T H E
J O U R N A L
OF
PHYSICAL CHEMISTRY Registered in U.S. Patent Office 0 Copyright, 1974, by the American Chemical Society
VOLUME 78, NUMBER 8 APRIL 11, 1974
A Comparison of Approximate INDO and ab Initio Wave Functions F. A. Van-Catledge Department of Chemistry, University of Minnesota, Minneapolis, Minnesota 55455 (Received September 25, 7973)
The constraints on properties calculations with approximate wave functions based on neglect of differential overlap are discussed. First and second moments of the electronic charge distribution, along with dipole and quadrupole moments, are calculated from INDO and ab initio wave functions for Hz, Nz, CO, and HF. The effect of symmetric deorthogonalization on INDO expectation values is also considered. The molecular charge distribution predicted by these wave functions is examined more closely by the use of density difference maps and a new function A ( 6 p ) defined to facilitate comparison of approximate and nonempirical wave functions. It is concluded that deorthogonalization of the INDO wave function is necessary.-It is further suggested that INDO calculations may better approximate nonempirical results if the method of selection of core terms and Po values is revised.
Introduction The neglect of differential overlap (NDO) method developed for three-dimensional SCMO calculations by Pople, et al.,1 has proved extremely useful in providing semiquantitative information concerning molecular electronic structure. The formalism developed preserves invariance to local coordinate rotation, hybridization, and transformation to a symmetry-adapted basis. The integral approximations necessary to accomplish this are summarized in Table I. The most commonly employed levels of approximation have been CNDO (complete neglect of differential overlap) and INDO (intermediate neglect of differential overlap). Molecular properties that have been calculated using one of these two methods include dipole moments,l-S spin densities,1,2 nuclear hi el ding,^-^ ir intensities,lO-lz molecular p ~ l a r i z a b i l i t i e s and , ~ ~ quadrupole rnoments.l4-16 These methods have also been used for uv spectral calculations.17-2z Most recently INDO wave functions have proven to be potentially useful for obtaining static potentials to be used in electron-scattering calculat i o n ~ The . ~ ~NDDO (neglect of diatomic differential overlap) method has not proven popular, due to increased computational complexity, but it has been studied briefly by Allen, et al., in connection with predictions of geometries and rotational barriers in hydrocarbons.24.25 Comparison of the available specification for NDDOZ5with the Wiberg version of CNDOZ6showed that the latter tended to be more reliable for the properties investigated. It is important to note that while the original specifications for CNDO and INDO were designed to mimic ab initio calculations,l several groups have modified these to obtain more accurate predictions of one or two propert i e s . 8 ~ 9 ~ z 0 ~ 2Such 1 ~ z 6modified schemes we choose to desig-
nate as semiempirical, while considering the original specifications as approximate or “ ~ e m i r i g o r o u s . ”A~ ~key feature of the approximate methods is that the repulsion integrals retained are calculated using basis functions from the type of ab initio calculation one wishes to mimic (STO’s, etc.). The semiempirical methods generally resort to a Pariser-Parr approachz8 for these integrals and, for spectral calculations, abandon hybridization invariance. Though the energy (relative) and properties calculations mentioned have been generally successful, there remain some difficulties whose origins are not well understood. The low-energy orbitals for most molecules are too strongly bound, relative to STO calculations. Orbital orderings for medium-sized and large molecules do not correspond to those obtained from nonempirical calculations and, as a consequence, spectral calculations using the original specifications for CNDO and INDO ascribe a great deal of u u* character to excited states that have been preA* in n a t ~ r e . This ~ ~ -fact, ~~ viously understood to be K while disappointing, is an acceptable limitation given the complexity of the problem being addressed. Both of these factors seem to infer that splittings within the u manifold are too large. It is much more serious, however, that the calculation of quadrupole moments using the original prescriptionlb are rather un~atisfactory.l~-~6 This difficulty has been overcome by resorting to deorthogonalization of the wave function prior to calculation of this property.16 This approach has also been used in the calculation of dipole moments? and rotational ~ t r e n g t h s . ~ ~ .The Z 9 justification for this approach is the interpretation of NDO calculations as involving symmetrically orthogonalized AO’s (A’s), which are related to the nonorthogonal basis (x’s)by the t r a n ~ f o r m a t i o n ~ ~ -+
-
763
764
F. A. Van-Catledge
TABLE I: Summary of Integral Approximations for NDO Calculationsa Integral type
NDO specification
(PAPAlrASA)
(PAIAcorelQB)
(PA4BlrCSD)
terms
(12) ( x ~ ~ ~p xon~atom ~ ) A; q on atom B The necessary modification for the approximation (10) to hold is (XPAI~/X~B)J.~B
( ~ ~ A I ~ I E A ~ J
a
See ref le.
PPo
E
PAO
= an atomic bonding parameter.
Y A B = (SASA~SBSB).
an orbital bonding parameter.
(1)
= xs-1/2
where
s = xtx
(2) A’s obtained in this and the approximate to be true.l Davies, NDO calculations in
Lowdin has demonstrated that the way most closely resemble the x’s,3l specifications presume the converse on the other hand, prefers to regard general as semiempirical. We propose to examine in detail selected properties of the charge distribution as predicted by INDO calculations and compare them to those obtained for an ab initio wave function for the same basis (STO). Both bases ( A and x) will be considered for the INDO results. We hope to (1) determine the more appropriate interpretation of the calculation and (2) ascertain (in so far as possible) the effects of neglect of differential overlap on the predicted charge distribution. Properties Calculations
Once the wave function has been obtained, properties may be calculated in the following manner. The MO’s, ($’ti), may be represented as =
xu
=
xv
( 3)
(4) TWO density matrices may be defined as P = V p R=UU’ (5) where P is the NDO bond-order matrix and R is the density matrix in the nonorthogonlal basis. Now for the expectation value of the operator 0 we have the following expressions
(6)= Tr(P0’)
Ox =
X’6X
(6)
or alte,rnately
(6)= Tr(ROX)
Ox= x’6x
For the special case where 0 = 1, we have total number of electrons and Tr(P0’) = Tr(ROX) = n
@=E
Ox=S
(7)
(0)=
n, the (8) (9)
Now it is customary in NDO properties calculations to make the approximation
( x p ( 6 1 xQ)= (xpl6lxCl)
(10) a consequence of assuming the converse of Lowdin’s proof.31 This necessitates further constraints since, in general
Tr(PS)
>
n
(11)
The principal contribution to this inequality from the The Journal of Physical Chemistry, Vol. 78, No. 8, 1974
(13)
(One might have also introduced 6pq in (13) but Pople, et ul., found the intraatomic crossterms to be important for dipole moment calculations.lb) We shall use both approaches to properties calculations. Properties calculated according to eq 6 and 13 will be labeled INDO. We shall obtain R as
R = s-1/2pS-1/2
(14)
and properties calculated according to eq 7 will be labeled INDO-D as suggested by Bloor, et al. Retention of the intraatomic crossterms in NDO dipole moment calculations requires designation of the basis set. The same is true for quadrupole moment calculations even if these crossterms are dropped. Inconsistencies in interpretation arise if STO’s are used for these matrix elements in a semiempirical specification. Semiempirical calculations imply a symmetrically orthogonalized, undefined A 0 basis whose presumed repulsion integrals are markedly different from the STO values. It is reasonable to assume that similar differences obtain for the other types of integrals. Against the background laid out in the preceding discussion, we will examine the components of the second moment of the charge distribution for Nz, CO, and HF as predicted by INDO and INDO-D. The reference wave functions are the STO calculations reported by R a n ~ i 1 . ~ ~ For CO and HF we do not reproduce previously reported values for the dipole moment and as a consequence the values for the quadrupole moment are slightly off. The reasons for this discrepancy are unclear. We have, therefore, included in the Appendix a brief description of the calculations, along with expressions for the relevant matrix elements as we have derived them.33 The qualitative aspects of our results will be unaffected by the possible error. In our studies of static potentials, we found it useful to include the 1s orbitals as symmetry orbitals in certain properties calculations.23 This permits direct cbmparison of the values with nonempirical results. This modification is designated INDO/ls for the A basis, or INDO/ls-D for the x basis. Another probe giving insight into the charge distribution is the function introduced by R O U X ~ ~ 6p(r) = pi&) - PA(^> where p M ( r )is the electron density at the point r for the molecular charge distribution and p A ( r ) is the electron density appropriate to the noninteracting atoms at the same internuclear separation. As is customary, p A ( r ) is computed using spherically averaged p orbitals. thereby circumventing the problem of arbitrarily assigning orbital occupancies. PA
=
[nip18 + nzpzs
+ (2 -
ni - n2)i&pI
(15)
where Pns = XnstXns P2p
= (1/3)(Pzp,
+
Pzp,
(16)
+
mp,>
(17)
This function has proven exceedingly helpful in understanding the differences among various types of molecular wave functions.35 NDO-type wave functions have been
765
Comparison of Approximate INDO and ab Initio Wave Functions studied in this way by Boyd who studied density difference maps for adenine derived from extended Huckel (EH), CNDO/Z, and CNDOIB-D wave functions.36 In this work36 it was found informative to examine differences among the various charge distributions. We cannot make direct comparisons of the charge distributions derived from INDO (and INDO-D) with those from STO, since in the latter case the 1s STO and 2s STO (2s’) are made orthogonal (explicitly in the atom, implicitly in the molecule).
xls = (5+?/rj1’’ exp[--&rl xzSt= (5+z5/3~)1’zr exp[-LrI
s=
(18)
- SXlJ
(21)
Ransil and Sinai37 have recently examined the differences among different “Hartree-Fock-limit” atoms and found them to be significant and, therefore, problematical in comparing 6 p ( r ) functions for different molecular calculations. For this reason we have found it convenient to define the function =
GpCINDO-D)
+
+
(20)
(XlSlX2S’)
x z s = (l/lvF-F)(x2sf
A(&)
+
(19)
- Gp(ST0)
/
/’
(22)
which permits us to compare the two difference maps without running afoul of the differences in starting atoms. (It will be recalled that NDO calculations employ valence STO’s without orthogonalizing them to the inner shell orbitals.) This function becomes extremely important for the analysis of Nz and CO as the difference maps themselves are superficially quite similar. Contours of constant electron density are drawn a t 0.0, hO.1, fO.O1, and f O . O O 1 electrons per cubic atomic unit. The following sets of orbital exponents were used throughout: H, 1.2; C, 5.7, 1.625; N, 6.7, 1.95; 0, 7.7, 2.275; F, 8.7, 2.6.
Results Hydrogen Molecule ( R = 1.402 a u ) . Insight into the need for deorthogonalization is gained by considering calculations on Hz. The molecular orbitals are completely determined by symmetry, thereby circumventing any problems arising from integral approximations. For the INDO wave function, the quadrupole moment is zero and Sp(IND0) is zero for all r unless different orbital exponents are used for the atom and molecule. When deorthogonalization is carried out, however, we obtained a nonzero guadrupole moment, and the density difference map becomes consistent with chemical ideas regarding bond formation. The results for quadrupole moments are summarized in Table 11. Nitrogen Molecule ( R = 2.068 a u ) . The molecule Na has served as a benchmark for the testing of SCF calculations. Its symmetry eliminates problems of imbalance in the basis set and, unlike Fz, a minimal, unoptimized STO basis will predict a bound ground state. A rather extensive study of properties of the charge distribution predicted by various wave functions for Nz has been reported,35a and provides useful guidelines in the interpretation of our results. The density difference map, Sp(INDO), shows extensive buildup of electron density in the lone-pair regions (Figure 1). There is, on the other hand, a diminution of electron density in the entire internuclear region. Thus, in the absence of any data on energetics, one might associate such a map with a repulsive state. The map Gp(ST0) shows an annular region of positive density difference but
Figure 1. Density difference maps for N Z (R = 2.068 au). Contours are drawn at 0, fO.O1, and hO.001 electrons per cubic unit. The representations are positive contours (- - -); negative contours (---); null contours (-). bp(lND0): The density difference map resulting from the INDO density matrix, excluding two-center atomic charge distributions. bp(lND0-D): The density difference map resulting from the symmetrical deorthogonalized I NDO density matrix including ail atomic charge distributions. bp(ST0): The density difference map resulting from the density matrix of an ab initio calculation using STO’s with exponents determined from Siater’s rules. A ( & ) : Sp(lND0 -D) - bp(STO), a map showing the difference in charge reorganization predicted by INDO-D and STO. is negative close to the internuclear axis. More elaborate basis sets yield a Sp function more in keeping with chemical intuition,35 but 6p(STO) must be our basis for comparison. The function Gp(IND0-D) is superficially quite similar to the latter, especially in the internuclear region. In fact, somewhat more buildup of electron density is predicted for the annular region previously noted. The function A ( b p ) confirms this in that it is positive between the nuclei and negative in the lone-pair regions. The results of the second moment calculations reflect the features that The Journal of Physical Chemistry, Vol. 78, No. 8, 1974
F. A. Van-Catledge
766
TABLE 11: Properties of the Charge Distributions for Selected Molecules f r o m Various Wave Functions.
(4
Wave function
p p INDO-D b INDO
r
= = = =
(4
c
1,3888 2 .0000 1 ,4668 2 .0882
Experimental
6.5191 6.7495 6.8175 7.7009
Experimental
5.1464 6.9800 5.0645 6.9079
0.00 0.00 0.32 0.34 0 . 64b
24.6455 23.0785 23.6855 23.6169
37.6837 35.5775 37.3205 39.0186
-3.35c -1.61d -3.17,e -3.20C -1.36 - 1,90e,' -0.95f -1.13b
co -0.3605 -0,0254 -0.1316
STO. LMSSi
Experimental (C -0 f)
STO
2.3688 2.9800 2.1308 2.7315
eZZ
Nz
CNDO/2 CNDO/2-D INDO/lS INDO/lS-D STO LMSS (HF 1imit)g
CND0/2 CNDO/2-D INDO/lS INDO/lS-D
(4
Hz
1.2 1.0 1.2 1.0
CNDO/2 CNDO/2-D INDO/lS I N DO / 1s-D STOe
(4
0,366 0.032 0.139 0.287h -0.071 -0,108 0.044i
6.5941 6.6899 6.7944
25,6755 24.0519 24.9386
38.8637 37.4317 38.5274
-3.79c -2.1Bd -3.50,C -3.49" -1.77 -2.56 -2.40 -1.53i -1.59( -1.4gb
HF 0 . 82c
-0.0039 -0.0738 -0,0940 1.3876
(1062[42)2
Experimental" (CH +F-)
-0.856 -0,786 -0.766 -0.345 -0,756 -0.683
3.4952 3.3744 3.5874 3.6024
5.2256 4.7868 5.7058 5.8603
12.2160 11.5356 12.8806' 13.0651 13.7180
1.16d 1.04 1.36 0 . 66e 1.82k 0.94
a All values are in atomic units. W. H. Flygare and R. C . Benson, Mol. Phys., 20, 225 (1971). Reference 14. Reference 16. e This work. f Reference 34a. A. C. Wahl, private communication. See P. E. Cade, K. D. Sales, and A. C. Wahl, J . Chem. Phys., 44, 1973 (1966), for the wave function. Reference 34b. a W. M. Huo, J. Chem. Phys., 43, 624 (1965). j B. Rosemblum, A. H. Nethercot, Jr., and C. H. Townes, Phys. Reu., 109, 400 (1958). R. P. Hurst, M. Karplus, and T. P. Das, J. Chem. Phys., 36,2786 (1962). l J. W. Moscowitz, D. Neumann, and M. C. Harrison, in "Quantum Theory Atoms, Molecules, and the Solid State," P. 0. Lowdin, Ed., Academic Press, New York, N. Y . , 1964, p 227. " D. F. Smith, Proc. U.N . Conf. Peaceful Uses At. Energy, 2nd, 1958, 28, 130 (1958).
have been noted in the density difference plots (see Table I). For INDO/ls the values of (x2) and (9) are, respectively, smaller and larger than those from STO. The unsatisfactory value of eZzis seen to be a direct consequence of the omission of interatomic terms in the properties calculation. The INDO/ls-D values are more satisfactory in this regard, though ( x 2 ) and ( 9 )are both smaller than those from the STO calculation. Similar 'results are obtained from CNDOI2-D calculations.16 Indeed, the value for the quadrupole moment is closer to the experimental result for INDO/ls-D than for STO. This result, however, reflects deficiencies on the part of the approximate calculation. Carbon Monoxide ( R = 2.132 au). The CO molecule is also a useful checkpoint as it is isoelectronic with Nz,but introduces a small difference in electronegativity as the perturbing influence. Previous studies of basis dependence of the function 6 p for CO have shown the same qualitative changes as noted for N2.S4b The various 6 p plots that we have obtained (Figure 2) follow in a reasonable fashion from the shift of a proton from one nucleus to the other in Nz. The INDO calculation continues to emphasize lonepair formation at the expense of bonding, while INDO-D is similar, but not identical with STO. In this case it should be noted the function A(6p) reflects the asymmetry in nuclear charges perhaps more strongly than the 6p The Journal of Physical Chemistry, Vol. 78, No. 8, 1974
functions, as it is negative along the internuclear axis from the carbon atom to about the midpoint of the bond. The second moment calculations (see Table 11) parallel those for Nz,also. INDO/ls predicts too large a quadruvalue. INDO/ pole moment by virtue of an overlarge (9) 1s-D exhibits the same types of discrepancies as noted for Nz, resulting in a value for OZz that is closer to experiment, but smaller in absolute value than that from STO. As the center of inversion has been lost on going from Nz to CO, we would expect to see some dramatic effect of the INDO integral approximations on the T system, where they would be easily recognizable. For this subset of orbitals the neglect of hybrid and exchange repulsion integrals is not as critical as one might have expected. The elements of the density matrices R(IND0-D) and R(ST0) are within -5% of each other for these AO's. It would seem that, for the a system at least, any errors incurred by these omissions are being compensated in some other way. Hydrogen Fluoride ( R = 1.733 au). We felt that further insight might be gained by examination of a first-row hydride, as absence of orbitals with n = 2 on one of the atoms could help to localize the sources of difference between INDO-D and STO. Our choice of HF was based on a desire to maximize the difference in electronegativity while maintaining covalent bonding. As our density differ-
767
Comparison of Approximate INDO and ab Initio Wave Functions
,”-.
I P! IN D O D) ~
-
I//
.
\--
,
Figure 2. Density difference maps for GO ( R = 2.132 au). Contours are drawn at 0, hO.1,fO.O1, and fO.OO1 electrons per cubic unit. The representations are positive contours (- - -); negative contours null contours (-). Gp(lND0): The density difference map resulting from the INDO density matrix, excluding two-center atomic charge distributions. 6p ( I NDO-D): The density difference map resulting from the symmetrical deorthogonalized INDO density matrix including all atomic charge distributions. 6p(STO): The density difference map resulting from the density matrix of an ab initio calculation using STO’s with exponents determined from Slater’s rules. A ( 6 p ) : Gp(lND0 -D) - 6p(STO), a map showing the differences in charge reorganization predicted by INDO-D and STO.
Figure 3. Density difference maps for HF ( R = 1.733 au). Contours are drawn at 0, fO.l, fO.O1, and fO.OO1 electrons per cubic unit. The representations are positive contours (- - -); negative contours (---); null contours (-). &(INDO): The density difference map resulting from the INDO density matrix, excluding two-center atomic charge distributions. 6p(lNDO-D): The density difference map resulting from the symmetrical deorthogonalized INDO density matrix including all atomic charge distributions. Gp(ST0): The density difference map resulting from the density matrix of an ab initio calculation using STO’s with exponents determined from Slater’s rules. A(6p): Gp(lND0 -D) - 6p(STO), a map showing the difference in charge reorganization predicted by INDO-D and STO.
ence plota show (Figure 3), some of the features noted for N z and CO are not apparent here. The INDO wave function, while still overemphasizing lone-pair buildup, is more similar to STO than INDO-D. The latter is chhracterized by a lack of significant u lone-pair formation. Since the p orbitals have been spherically averaged, development of the u lone pair in STO does not produce the marked change in this region previously seen. The striking feature of Gp(IND0-D) is that it is negatiue in this region. It is more than a little surprising, therefore, that the function A(6p) exhibits the same feature that we have seen in
the other molecules, namely, excess electron density in the internuclear region. Despite the unexpected features of the density difference maps for HF, the trends in the moment calculations are consistent with those for N2 and CO. Deorthogonalization of INDO (or CNDO) yields a more positive value of e,,, this value in turn being less negative than that obtained from STO.
(-a-):
Discussion One of our original aims was to determine if it would be possible in principle, by straightforward modifications, to The Journal of Physical Chemistry, Vol. 78, No. 8, 1974
F. A. Van-Catledge
improve upon the electronic charge distributions given by the simpler NDO methods. This requires that we attempt to establish the origins of the noted discrepancies. We may eliminate from consideration such factors as spherically averaged repulsions and atomic us. orbital P’s. These two factors should be retained for preservation of invariance though the latter, while desirable, is not essentia1.l CNDO and INDO were designed to mimic ab initio calculations,. but the parameterizations are based in part on experimental data which we suspect may detract from the comparability we seek. We shall examine some possibilities within the framework of INDO parameterization. T h e Nodeless 2s‘ Function. The recurrent feature in INDO-D for the molecules we have studied has been improper s-pa hybridization. (Clearly, one reason is the diminution of y by virtue of the spherical averaging.) It is possible, however, that this phenomenon might be related to the fact that the function x Z s ’ (eq 19) does not extend quite as far from the nucleus as x z S (eq 21). This is shown in Figure 4, where we have plotted the radial distribution difference function for the nitrogen atom where
6p(2s’ - 2 s ) = p2s’ - pzs (23) This is also reflected in the various integral values displayed in Table 111, where all integrals but ( r - l ) change as would be anticipated for a more extended orbital. (The anomaly in this latter term arises from the contribution of the 1s function near the nucleus.) While this proper 2s function may, therefore, appear attractive, we have found that use of this function (set B) requires changes in the core terms, Us, and Upp, that offset any benefits gained (see Table IV). This approach is also unsatisfactory if hybridization invariance is to be preserved, as it would require the radial functions for the valence shell orbitals be different for different 1. While this is in fact true, such a modification is strictly allowable only at the NDDO level of approximation. Klopman and Pollk3* have made some preliminary studies along these lines. Use of Experimental Exchange Integrals. If we relate hybridization to s p promotion, it seems clear that the changes in the integrals effecting the energy required for s m - l p n + l will alter the hybridization the process smpn implied by INDO calculations. We have, consequently, examined the effect of replacing the Slater-Condon terms, G1 and F, with the appropriate values derived from STO’s (set C). As can be seen (Table IV) such a change results in a slightly decreased promotion energy for the nitrogen atom. The point at issue now is whether or not the observed change is qualitatively able to produce the
-
Figure 4. The radial distribution corresponding to the nodeless
2s orbital minus the radial distribution function for the Schmidtorthogonalized 2s function for the nitrogen atom ({, = 6.7; {* = 1.95).
TABLE 111: Comparison of Expectation Values for the Nodeless and Orthogonalized 2s STO’s of Nitrogen. Integral
Nodeless 2s
(r) (r2) (r-9
1.282051 1.972387 0.975000
Orthogonalized 2s
1.313891 2.054832 1.079566
-
All values are in atomic units.
changes desired. In order to cause A(6p) to more closely approximate a null map for N2, let us say, we need to decrease the p character of 2ag and 2oU orbitals while increasing the p character in 3ag. Such a change must, at this level of approximation, be associated with an increase in the promotion energy. Use of STO-derived exchange integrals does not have the desired effect (Table IV). We should also note at this time that the core terms required for the various combinations of repulsion and exchange terms are remarkably similar. This suggests that we have not yet addressed ourselves to the cause(s) of the excessively low energies obtained for the more strongly bonding orbitals .20* Use of Experimental Ionization Data. It has long been customary to use experimental ionization potentials and electron affinities in the parameterization of semiempiri~ ~ ~question we wish to raise is whethcal c a l ~ u l a t i o n s . 3 9The er it would be better for approximate valence-shell calculations to take these data from theoretical atomic calculations within the particular basis chosen. Such an approach will ensure that, despite the INDO approxima-
TABLE 1% Dependence of INDO Core Integrals and Promotion Energy upon Exchange Integrals and Ionization Energies for t h e Nitrogen Atoma Set Ab
I,
A,
1, A, JS, JSZ
K,, Jw US,
UP,
AE(szp3-+ sp4)
0,940096’ 0.479677f 0,4847481 0.0492471 0,708398 0.708398 0 . 115343h 0,690874& - 3.669989 - 3.295636 0.437124
set B
C
0.940096f 0,4796771 0,4847481 0.049247 0.689754 0,697844 0 . 115343h 0. 690874h - 3.583191 - 3.274528 0.411186
Set De
Set Cd
0,940096f 0,479677’ 0.484748’ 0,0492471 0,708398 0.708398 0.156576 0,680977 - 3.584526 - 3.229660 0,429179
0.9406150 0,2508610 0,40111O0 - 0,2798670 0.708398 0.708398 0.156576 0,680977 - 3,470377 - 3,023284 0.521406
All values are in atomic units. The original INDO parameter (see ref Id). An orthogonal 2s orbital was used. The exchange integrals are taken from STO’s. e The exchange integrals and ionization energies are from STO’s. f These are the experimental values from ref lb. These are the theoretical values taken from the STO atom. These are the experimental values from ref Id. @
The Journal of Physical Chemistry, Vol. 78, No. 8 , 1974
Comparison of Approximate INDO and ab lnitio Wave Functions tions, the core terms will approximate the net potential seen by the valence electrons more closely and, thereby, facilitate direct comparison of the approximate and a b initio results. (This necessitates a philosophical decision that the purpose of the calculation is maximum comparability with ab initio results.) We have derived, for the nitrogen atom, average ionization potentials, and electron affinities based on an STO atom. The INDO core terms and promotion energy derived therefrom are presented in Table IV as set D. The outstanding feature in this set is the increase in the promotion energy of -0.1 eV, which should produce the desired effect. Further, both core terms are now smaller in magnitude and should ameliorate to some extent the orbital energy problem. Selection of Atomic B’s. In the original specification1 pA0 was taken to be a constant for atom A and the H p ~ g ~ C calculated ore using a simple average of the atomic p’s. This simple approach may be another source of difficulty. In anIND.0 study of geometries of hydrocarbon^,^^ we have found that &Ho was best chosen as being different from the mean of pcco and p H H 0 . This implies the need for an evaluation of the terms. We suggest that the most appropriate approach for approximate calculations would be to select a value of @AB” that best reproduces the energies of the occupied orbitals in the relevant ab initio calculation. While this ultimately requires more values for Bo, the usefulness of the method should be considerably enhanced where good approximations to nonempirical calculations are sought. We do not favor in this context distinguishing between s and p orbitals as seems to be necessary for spectral calculation^.^^-^^ Useful results have also been obtained by introducing symmetry dependence (u, ?r, etc.) for p A B 0 . 1 9 Neither approach is fully consistent with the invariance requirements, however, and should be avoided in “semirigoro~s”~7 calculations. Conclusions As previously stated we prefer to interpret NDO calculations as approximations to ab initio calculations, rather than semiempirical calculations of specific properties. Our properties calculations clearly indicate that some alteration of INDO parameterization will undoubtedly increase comparability. It appears most important that (1) the average ionization energies be taken from the atoms appropriate to the ab initio calculation and (2) all two-electron integrals be evaluated for these atoms. [The same is probably true for nuclear attraction terms, which we have not discussed explicitly. The current method of evaluating these is based on considerations of H2 (32u+),1c which is atypical in these calculations (see Table II).] We further suggest that the resonance parameter DABo for a given atom pair is unique and not simply related to BA0and pB0. It is our contention that reparameterization as outlined should lead to a method yielding highly useful approximations to ab initio wave functions.
Acknowledgment. We wish to express our thanks to the University of Minnesota Computer Center for a grant supporting the calculations reported herein. Supplementary Material Available. The Appendix, containing the description of the first and second moment calcutations along with expressions for the A 0 matrix elements, will appear following these pages in the microfilm edition of this volume of the journal. Photocopies of the supplementary material from this paper only or micro-
769
fiche (105 x 148 mm, 24X reduction, negatives) containing all of the supplementary material for the papers in. this issue may be obtained from the Journals Department, American Chemical Society, 1155 16th St., N.W., Washington, D. C. 20036. Remit check or money order for $3.00 for photocopy or $2.00 for microfiche, referring to code number JPC-74-763.
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The Journal of Physical Chemistry, Vol. 78, NO. 8, 1974
