13-,etc.) were discussed in the earlier section on Xe LIGANDS r AB, molecules. A linear geometry was predicted. Results of a calculation on an AX3E2molecule ClF3 are shown in Figure 5. The most stable geometry was "T-shaped" with a bent axis (top of the "T") and long axial bonds. The angular dependence of the eigenvalues as the axis is bent is shown in Figure 5 along with the expansion coefficientsfor the occupied orbitals. It is obvious from the figure that the primary contributor to the stabilization of the bent axis is the 3a1 orbital. The bending of the axis significantly reduces the antihonding character of the orbital and stabilizes the deformation. It should be stated, however, that the present H M O model exaggerates deformations from ideale. e, ized geometries. The bond orders listed in Figure 5 are more less for a deformation of 10' from the reference geometry stable stable even though the minimum enerw I *I -" was obtained for a 1 "I deformation three times as large. Figure 6. Geometries for xenon tetrafluoride. 1.1 Summetry orbitals and assume that mo'ecular correlation diogrom for o square plonar array. (bl Deformed square geometry is determined by the lone pairs and bonding p h o r stwclures of ungerode and gerade symmetry. -zo
Clirl CNP,) CI~P,~ CI(P,I
0.80 0.00 0.15 0.00 0.32 0.37 0.32
-
.
Volume 46, Number 7, July 1969
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417
trigonal bipyramid configuration of Dab symmetry is roughly 2 eV less stable than the C4, model and is orbitally degenerate. The eigenvectors and eigenvalues for a HMO calculation on XeF&+are.given in Figure 7. The square pyramid structure is pictured in the VSEPR model as an octahedron in which one position is occupied by a lone pair of electrons. The equatorial ligands are bent away from this lone air and should h&e longer than average bond lengths. The HMO scheme predicts identical structural features without using the localized electron pair concept. As may be seen from Figure 7, the Pauli principle requires that the levels through 3a1 should be doubly occupied. The deformation of the four fluorines toward the axial fluorine is stabilizing because it reduces the antibonding interactions in the 3a1 level.
E(ev) -20
-30
so
85
80
W F , Xeq) Eigenvecton for XeFr+, 0 = 85'
r
1.q
Xelrl X~IPZI Xelp,l Xelp,l
0.79 0.00 0.00 O.10
0.00 0.00 0.56 0,00
0.00 0.56 0.00 O.OO
Ft
0.26 0.26 0.26 0.26 0.30
0.59 O.OO -0.59 0.00 0.00
0.00 0.59 0.00 -0.59 0.00
FZ
F"
F4 -5
lo
1e
291
br
301 0.00 0.00 0.00 0,00
-0.47 0.00 0.00 0.58
-0.17 0.50 ~ O . 1 7 -0.50 -0.17 0.50 -0.17 -0.50 0.80 0.00
0.32 0.32 0.32 0.32 -0.13
-0.13 0.00 0.00 0.47
Figure 7.
Angular dependence of the eigenrtates for an ABsE molocule os Eigenvecton for XeFaC with on axial-equatorial bond angle of 8 9 . a function of the axial-equatorid bond mgle.
large changes in the 2al, antibonding level. For the e , configuration, bond orders for Xe--& and X e F , bonds were lower than those for Xe-Fe and Xe--Fa bonds. I n localized electron pair terminology the geometry and the bond orders imply a structure with four bonding electron pairs (to atoms F1, F2, Fa, and F,) and one lone pair (localized between F1 and F4). The stabilization for the e , structure was small and may not appear in a more refined calculation. Even if a more accurate calculation does predict a square planar geometry, however, it is quite likely that the calculation will also ~ r e d i cat verv low force constant for the motion designated e,. The HMO model predicts a square pyramid (C4. symmetry) structure in which the four base fluorines are distorted toward the axial fluorine for AX
