William C. Herndon and Ernest0 Silber Texas Tech University
Lubbock,79409
Simplified Molecular Orbitals for Organic Molecules
The Hiickel molecular orbital method (I), as applied to a-electron organic systems, provides a suitable elementary theoretical basis for the correlation of many chemical and physical properties with calculated numerical quantities. Correlations of this kind are detailed in Streitwieser's well-known text on the Hiickel MO method ($), and in other books on MO theory (3-8). Hiickel MO theory is quite straightforward and easy to apply, but the theory is not usually presented to students of organic chemistry a t the undergraduate level. Even at the graduate level, "resonance theory" (9) seems to occupy a more predominant position in discussions of organic structures and reactivities. A possible reason for this situation may be that molecular orbitals are usually presented as a table of eigenvectors (coefficients) and eigenvalues (energy levels), ordinarily obtained as the output of a ready-made computer program.' The interposition of a computer during applications of MO theory is unfortunate in several ways. The price of computer time, the necessity for preparation of input data cards, and the increasing turn-around times a t computer facilities may inhibit usage of .the computer. More important, a student may not be able to obtain an appreciation of the significance of the calculations if his personal participation is limited to computer-related activities. Consequently, a student may be required to perform several manual MO calculations during his apprenticeship in quantum mechanical organic chemistry. Heilbronner and Straub (10) pointed out that these manual calculations are tedious, and that the usual procedure for obtaining molecular wave functions and energy levels can only be applied to a few symmetrical molecules if the calculations are to be carried out by hand. They suggested a different procedure, based upon a set of rules for depicting the nodal character of molecular wave functions, which allows one to write a wave function and its associated energy value simply by inspection of the topology of the molecular system. The MO's were termed "qualitative MO's" and they were considered to be approximations to the exact Hiickel MO's. Heilbronner and Straub then demonstrated a method for correcting the node-based wave functions by second-order perturbation theory, to obtain closer approximations to the exact Hiickel molecular functions. In this paper, we hope to demonstrate the usefulness of the initial node-based wave functions, here renamed 1 Several types of computer programs are available from the Quantum Chemistry Program Exchange, Indiana University, Chemistry Department, Bloomington, Ind. 47401.
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Journal of Chemical Education
simplified molecular orbitals (SMO's). Some new rules for writing these wave functions will be introduced, a simpler method for improving the wave functions will be explained, and many examples will be given. Since these SMO's can be written down at sight, their utility might be questioned in comparison to wave functions obtained by more traditional methods. We will show that the SMO's can be as useful as the Hiiclcel MO's by delineating the correspondence of SMO calculations and deductions with the analogous Hiickel quantities. In many cases, the results of an SMO calculation are more qualitative in nature than the analogous results from Hiickel calculations. We do not believe that this constitutes a handicap, especially from the pedagogic standpoint. Finally, these ShlO's also form a good basis for discussing the orbital-symmetry conservation rules of concerted reactions which have been laid down by Woodward and Hoffmann (11). They can also provide the starting point for reactivity discussions based on first and second-order perturbation theory (la). Mathematical Formulism
In this section we will compare the Huckel MO calculation procedure with the SMO procedure, and we will illustrate the procedural differerices by examining the a MO's of 1,3-butadiene. Huckel MO Theory
The a molecular orbitals are written as linear combinations of the p-atomic orbitals (LCAO) on each carbon atom, eqn. (1).
The energy of a particular molecular orbital is obtained by solving the integrated form of the Schrodinger wave equation, eqn. (2). E = f SH$dr (2) Substitution of a molecular orbital, eqn. (I), into eqn. (2) gives rise to integrals involving atomic orbitals of the types f & H + , d ~ = a and f &H&ds = 6, coulomb and resonance integrals, respectively. The coulomb integral approximates the energy of a 2p electron near the nucleus, and all coulomb integrals for a a carbon system are assumed to have a common negative value. The resonance integral, 6, represents
the interaction energy that an electron creates by bonding two orbitals together. The optimal value of the coefficients in eqn. (1) are those values which give the most stable molecular orbitals. This requirement allows one to write a set of secular equations involving the unknown coefficients, the molecular energy values, and the atomic orbital integrals, eqns. (3)-(ti). cl(a
c10
- E ) + cr0
+ cda - E ) + ca0 ezB + cda - E ) + CIP Cr0
=
0 (3)
=
0 (4)
=
0 (5) 0 (6)
+ c4(a- E ) =
If the set of secular equations is to have nontrivial solutions (the trivial solution has all c( = 0) the secular determinant, eqn. (7) must be satisfied.
A solution of this secular determinant gives four values of the energy E in terms of the integrals a and 6. Substitution of any one of the energy values back into the secular equations (3)-(ti), gives a corresponding set of coefficients and therefore determines one of the wave functions for the system. The four resulting wave functions and energy values are a set of functions and energies which are approximations to the actual wave functions and energy levels of the a 140's of butadiene. The form of the secular determinant is often simplified by introducing the parameter X = (or - E)/P. The secular determinant is then written for any a system from a visual inspection of the bond connectivity of the system. The determinant simply has X's along the diagonal and 1's inserted where a bond exists. The a electrons occupy the lowest molecular orbitals doubly, and the total a electronic energy is found by summing over the occupied orbitals as shown in eqn. (8), where k is an occupancy number, either one or two.
The resonance energy of butadiene is calculated by comparing the total a energy with that calculated assuming no interaction between orbital 2 and 3 of the a structure, i.e., the a energy of two isolated double bonds. Strictly speaking, the set of wave functions obtained by this procedure should constitute an orthonormal set. This imposes the condition that the wave functions be normalized, that eqn. (9) should hold.
The values of the normalized coefficients lead to several valuable concepts, including the electronic charge density and the bond order defined by eqns. (10) and (ll), respectively. OEC
9. =
C m
k-+%P2
(10)
The absolute values of the a and @ parameters can be defined by theoretical calculations, but they are usually established empirically with regard to agreement of calculated molecular orbital quantities with some experimental measurement. The Hiicltel MO theory is therefore highly empirical. Several approximations, perhaps unjustifiable, are made in the application of Huckel 110 theory to a molecular system. Nevertheless, it works, and fruitful speculations and experiments have been engendered by its application. Simplified MO Theory
SMO wave functions are defined solely by their nodal characteristics, and in this way they resemble the wave functions of free-electron theory (13). The most stable molecular energy level for a a networlc of electrons is characterized by a wave function which has positive amplitude a t each orbital in the network. In other words, there are no nodes in the wave function for the most stable 110 of the hutadiene r system. In analogy to the usual characteristics of macroscopic standing wave systems (vibrating violin strings, etc.) higher energy levels have increasing numbers of nodes in the amplitude function. A wave function for each molecular energy level can now be written as an LCAO with coefficients either 1, -1, or zero depending upon the position of the nodes in the level under consideration. The two bonding levels, I and 11, are depicted below
and the normalized wave functions are given by eqns. (12) and (13).
The reader may easily verify that the energy values associated with and J.z are as shown by substitution of J.1 and J.2 into the Schrodinger equation, eqn. (2). The analogous Huckel MO's are given by eqns. (14) and (15).
Both sets of MO's give uniform charge densities, q, eqn. (lo), equal to unity, and the bond orders are PI,P(SILIO) = 1.00, pz.s(SM0) = 0.00, pl,z(HMO) = 0.89, and p2,s(HMO) = 0.447. The HMO theory gives a resonance energy of 0.4728 for butadiene. The SMO orbitals yield no resonance energy for butadiene. This agrees with Dewar's contention that open-chain polyolefins are not stabilized by resonance (Id), but the agreement is fortuitous since higher polyenes are calculated to have small delocalbation energies. It is not necessary to actually carry out the process of substituting an SMO into eqn. (2) to determine the associated energy level value. The energy is given by eqn. (16) Volume 48, Number 8 , August 1971
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503
and is determined by an elementary counting process. N is the number of atomic orbital centers which have a non-zero coefficient in the wave function (nodes do not cross the orbital center); B is the number of bonding interactions (a node does not pass between two orbital centers); and A is the number of antibonding interactions (a node passes between two orbital centers). Rules for Depicting the Nodal Character of Molecular Wave Functions
Molecular orbitals which are determined from considerations of topology have several useful characteristics which can be used to depict the nodal character of the SMO wave functions. More complete lists may be compiled by consulting the literature (#-8,10, 13, 15), but the rules listed below are sufficient for our purpose. 1) F orbital networks may be divided into two classes, alternant and nonalternant systems (16). The orbitds in alternant S ~ S I P I I I Yrat, be divided intu two pnups by stilrnng ~ l ~ e r w t ornt bital+, i t , surh n wn) thnr r w o t n r r d or1,iraL are nor linkrd by a Iru~d. Exnntrrles 111 and I\' are ~lterlr>urt and nonnltt~r~~ant, re spectively.
Figwe 1. c, fulvene;
Simplifled molecular orbital%
o, Odatetrone; b, naphthalene;
d, indenyl radical.
Some Examples
2) Alternant odd or even systems have equal numbers of bonding and antibonding molecular orbitals. Odd alternants brave a nonbonding level. Some even alternsnts may have two nonbonding levels. 3) I n alternrtnt systems, each bonding orbital has a. conjugate antibonding orbital, which has the same absolute values of the enerav and coefficients, but in which the signs of the unstaned orbicil coefficients have been changed. 4) For any F m~lecularsystem the orbital of lowest energy has no nodes. Orbitals of increasine enerev have increasine
of symmetry. 5) The number of atomic orbitals within domains defined by nodes is roughly equivalent. 6) The highest filled MO (HFMO) often corresponds to the valence-bond structure if a single structure can be written for a. molecule. That is, there are nodes across the bonds which are written as single bonds in the valence bond structure. The lowest vacant MO (LVMO) has nodes which cross formal double bonds of the valence-bond structure. 7) If parallel movement of a node one-half of a bond length gives a lower energy, then the lower energy is a better approximation to the eigenvdue than the initial energy. This rule has priority and may conflict with rules 5 and 6.
One consequence of the above rules is that one does not need to separately carry out calculations for all orbitals of alternant systems. The energies and wave functions of antibonding levels are given by their conjugate relationships to the bonding orbitals. The treatment of nonalternant systems cannot be simplified in this manner. We also note that one can separately identify MO's of interest, like the HFMO's of a series of molecules, and one can calculate separately those particular energy values. It is not necessary to solve a secular determinant for all energy levels. 504 / Journal of Chemical Education
Figure 1depicts the SMO's for (a) a linear alternant polyene, octatetraene, (b) a bicyclic alternant aromatic, naphthalene, (c) a nonalternant cyclic polyolefin, fulvene, and (d) a nonalternant radical, indenyl. Hiickel MO calculated quantities are given in parentheses. Octatetraene is a one-dimensional a-system in essence, and nodes are introduced perpendicular to the T-molecular axis. The bond orders alternate large and small, and the highest filled level is less stable than the highest filled of butadiene as expected. A small resonance energy of 0.6670 is calculated for this open chain polyene. Naphthalene has two orthogonal planes of symmetry, excluding the molecular plane, and a single node can therefore have two orientations. The wave function with a node perpendicular to the longer axis of the molecule has the lower energy, and this is generally true for two dimensional molecules. Orbitals 4 and 5 each have two nodes. The calculations show that the wave function with crossed nodes is of higher energy. Naphthalene has a ?r delocalization energy of 2.4p in this SNIO approximation. The antibonding orbitals for octatetraene and naphthalene are not shown in Figure 1 since they are determined according to rule 3 of the previous section. However, fulvene and indenyl are not alternant hydrocarbons, and the pattern of energy levels is not expected to be symmetrical for these molecules. Fulvene does show one interesting feature. In this approximation the highest vacant level has two perpendicular nodes. The next to highest vacant level has three parallel nodes. The Hiickel wave functions reverse this pattern. If the SMO nodes were not constrained by the choice of coefficients (+I, 0, or - 1) to lie across the midpoints of
bonds and orbitals, a closer correspondence between HMO's and SMO's would emerge. The highest filled MO, $8, of fulvene has exactly the same nodal character, energy, and coefficients as does the HFMO of butadiene. We therefore expect a close correspondence between electron-donor properties of fulvene and butadiene. However fulvene bas a lowlying vacant orbital, $4, and it should be a better electrophile than butadiene. The five occupied orbitals of indenyl radical are listed in Figure Id. There are two reasonable choices for the positions of the nodes in $5 of indenyl, shown in structures V and VI.
Structure V follows from rules 5 and 6, but structure V I is a better approximation to a correct wave function for the molecule since it leads to a more stable energy for the half-occupied orbital. The results of manipulating the nodes suggest that, in this case, an exact function
Table 1. Mnlerule
SMO
for $5 would have electron amplitude distributed over eight orbitals as in structure V, but the node would be close to the central bridging bond as in structure VI. Energy of the Highest Filled Molecular Orbital
The magnitude of the Hiickel eigenvalue of the HFMO of unsaturated hydrocarbons correlates with several measured properties, including ionization potentials, the frequency of a-s*transitions, the stabilities of cbarge-transfer complexes, and polarographic oxidation potentials (9). The correlations are usually quite good for a related series of compounds, but discrepancies are evident when very dissimilar compounds are compared. SMO-HF eigenvalues for similar series also correlate well with the same experimental properties. One could demonstrate this fact with several graphs, but a single plot of SMO values versus Hiickel MO values will also serve the same purpose. This plot is given in Figure 2. The compounds are identified and the numerical values are listed in Table 1. These 30 compounds were arbitrarily chosen as being representative of several different types of a systems. No odd alternant hydrocarbons are included in the table since the eigenvalue of HFMO for such a compound is always
Eigemalues of Highest Filled Molecular Orbitals Hiickel SMO MO lmodifiedl
Molecule
SMO
Hiickel SMO MO (modified)
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505
Table 2.
Molecule
7
Eneraies of C Com~ounds
SMO
Hiickel MO
DE
DE
(SMO)
(HMO)
Figure 2. Highest-fllled molecular orbitalt (HFMOI for compoundr in Table 1 (units of 8).
identically zero. The table also contains modified SMO values, which will be defined in a later section. belocalization Energies from SMO Calculations
The energy value, eqn. (I), in quantum mechanical calculations is an integral, that is, it is a sort of average value. A wave function which has the nodes in the wave function correctly spaced will give a realistic value of the energy for that function, even if the coefficients are arbitrarily chosen within certain limits. The eigenvalue is not very sensitive to the absolute values of the coefficients as long as the correct number of nodes is present. Consequently, one could expect that the total Hiickel a energy of a particular molecular system, being a sum over several energy levels, would he more or less correctly determined by an SMO calculation. We calculated the total SRI0 a energies for all Cg systems which are listed in the Coulson-Streitwieser Huckel MO tables (16). These energies are compiled in Table 2. The delocalization or resonance energies (DE), defined as E, - 80, from the two methods are plotted against each other in Figure 3. One can see that the Huckel delocalization energies can be determined within about 0.1 of a 0 unit by an SMO calculation. The molecules are arranged by the SMO calculation in the same order as obtained from a Huckel MO calculation. If one wishes to base chemical arguments upon calculated a energies, the SR'IO results are as useful as Hiickel results. The folly of comparing calculated a energies of very different structures is illustrated by the fact that pentalene is calculated by Huckel MO theory to have a greater delocalization energy than styrene.
I
0.8 Figwe 3. 8).
I
;i(sMd'G
I
20
Delocaliration energies (DE) for Cs v-orbital systems (unib of
only involves the simplest algebraic and arithmetic operations, so SMO's for quite complicated molecules can be modified in seconds with the use of a slide rule. The technique is best illustrated with examples. Butadiene. SMO's and associated eigenvalues are depicted in VII and VIII.
Rectification of Simplified Molecular Orbitals
SMO's may be modified by an iterative process which uses the SMO eigenvalue to calculate new values of the eigencoefficients,different from *1 or zero. These new coefficients can then be used to redetermine a new eignevalue by substitution in eqn. (16). The iteration 506
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Journal of Chemical Education
The secular equations involving the coefficients, are given by eqns. (3)-(6). We recommend they be rewritten in the form of eqns. (17) and (18).
The general rule for writing a secular equation in this form is that the coefficient of an orbital times the eigenvalue y equals the sum of the coefficientsof bonded orbitals, using eqn. (17) and eigenvalue El = 1.500, 1 . 5 ~= b or 3a = 2b. The coefficients and new eigenvalue are calculated as shown below.
The rectified MO eigenvalue is 1.615 which compares well with the Hiickel 140 value of 1.618. The energy is still calculated with eqn. (16), but now N is the sum of the squares of the coefficients of the wave function, and the bonding interactions B and the antibonding interactions A are modified by the relative values of the new coefficients, as illustrated in eqn. (19). In many cases, it will not he necessary to actually write the secular equations as in eqns. (17) and (18). If a molecular system has a terminal orbital, as orbital a is in VII and VIII, then the relative values of the attached orbital to the terminal orbital is simply the eigenvalue, b/a = y. consequently the rectified coefficients for a simple system like butadiene can be written for any value of the eigenvalue without solving the secular equations. E($x)
E
2 68
= - (30
= 0.500
2 - 9) = 468 - = 0.618
The Huckel eigenvalue for fizis 0.6180. Pentalene (HFMO). The nodal character of the highest filled MO of pentalene is given by IX. The nodes are close to orbitals b since X gives a more stable eigenvalue.
The Huckel MO energy is 0.4710 and the coefficients are correct within 6%. One additional cycle mill give coefficients which are identical to Huckel coefficients (three significant figures). The foregoing method was used to calculate a modified value of the HFMO for the compounds in Table 1. The total time required to calculate all 60 eigenvalues in Table 1 was a little less than two hours. Less time was required for the calculations than the time which would have been required for the preparation of the punched computer cards for machine calculation. The SMO Eigenvedors
The correlations between the eigencoefficients of HFMO and LVMO with hyperfine splitting constants observed in aromatic hydrocarbon radical cation and anions is considered to he one of the remarkable snccesses of Hiickel MO theory. Of course, unrectified SMO's cannot yield a correlation of this type since all coefficients are either zero or 1. Even after a first or second rectification the amplitude coefficients themselves may still be quite unrepresentative of the exact Huckel molecular orbitals. Practically, this requires that chemical arguments based on coefficientsobtained by the foregoing method he considered as tentative.
*
Conclusion
What are these simplified molecular orbitals useful for? Our contention is that SMO's are as bad as Huckel MO's. Like Huckel MO's, they can be used to get rough ideas about energy levels and electron distribution in s molecular systems. Unlike Hiickel MO's, they can he obtained quickly without the use of a computer. The SMO's are useful in the classroom, since wave functions and energies can he written for simple molecules, almost as quickly as one draws the stmcture of the molecule. A caution should he interposed a t this point. The SMO's are useful for discussing those properties which correlate well with Huckel MO calculations. In general, chemical reactivities correlate well with Huckel MO calculated charge densities, bond orders, etc., only if the molecular system is an alternant hydrocarbon. We therefore infer that SMO calculations should not be used in discussing questions related to chemical reactivity except in a qualitative manner. The perturbational MO method is better suited to examining questions pertaining to the relative energies of transition states. Acknowledgment
The necessary arithmetic is summarized below. 0.5a
=
24
0 . 5 ~= 26
a
=
4b
+ e, c = 4b
The financial support of the Robert A. Welch Foundation is gratefully acknowledged. We also wish to thank our colleagues, Joe. A. Adamcik and Lynn S. Marcoux, for their encouraging interest in the development of the concepts described in this paper. Literature Cited (1) H U c s s ~ E , .."GrunduOga dar Thaorie Ungeailttieter und Aromatiaohea Verhindungen," Veda! Chemie, Heiddbsrg, 1938. (2) S~nammrser;n,Jn., A., Molaaular Orbital Theory for Organic Chamista," John Wiley & Sons, Inc., New Yaik, 1961.
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(3) RonEnrs. J. D.. "Notes on Molecular O ~ h i t s lCalculations." W. A. Benjamin. Ino., New York, 1961. E . A N D MOSER,C.. " Q u B ~ Chamistry." ~ u ~ (4) DAUDEL.R.,L E F ~ B V R R.. Interscience Puhliahers, Ino.. New York, 1959. (5) SALEM.L.. "The Molecular Orbital Theory of Conjugated Systems," W.A. Benjamin, Inc.. 1966. (6) PILAB,F. L.. "Element&~yQuantum Chemistry:' McGraw-Hill Book Co.. New York. 1968. (7) F ~ m n rJn., . R. L.. "Moleoular Orbital Theories of Banding in Organic Moleoules," Marcel Ihkker. Ino.. New York. 1968. (8) D E ~ * R : , M J. . 8.. "The Moleoulhr Orbital Theory of Organic Chemistry. MoGraw-Hill Book Co.. New York. 1969. (9) WXE'**D. 0. W.. "Resonance in organic Chemistry:' John Wiley & Sons, Inc.. New York. 1955. n .A,. Tctioher(ron. 23, 845 (1967). (10) HerLnnoaNEn. E.. A N D S ~ ~ n u P. (11) W o o ~ w * n o ,R . B., *NO HOFPYANN,R., .'The conservation of orbital
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Journal o f Chemical Education
.. (12) D E ~ A RM. .
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J. 8.. J . Am-. Cham. 800..74,3341 (1952); Funur, K., i n
"Moleoular Orbitala i n Chemistry, Physios, and Biolopy," (Ediloia: LOWDIN. P. 0.. AND PULLMAN. B.. Academic Press. Ino.. New York. W. c.. A N D HALL, L. II., fieorat. L him: 513-37; HERNDON, 1964, Ada (Bed.), 8 , 165 (1967): SAGEM,L., J . Amer. Chem. Soc., 90,543
\.""-,.
(1ORI1)
(13) PGATT, J. R.. e t al.. "Free Electron Theory of Conjugated Moleoales:' John Wiley & Sons, Inc., New York. 1964. (14) DEWAR.M. J. 8.. "Hyr)erc~nj~ge.tion,"The Ronald Press Co., New York.1962. (15) C o u ~ s o l r C. , A,, m n R o s n n s o o r ~ G. . S.. PTOC. Cambridge Phil. Soc.. 36, 193 (1940). (16) C o u ~ a o C. r A,. AND Smnmwmsen. A,, Jn.. "Dictionary of r-Electron Caleul~tians." W. H. Freeman and Co., San Francisco. 1965.
