3012
RICARDO FERREIRA
Bonding Properties of Diatomic Molecular Orbitals1 by Ricardo Ferreira Chemistry Department, Earlham College, Richmond, Indianu 47374
(Received January 6 , 1971)
Publieatwn costs assisted by Earlham College
In the LCAO-MO approximations, atomic orbital energy matching is a necessary criterion of bond strength for half-filled molecular orbitals. Filled molecular orbitals built from atomic orbitals of differing energies may be strongly bonding due to the interatomic Coulomb energy. Implications of these principles are discussed in relation to the bonding properties of some molecules and free radicals.
Introduction The differences in bonding properties between oneelectron and electron-pair bonds have been discussed from the very beginnings of the valence bond (VB) theory by Heitler and London, Pauling, and others.2 In VB theory, a necessary condition for the formation of a strong one-electron bond is a large value for the resonance integral between structures XA(~)XB(O) and XA(O)XB(~). This condition obtains only if the two structures have comparable energies, that is, if the energies of the valence orbitals XA and XB are very close. No such restrictive condition exists for the formation of strong electron-pair bonds, which are consequently of much more widespread occurrence. We have recentlya returned to this problem from the molecular orbital viewpoint, stressing the differences in bonding properties between half-filled and filled MO’s in the case of large differences in the diagonal elements. The purpose of the present paper is to discuss this problem in more detail and to analyze its implications to the rationalization of chemical phenomena. I n a two-center MO described by P = CAXA CBXB, if the diagonal matrix elements HAAand HBBare such that (HAA- H B B>> ~ ~HAB - H A A ~ Aand B ([HaA( > ( H B B ( , the solutions of the secular equation can be approximated to
+
elements. We shall now consider the implications of a large value for ( H A A - HBB) in two distinct situations. General Formulation. Case 1. Half-Occupied Bonding MO’s. Included in this case are mainly molecule ions (Hz+, LiH+, HCl+, CO+, etc.) but also radicals (CN, BO, etc.) and metals. Mulliked has shown that for a half-filled NIO we can write
HAA= HBB
(5)
EA +.f”BXA*XAdT
= EB
+f”AXB*XBdT
(6)
where EA and eB are the atomic orbital energies, and a B and a~ are the potential energy operators for one electron in the field of the cores B+ and A*. The dissociation energy for the process AB+(g) --t A(g) B+(g) is
+
De
= -€b
+
EA
-
(7)
Vaa
where V,, is the nuclear (or core) repulsion energy. From (3) and (5)
De=
-EA
- faBXA*XAdT
+
EA
- VCO
(8)
I n the perfect screening approximation6 .faBxA*xAdT = -e2/rAB. The core repulsion term may be represented by Ti,
e2
= TAB
+R
(9)
where R is a noncoulombic term that vanishes rapidly ea* =
HBB-
(HAB- HBBSAB)’ HAA- HBB
(2)
Eb and E&* are the bonding and the antibonding orbital energies and do not include the internuclear repul~ion.~ Further approximation gives Eb E&*
=
HAA
(3)
=
HBB
(4)
which corresponds to diagonalization of the determinantal equation. The orbital energies of the bonding and antibonding states become equal to, respectively, the lower (HAA) and the higher (HBB) diagonal The Journal of Physical Chemistry, Vol. 76,No. 19, 1971
(1) Work supported by the Research Corp. (2) For a lucid presentation see L. Pauling, “The Nature of the Chemical Bond,” 3rd ed, Cornel1 University Press, Ithaca, N. Y . , 1960, pp 21-23. (3) R. Ferreira, Chem. Phys. Lett., 2, 233 (1968). (4) As pointed out by G. Doggett [Mol. Phys., 10, 225 (1965)], F. E. Harris [J. Chem. Phys., 51, 2779 (1968)], and ourselves [R. Ferreira and J . K. Bates, Theor. Chim. Acta, 16, 111 (1970)1, in Htickel-type calculations in which the matrix elements are dependent on the net charges, the H matrix elements of eq l and 2 are not identical with the F matrix elements of the SCF eigenvalue equation ( F - eifJ)ci = 0. The corresponding operators are related by the expression H = F - ‘/nG, where G is the electronic interaction operator. (5) R. S. Mulliken, eq 97 and 98 of J. Chim. Phys. Physicochem. Bbl., 49, 497 (1949). (6) J. A. Pople, Trans. Faraday Soc., 49, 1375 (1953).
BONDING PROPERTIES OF DIATOMIC MOLECULAR ORBITALS
3013
Table I : Bonding in Half-Filled and Filled MO's fA/eB
Molecule
>1
,,--
Molecule
De, eV
ion
De, eV
LiH
2.5"
LiH+(B+)
> l H A B - H A A S A B l correspond to De = 0. I n other words, for half-filled MO's, strong bonding can only occur if HAA Z HBB, or, since J'BXA*XAdT = J'AXB*XBdT, if EA EB. This is the atomic orbital energy matching criterion for strong bonds.8a It should be emphasized that this is only a necessary condition: one-electron bonds between atomic orbitals of the same energy may be quite weak (see the discussion on the halogen molecules). Case 2 . Doublg Occupied Bonding MO's. This case includes mainly neutral molecules and some states of molecule ions (HCl+, Clz+, %; etc.). Equations 3 and 4 are again valid if ~ H AA HBB~ >> ~ H AB HAASABI, but now the same approximations that led to expressions 5 and 6 givesb HAA
=
EA
+ cA2/2(IA - AA) -
I A , IBand AA, A B are the ionization energies and the electron affinities of orbitals XA and xB, and EA = -IA, EB = -IB. I n this case the dissociation energy B(g) is for the process AB(g) + A(g)
+
+
De = -2Eb EA EB - V c c (12) Since ~ H AA H B B>> ~ ~ H AB HAASABI, the coefficient CA is close to unity and from (3) and (10) we may write
which is the correct expression for the bond energy of an ionic bond. It is seen that in the case of filled n/IO's, even when ~HAA - H B Bis~large, the bond A-B may be strong on account of the interatomic coulomb energy. Atomic orbital energy matching in these cases is not essential for strong bonding.
Discussion We have collected in Table I some data that show the difference between singly and doubly occupied bonding MO's and its relation to the value of EA - EB. It is seen that strong one-electron bonds occur only in systems for Which EA - EB S 0. The halogen molecules represent an interesting case : Cornford, Frost, McDowell, Ragle, and Stenhouseghave recently reexamined the photoelectron spectra of the halogens and confirmed earlier resultslO for C&+,Brz+, and Iz+showing that the %,+ states are unstable and higher than both the and states. They have also shown that the Q,+state of Fzf cannot be observed up to 21.2 eV (their cutoff energy) and that the previously reportedlo state of F2+at 17.35 eV should be attributed to nitrogen impurities. It seems therefore that the orbital sequence of the outer electrons of all halogen molecules is (nsu,) z(nsuU)z(npug)z(nprU)4( n p ~ , ) ~]2,+. , It should be pointed out that Wahl's SCF calculations of the F2systemll put the 3ug orbital of F2(lZg+) between the lII, and the 'IT, orbitals, and application of Koopmans' theorem, with its assumption (7) There may be some question as to whether we should call these orbitals "bonding." However, we will continue to do so throughout this paper, since the orbital ea* is definitely antibonding. There is no consistent definition of what constitutes a bonding orbital. For example, the doubly occupied 2 u g orbital in Liz is formally bonding, but removal of one electron strengthens the bond. (8) (a) C. A. Coulson, "Valence," Oxford University Press, 1952, pp 71-73; (b) R. S. Mulliken, ref 5, eq 107 and 108. (9) A. B. Coraford, D. C. Froat, C. A. McDowell, J. L. Ragle, and I. A. Stenhouse, J . Chem. Phys., 54, 265 (1971). (10) D. C. Frost, C. A. McDowell, and D. A. Vroom, ibid., 46, 4255 (1967). (11) A. C. WaN, {bid., 41, 2600 (1964); unpublished results, 1970. The Journal of Physical Chemistry, Vol. 76, N o . 10, 1071
RICARDO FERREIRA
3014 that the orbitals of the ion are the same as those of the neutral molecule, would lead to an inversion of the ionization energy assignments. Recently Bertoncini, Das, and Wahl12 made a SCF calculation of the NaLi molecule, arriving at the values D,(NaLi, Q,+) = 0.852 eV and D,(NaLi+, zZg+) = 0,919 eV. This result is compatible withour arguments, since ELi S EN^. Table I1 compares the dissociation energies of the 2Z+ states of the hydrogen halide ions, HX+, with those of the neutral molecules, HX, lZ+. The values of D,(HX+, ?Z+) refer to the dissociation to H+('S) X(2P).13 It is seen that for the 22+ state of the H X + ions the bond energies are in the reverse order of that of the molecules HX, '2+. The bond energy trend for the molecule ions correlates well with the differences in the diagonal elements. For the ionic species these are identified with the VSIE's.14 For the neutral molecules IHAA- H B B l corresponds to the differences in the valence-state electronegativities. l 4 These results show how remarkably large the interatomic coulomb energy is in the H F molecule.
+
Table I1 : Dissociation Energy of the Hydrogen Halides
IHAA Moleode
De, eV
HF
5.87"
HC1 HBr HI
4.43" 3.75" 3.08"
-
HBB~, eV
IHAA HBBI,
Molecule ion
De, eV
5.06
HF+(%+)
7.38
2.19 1.38 0.69
HClf(2Zf) HBr+(ZZ+) HI+(Q+)
0.87a (0.72)" 1.80" 2.06"
2.81a
0.93
eV
1.49
0.12
5 H . J. Lempka, T. R. Passmore, and W. C. Price, Proc. Roy. SOC.,Ser. A , 304, 53 (1968).
It should be pointed out that, except for HF+, the lowest possible dissociation limit for the z2+states of the H X + species is H(2S) X+(3P). However, the only X+(3P)is the zII bonding state arising from H(2S) state. The bonding state 2 2 + arises either from H+(lSS) X(zP)or from H(2S) X+(lD), the former combination being the more stable one for H F + and HCl+, the latter more stable for HBr+ and HI+. In order to make a meaningful comparison, the dissociation energies shown in Table I1 refer, as stated before, to H+('S) X(2P). The case of the CH radical and the CH+ molecule ion, discussed by Mulliken, lrj is of considerable interest. If CH(a2a) is excited to CH(a.n2), both re and we remain essentially the same. Again, if CH(a2a, is ionized to CH+(az,l2+),Are 0 and Awe S 0. On the other hand, if CH+(az, lZ+) is excited to CH+(aa, %), re increases and we decreases sharply. Also, the dissociation energy of the process CH+(a2, lZ+) + H(?3) C+(zP) is 3.6 eV, whereas for the process CH+(an, TI) 4 H(%) C+(2P)it is only 0.7 eV.
+
+
+
+
+
+
+
The Journal of Physical Chemistry, Vol. 76, hTo. 19, 1971
These facts can be rationalized from the assumption that the 3a and the I n MO's are bonding when doubly occupied but nonbonding when singly occupied. This is expected from the large difference in the diagonal elements of hydrogen and carbon, indicating a large coulomb stabilization. This seems to remove the contradiction discussed by Mulliken. E The importance of these considerations for the rationalization of chemical behavior can be seen in some further examples. Although the dissociation energy of CO is higher16 (11.11eV) than that of Nz (9.76 eV), the reverse is true for the dissociation energieslB*l7of CO+(%'+) (8.36eV) andNz+(zZ+)(8.69eV). The dimerization of C N ( g ) ( 9 ) to cyanogen, and the polymerization of BO(g)(2Z) to solid (BO), can be rationalized in terms of the weakly bonding halfoccupied 3ab 340 in these radicals. The dissociation energy1*of CN(g) is the same as thatlgof BO(g), 7.5 eV. Since the 3Ub hiI0 should be less bonding in BO(g) than in CN(g), we predict that D(BO+) is greater than D(CN+). From the ionization energy'l of CN(g) to CN+(lZ), 14.5 eV, we obtain D(CN+) = 4.25 eV. We predict that D(BO+) > 4.25 eV and that I(B0) < 12.54 eV. The only values found in the literature20 are those estimated by W. A. Chupka (3.9 eV for D(BO+) and 12.8 eV for I(B0)) and it is possible that they should be revised. I n our previous note3we pointed out that the stability of the HF2- ion is due to the large interatomic coulomb term of the doubly occupied bonding ag = c1 1s cz (2pz, - 2pz,) orbital. We also indicated that HeFz cannot be stable since X H >~> XF and therefore no significant interatomic coulomb term occurs in the latter molecule. We can also predict that the 2Zg state of HF2 is unstable. However, from the data of Lempka, Passmore, and Price13 on the H X + ions, the state of HFz could be weakly bonding. These qualitative predictions have now been confirmed by the calculations of Noble and Kortzeborn.21 Acknowledgments. We thank Drs. C. A. McDowell and A. C. Wahl for sending their recent results prior to publication and for the benefit of correspondence. We are grateful to the referees for valuable comments.
+
(12) P. J. Bertoncini, G. Das, and A. C. Wahl, J . Chem. Phys., 52, 5112 (1970). (13) H. J. Lempka, T. R. Passmore, and W. C. Price, ref a in Table
11. (14) J. Hinze and H. H. Jaffe, J . Amer. Chem. Soc., 84, 540 (1962); J. Chem. Phys., 38, 1834 (1963); J. Phys. Chem., 67, 1501 (1963). (15) R. S. Mulliken in "Quantum Theory of Atoms, Molecules, and the Solid State," P . 4 . Lowdin, Ed., Academic Press, New York, N. Y., 1966, p 231. (16) G. Herzberg, ref a of Table I. (17) J. L. Franklin, et al., ref b of Table I. (18) J. Berkowitz, J. Chem. Phys., 36, 2533 (1962). (19) A. A. Mal'tsev, D. I. Kataev, and V. M . Tatevski, Fiz. Probl. Spektrosk., A k a d . N a u k SSR, 1, 194 (1960). (20) J. Berkowitz, J . Chem. Phys., 30, 858 (1959). (21) P. N. Noble and R. N. Kortzeborn, ibid., 52, 5376 (1970).
