FRANK 0. ELLISON
2294
Vol. 66
SOME APPLICATIONS OF SEMI-EMPIRICAL VALENCE BOKD THEORY TO SMALL MOLECULES' BY FRANK0. ELLISON Department of Chemistry, Carnegie Institute of Technology, Pittsburgh IS, Pennsylvania Received M a y $8, 1068
A semi-empirical theory of the low-lying electronic states of diatomic molecules is described and applied to H P JLiH BeH, and BH. Ionic and covalent valence bond structures are utilized as a basis; the atoms-in-molecules method, modified to account for intraatomic energy changes on deflation or inflation of atoms and ions in molecules, is employed. Coulomb parts of the interatomic energy are obtained theoretically using integrals over Slater orbitals. A new approach is proposed for estimating exchange integrals in terms of overlap and empirical atomic properties. In the over-all theory, all empirical quantities admitted are strictly atomic; no calibration parameters dependent on molecular properties are present. Encouraging results for dissociation and excitation energies, especially for H2) LiH, and BeH, indicate that this rather simple model is fairly sound. However it is concluded that further applications will require molecular parameters in order t o obtain semiquantitative predictions.
One of the major devices which chemists have long been seeking through quantum mechanics is a scheme for the quantitative theoretical analysis of known molecular energies and for the quantitative theoretical svnthesis of unknown molecular energies. Although the universal reagent (the Schrodinger wave equation) is known, only dilute solutions (molecular orbital and valence bond theories) have been practicable to any extent. The yield of qualitative understanding of molecular stabilities has been impressive; unfortunately, quantitative successes have been few. One main obstacle in the way toward obtaining and interpreting accurate values is the profound effect of electron correlation, especially of the intraatomic kind, on molecular energies. Really theoretical accounting of this correlation energy requires an extremely large set of configurational functions in the conventional variational calculation. However, Moffitt2 has shown that it is possible to allow for intraatomic correlation by admitting experimental energies of atomic and ionic states in a particular way. The wave function for a diatomic molecule is expanded in terms of functions (Wn),where $n is a product of eigenfunctions for the two atoms involved and a is the normalized antisymmetrization operator applied to assure satisfaction of the Pauli principle
+
+
(1) In practice, appropriate products of ion states also are included in the expansion; Le., $iA and $jB may be eigenfunctions for A+ and B-, etc. To find the minimum expectation value for the total energy, the secular equations Hc = ScE must be solved. In the atoms-in-molecules approach, the total Hamiltonian operator H is first rigorously divided into two main parts H$n
=
= Zncn(E+n)j $n = $iA
+ HB' +
(Hao
Vn)$n
+
=
+jB
+
[TVio(A) Wjo(B) V n ] $ n ( 2 ) HAO and HB' contain all intraatomic terms, Vn all remaining interatomic terms (the particular division of H used depends upon $n whether it in(1) Supported in part by a grant from the National Soience Foundation. (2) W.Moffitt, Proc. Roy. SOC.(London), 8210,245 (1951).
volves eigenfunctions for neutral atoms, or for A + and B-, etc.). The elements of the H-matrix turn out to be Hnm
= 1/2[Snm(Wno
-k )',/IT
where Wno = TVio(A) atomic state energies Snrn =
S(a+n)$nid7,
+
Vnm
+
VmnI
(3)
+ Wjo(B)are experimental
Vnm
= J(a$n)Vm+ind7
(4)
Moffitt suggested that the Vn, and Sn, could be satisfactorily assessed using approximate wave €unctionsof the orbital type. Now it is well known that the effective nuclear charges for valence electrons in atoms and ions often must be modified in order to yield good approximations of interaction energies. hssuming that such modifications do not appreciably affect intraatomic correlation energies, Hurley3 has developed a new and quite successful method for correcting ab initio orbital-type calculations for their neglect of correlation energy. As molecules become more and more complex, ab initio calculations become correspondingly difficult to carry through, to correct, and indeed to interpret physically. Until such time that this situation is improved, simpler approaches containing additional empirical elements are of value in providing at least semiquantitative answers to chemical questions. Some attempts at building such methods on more or less fundamental theoretical grounds are typified by the Magic Formula for the structure of bond energies developed by Mulliken4 and by studies of hydrides of first-row elements by Longuet-E-Iigginsb and by Companion and E l l i ~ o n . ~ . ~ Description of Method and Application to Hz.The work reported here represents an attempt to extend the hIagic Formula by explicit incliision of the coulomb energy, of ionic-covalent resonance, and by possible theoretical improrement of methods for estimating VB exchange iiitegrds. (3) (a) A. C. Hurley, Proc. Phys. Soc. (London), A69, 49 (1956):
(b) A. C. Hurley, J. Chem. Phys., 28, 532 (1958). (4) R. S. Mulliken, J. Phys. Chem., 66, 295 (1953). (5) P. C. H. Jordon and H. C. Longuet-Higgins, MOL Phvs., 6, 121 (1962). (6) A. L. Companion and F. 0. Ellison, J. Chem. Phys., 32, 1132 (1960). (7) F. 0. Ellison, ibid., 36, 3112 (1962).
APPLICATIOKS OF SEMI-EMPIRICAL VALENCE BOXDTHEORY TO SMALLL~OLECULES 2295
Dec., 1962
The starting formula is eq. 3 with the important qualification that the effective nuclear charges for all orbitals comprising and $iB are subject to modification. We shall make the simplification that a given atomic orbital will possess the same effective nuclear charge throughout all fin. From here on in this paper, the $,, in eq. 1 and 4 will be considered to be in the VB b a s h a Thus, the ground state of €Iz is approximated using covalent and ionic functions $q =
2-’/2
[la61 - ldbll
where
(ab:Vi :cd)
=
-
Vi = -I/rza V? =
J a ( l ) b ( l ) Vic(2)d(2)dv(l)du(2)
-1/?1b
l/Tlb
+
l/r2b
f l/fi
1/~12
+ I/R
(IO)
The two coulomb-type integrals may be expressed in terms of the basic two-center nuclear attraction integral ( A :bb) and electron repulsion integral ( a a : b b ) as well as the nuclear repulsion term l/R,1° or alternatively in the forms
(5)
( a a : V i : b b ) = 2a - /3 these are given here in terms of determinantal wave ( a a : V 2 : a a )= 2a - 1/R (11) functions, in which a represents a Is orbital with situated on where CY = 1,/R ( A : b b ) a n d b = l / R (aa:bb). arbitrary effective nuclear charge The Rlulliken approximation” for overlap disatom A; a bar over the orbital indicates @-spin, tributions is utilized in estimating exchange- and no bar indicates a-spin. hybrid-type integrals; that is, me assume I n general, eq. 3 is rewritten
Hnm
‘/’2[Snrn’(Wn
+
Wm)
+ Vnm’ + Vmn’]
(6)
where the primes on Snm’ and the Vnm’indicate that these are evaluated using approximate orbital wave functions. The absence of the 0-superscripts on the Wn (cf. eq. 3) means that these now represent semiempirical valence state energies of component hypothetical atoms or ions which have had their valence shells “inflated” or “deflated” to yield a best fit to the molecular eigenfunction. It has been shown* that if all electron coordinates in an exact atomic eigenfunction are multiplied by a factor y, the energy W,(A) of the modified eigenfunction is related to the true energy I’Vio(A) as
W,(A)
=
WI0(A)y(2- 7)
(7)
For example the energy of a modified H-atom is W(H) = - (13.61e.v.)c(2 - S). For H-, the orbital ) ~ function is 0.6875; exponent in the best ( 1 ~wave to modify this function so that it is of the same “size” as an H-atom with orbital exponent c, the electron coordinates in H- need to be multiplied by y = p/0.6875. Assuming that the best orbital description is in close correspondence to the true electron density of 13-, the “true” energy of modified €1- is W(H-I = W0(H-)y(2 - y), where y == r/0.6875. Thus in eq. G
TVl
=
2TIf(FT)
=
-(27.21 e.v.){(2 - p)
TV2 = Tt’(€I--)= -(14.33 e.v.)y(2
- y)
(8)
WO(H-) = -14.33 e.v. is the exact energy of I-I-.9 Using eq. 3 i,t is easy to show that SI]’= S22’ = (1 S2) and that Slz‘ = 2S, where S is the lsa-lsb overlap integral. Elements in the V‘matrix can be expressed as
+
+ (ab:Vi:ab), V22’= (aa:B2:aa) + (ab:V2:ab),
Vii’
V,j’
=
=
( 8 ) F. 0. Ellison,
(aa:Vi:bb)
2 ( a a : V j : a b ) ,i # j
Chem. Phus., to be published. (9) L. R. Hennch, Astrophy. J., 99, 59 (1944). J’.
(9)
l/?S[a(l)a(l)
a(l)b(l)
+ b(2)b(2)]
(12)
Thus
+
+
(ab:Vi : ab) = ’/&“ [(uu: 1’1 UU) (aa: Vi :bb) ( b b : V i :UU) (bb:Vi : b b ) ] (13)
+
I n addition to two-center interactions a, p, and 1/R, there appear in these new integrals additional interactions of the one-center variety; e.g. (aa:Vl:aa)
=
-(A:uu)
+ ( ~ u : u u-)
CY
(14)
But ( A : a a ) - (aa:aa) can be associated with the change in potential energy accompanying the process
H-
=
Ir7
+ le-,
AT7 = ( A : u a ) - (aa:aa) (15)
We shall denote this particular AB by the symbol Ih(h2) hereafter. Since all orbitals being used are assumed to contain the same effective charge, it would be improper to associate I h ( h 2 ) with twice the electron affinity (the electron affinity being AE for the process, which according to the virial theorem is equal to one-half AV). Rather, this AV is for the reaction involving hypothetical species which have been appropriately rescaled. It can be shownS that if all electron coordinates in an exact atomic eigenfunction are multiplied by a factor y,the potential energy V(A) of the modified eigenfunction is related to the actual potential energy V O ( X ) as
T‘(a)
=
VO(A)y = 2I?‘O(A)7
(16)
Using y = r/0.687,5 for H-, we obtain Ih(hz) = (14.54 e.v.)r. Similarly, for the process H = H+ le, A V = ( A : a a ) = Ih(h) = (27.21 e.v.)
