William A. BerneW 3M Company
St. Paul, Minnesota 55101
Localized and Delocalized Molecular Orbital Descriptions of Methane
Two of the most valuable concepts to the teaching and understanding of structure and reactivity in organic chemistry have been those of directed valence (localized two electron bonds and lone pairs) and of molecular orbitals (in which the electrons are delocalized over the entire molecule). For the student the relationship between these two concepts, that of localized molecular orbitals (LIMO'S) and of delocalized molecular orbitals (MO's or Dh401s), is rather vague, with perhaps the implication that they represent two different unrelated approaches to molecular understanding. These two concepts do have a very definite relationship, however, and the purpose of this article is to show that this relationship can be easily demonstrated for the case of methane, commencing with simple hybrid atomic orbitals (HAO's) as a basis or starting point. Background
The Hartree-Fock (HF) approximation to the solution of the Schrodinger equation considers each electron in a molecule to move in the field of the nuclei and the time average self-consistent field (SCF) of the other electrons, instead of their instantaneous field. This reduces the n-body problem to n one-body problems, neglecting only the instantaneous electron repulsions or electron correlation. The ground states of most molecules represent closed-shell electronic systems and the H F SCF wave function for a molecule (*) is represented by a single determinant, antisymmetrized (because of the Pauli exclusion principle) product of one-electron, normalized, orthogonal H F MO's (@'s), each orbital being doubly occupied. Thus
These 140's (4's) are found by the variation method
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and are those which minimize the energy. In view, however, of the difficulty of obtaining H F MO's for large molecules they can be approximated by MO's formed from a linear combination of atomic orbitals (LCAO). It was pointed out by Fock that a single determinant, many-electron wave function (*) is invariant to unitary transformation' of the MO's (1). It follows, therefore, that the total energy and the electron density are also invariant to such a transformation. While DMO's may serve as a basis for describing and understanding some molecular properties, such as ionization potentials and electronic and esr spectra, another basis is desirable for other concepts which are more characteristic of electrons belonging to chemical "bonds," such as bond energies, bond polarizabilities, bond refractivities and bond dipoles. I n line with this desire it has been found that the unitary transformation which, starting with DMO's, minimizes the sum of the interorbital electron repulsion terms (Coulomb and exchange interactions), or maximizes the sum of the intraorbital repulsion terms, gives orbitals which show maximum electron localization (@. These orbitals (LMO's) correspond to bonds, lone pairs, and inner shells. While it is, in general, necessary to use iterative procedures to find these localized molecular orbitals (3, 4),the presence of symmetry in a molecule often makes it possible to find the required unitary transformation using group t h e ~ r y . ~ The theory of symmetry equivalent Lh40's was developed in a series of papers by Lennard-Jones, Hall, Pople, and Hurley (g, 6). They found that while LMO's minimize the interorbital nonbonded interactions those remaining were predominantly due to Coulomb forces. At the same time LMO's are exwected to show the smallest interorbital correlation 'Orthogonal transformation when the orbitals are real functions. 2Magnasco and Perio have developed a localisation proced Ire based on the maximization of the sum of local Mulliken ovxlap populations (7).
effects, antisymmetrization keeping electrons with the same spin apart (but allowing those of opposite spin to come closer than their Coulomb repulsion actually permits). I t should be noted that the theory of LMO's also provides a basis for the Valence-Shell Electron-Pair Repulsion (VSEPR) theory (6). Description of LMO's, and Transformation to DMO's, for Methane
The relationship between LlMO's and DMO's can be easily demonstrated for the case of methane starting with familiar, simple hybrid atomic orbitals (HAO's). The bonding about the carbon center can be described in terms of a 1s A 0 inner shell and four equivalent spa HAO's which point to the corners of a regular tetrahedron, the latter each overlapping with one of the hydrogen 1s AO's. The normalized, orthogonal carbon HAO's (7's) are (8) YO =
c,.
y, = 1/2 (C2,
+ Czp, - C B ~-" C
=
1/2 (Cs,
73
=
1/2 (C*#- Cznr
y, = 1/2(C,,-
etc. x2*, x3*, x4* for the antibonding LMO's. The required matrix for the orthogonal transformation (since the A 0 coefficients are real numbers) of these CH4LILlO's to DMO's has been given by LennardJones and Pople (i4).& Using this matrix a D M 0 description of methane can now be obtained (row x column) 1
0
0
'12
0
0
- '1% -I/*
-'/z
I/%
C*,,
d
+ Cgu - CspJ - c,,,+ C,,,)
where the C's represent carbon A 0 wave functions. Localized molecular orbitals (x's) describing the two center C-H bonds can now be formed by taking linear combinations of the hydrogen 1s AO's (H's) and carbon HAO's (7's). The general form of these L M 0 wave functions is xi = ayi
+ bHi
(1)
Normalization of the wave function requires that
where the overlap integral =
SYi.,dT
The (+) sign refers to the bonding LMO's and the (-) sign to the antibonding LMO's. The overlap integral between an spa HA0 and sols A 0 for the methane C-H bond distance of 1.094 A can be found in Mulliken's tables of overlap integrals (9, 10) to be 0.685. Assuming a purely covalent C-H bond3 and, therefore, equal coefficients, a = b, normalization gives a = b = 0.545
a*
xr, xa, x4,for the bonding LMO's, and XI* = 0.630 (C2, + Cn., + CZ, + Cs,,) - 1.260 HI
+ Caps + Cm + C2.J
m
S
etc.
=
b*
=
1.260
bonding
and the antibonding DMO's are dl*
=
1.260 CX,- 0.630 (HI
+ Hz 4- Ha + H4)
+ Hn - HI - Ha)
4%. = 1.260 Cap,- 0.630 (HL
Pall [21d
etc. with their symmetries indicated on the right. These 6's represent a minimal A 0 basis set of orthonormal LCAO MO wave functions wherein each orbital is delocalized over the entire molecule except for the C,, inner shell, while the x's represent wave functions for orbitals which are localized between two nuclei, except again the CI, inner shell. Inner shells represent an important part of the total energy of the molecule, but act primarily to partially screen the nuclei and in both descriptions have little influence on chemical properties. Thus, it is seen that the above two descriptions of the bonding in methane, localized molecular orbitals and delocalized molecular orbitals, are equivalent descriptions related by orthogonal transformation.= Figures 1 and 2 show energy diagrams for the bonding and antihonding orbitals in the two descriptions. In the LMO description it is seen that XI, x2, x3, and xn are degenerate while in the D M 0 description +%,ba, and 64 are degenerate. I t should be pointed out, however, in reference to the Ln40 energy diagram, that while energy differences between DMO's
antibonding
Substitution of the 7's and the values of a and b into eqn. (1) gives the following expression for the methane C-H bond LMO's a Although the C-H bond does have a small dipole moment (11, id), a referee has pointed out that this does not necessrtrily mean that the bond is ionic, but rather, reflects the fact that a hybrid orbital has s significant dipole moment (IS). LMO's me transformed to DMO's usine the tranmose of the mat& used to transform DMO's to ~ ~ 6 ' For s . the case of CH, the two are identical. 6 It should be remembered that a single orbital by itself does not necessarilg show the correct spatial probability distribution of one electron relative to the others. The electron distribution in the molecule is determined hy the total wave function after the spin orbitals are multiplied together and antisymmetrized.
Figure 1. methane.
LMO energy level description of bonding in gmund rtdo
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Figure 2.
MO energy level description of bonding in methane,
can be experimentally determined by spectroscopic means LMO energy levels are artificial in the sense that, e.g., one cannot experimentally measure bond energies although the concept is useful for predicting heats of formation whieh can be experimentally determined. Discussion
Although it has been neglected up until this point the LAlO's obtained in the manner illustrated for methane are slightly nonorthogonal. I n order to correct for this and satisfy the exclusion principle, it is necessary to slightly delocalize the LnIO's by including in them small secondary eontributions from neighboring atoms (16-19). This can be seen in the LMO's obtained from the transformation of the SCF LCAO 310's of I'itzer for CH4 (18) xo 1.012 CI, - 0.041 C*, - 0.025 (HL+ HS + HJ + H,) xt = -0.052 CI, + 0.301 Ct8 + 0.277 (C2.' + Caa, + Csp,) +
-
+ Ha + H,)
0.569 HI - 0.066 (HZ
The small secondary eontributions from neighboring atoms to hond, inner shell, and lone pair LMO's are usually negative in sign. Ruedenberg and Edmiston have suggested that it is the interference or antibonding effect between the primary and secondary eontributions to a LA40 which accounts for nonbonded repulsions in molecules (4). This is illustrated in the ease of ethane where the negative secondary eontributions in the C-H hond LMO's are larger in the eclipsed than 'in the staggered conformation leading to the conclusion that the primary factor in the rotational barrier is the negative overlap interaction between hond orbitals (LMO's). (1 . 9)., I n the ease of conjugated molecules, although the u 310's can be loealized with very little electron density off the main bonds, the r MO's, even when "loealized," show considerable delocalization. I n aromatic molecules there are degenerate sets of LMO's, each set corresponding to one of the possible I'kkul6 structures (20). I n addition to illustrating the fact that a molecule ?an be quantum mechanically described in terms of two equivalent and intratransformahle deseriptions (DMO's and LA4O's) the above example of methane also illustrates a second important principle, namely, that just as atomie orbitals can form a basis for the formation of molecular orbitals, loealized atomic orbitals (HAO's) can form a basis for the formation of completely localized molecular orbitals. The earlier mentioned observation of Fock concerning the invariance of 748
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the total wave function applies to atoms as well as molecules. AO's, which are delocalized about an atomie nucleus, can by appropriate orthogonal transformation, he transformed into AO's whieh show a maximum localization, and turn out to be hybrid atomic orbitals (3, 4). The 4 X 4 matrix of the coefficients of the four sp3 HAO's n,y,, y,, and y4 represents that orthogonal matrix which transforms the 2,, 2,,, 2,, and 2, AO's into four symmetry equivalent localized AO's or HAO's. If the four orbitals in these two descriptions are all singly occupied with parallel spins or doubly occupied, then these two deseriptions are equivalent, i.e., they lead to the same total wave funetion, energy, and electron dens it^.^ Sets of HAO's are themselves often appropriately transformable into other sets of equivalent HAO's (21-23). Transferability of LMO's
Ideally one would like to find loealized molecular orbitals which are to a reasonable degree chemically invariant and show a maximum insensitivity to molecular environment. Such orbitals could be transferred from one molecule to another containing a given group (bond, lone pair, or inner shell) and would dlow one to then write down approximate single determinant wave functions for many moleeules without direct caleulation. Correction functions, including eorrelation, could then be calculated if higher accuracy or more exact wave functions were desired (24). The electron correlation effects in LAfO deseriptions can be broken into two types, interorhital and intraorbital. I n methane the ratio of the magnitude of the inter-intraorbital correlation energy has been found to be about 2: 1 (26). The intraorbital correlation effects (electron pair eorrelations) are expected to be nearly constant and insensitive to environment for a given LMO. The interorbital eorrelation effects (London dispersion forces due to mutually induced instantaneous dipole moments) are quite sensitive to structural geometry. Bond energies, for example, in saturated hydrocarbons are quite constant if interorbital correlation effects are corrected for (%), suggesting that localized molecular orbitals describing the C-H and C-C bonds might be generally transferable in these molecules. Although such transferable LMO's would only be approximately invariant (more so for non-conjugated than conjugated compounds), if their errors are small they could still be useful to calculate approximate values of certain molecular properties. Conclusions
I n accordance with Occam's razor (277, calculations should only be taken to the level of accuracy that enables one to answer the question asked. The LMO's approximated from HAO's in this discussion are, of course, not exact because they are based on a minimum A 0 basis set and neglect off-bond electron density or secondary contributions-still, for many purposes, particularly qualitative discussions, they are adequate. A simple example of this can easily he worked out by calculating the total wave function for the case ot two singly occupied s p HAO's and for a yingly occupied s and p orbital by evaluating the single Slater determinant of the spin orbitals for the case two electrons with parallel spins.
This illustration for the case of CHa demonstrates how the standard two electron bond model for this molecule, based on sp3 hybrids, can be easily related to the concept and theory of delocalized molecular orbitals. I n general it is seen that the concept of hybridization really represents a zeroth-order approximation to localized molecular orbitals which in turn are transformable into delocalized molecular orbitals. Even a t this level of approximation, therefore, it is not surprising that simple HAO's have served successfully as a useful basis for explaining many facets of organic chemistry, with sp3, spZ, and sp HAO's generally being considered transferable from molecule to molecule for simple models of single, double, and triple bonded carbon. Literature Cited
( 1 ) Focrc, V., Z . Physilc, 61, 126 (1930). J., P ~ o c ROY. . Soe. (London), A198, 1, 14 ( 2 ) LENNARDJONES,
,'"T",. ,,"no\
( 3 ) EDMISTON, C., A N D RUEDENBERG, K., Rev. Mod. Phys., 35.4.57 (1963). C., A N D RUEDENBERG, K., J . Chem. Phys., 43, ( 4 ) EDMISTON, S97 ... fIQfi5)~ - - - - ,. ( 5 ) HURLEY,A. C., LENNARDJONES, J., A N D POPLE,J. A,, Proe. Roy. Soc. (London),AZZO, 446 (1953). ( 6 ) GILLESPIE,R. J., J . CHEM.EDUC.,40, 295 (1963). V., AND PERICO,A., J . Chem. Phys., 47, 971 ( 7 ) M.~GNASCO, (1967); 48, 800 (1968). H.,WALTER,J., AND KIMBALL,G. E., "Quantum ( 8 ) EYRING, \
Chemistry," John Wiley & Sons, Inc.. New York. 1944. p. 222.
(9) MULLIKEN, R. S., RIEKE,C. A., ORLOFF,D., AND ORLOFF, H., J . Chem. Phys., 17,1248 (1949). (10) WIBERG,K., "Physical Organic Chemistry," John Wiley & Sons, Ino., New York, 1964, Appendix 4. (11) ROLLEFSON,R., A N D HAVENS,R., Phys. Rev., 57, 710 (194n) - - - - ,. (12) THORNDIKE, A.M., J . Chem. Phys., 15,868 (1947). G. P., GUIDOTTI, C., MAESTRO, M., MOCCIA, R., (13) ARRIGHINI, A N D SALVETTI, O., J. Chem. Phys., 49,2224 (1968). E SA, N D POPLE, J . A., Proc. Roy. Soe. (14) L E N N A R D ~ O NJ., (London), AZOZ, 166 (1950). (15) KALDOR, U., J . Chem. Phys., 46,1981 (1967). (161 COUISON,C. A,, Trans. Faraday Soc., 38,433 (1942). O., J . Chem. Phys., 49, 65 (17) TRINDLE,C., AND SINANO~LU, \
11969) \-"--,.
(18) PITZER,R . M., J . Chem. Phys., 46,4871 (1967). (19) (a) PITZER,R. M., J . Chem. Phys., 41, 2216 (1964); (b) S ~ V E R 0S ,. J., KERN,C. W., PITZER,R. M., AND KARPLUS,M., J. Chem. Phya., 49, 2592 (1968). (20) HALL, G. G., A N D LENNARDJONES, J., PTOC.ROY. SOC. (London),AZ05.357 (1951). (21) MULLIKEN, R. S., Tetrahedron, 6 , 68 (1959). W . A,, J. CHEM.EDUC.,44, 17 (1967). (22) BERNETT, W . A,, J. Org. Chem., 34,1772 (1969). (23) BERNETT, (241 BOYS,S. F.,Rev. Mod. Phys., 32, 296 (1960). 0.. AND SKUTNIK. (25) SINANO~LU, . B... Chem. Phus. Letters. 1. 699 (1968). (26) PITZER,K. S., AND CATALANO, E., J. Am. Chem. Soe., 78, 4844 (19561. , , (27) RUSSELL,B., "History of Western Philosophy," George Allen and Unwin, London, 1946.
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