Localized atomic orbitals for atoms in molecules. II. Diatomic molecules

Localized atomic orbitals for atoms in molecules. II. Diatomic moleculesKeith H. Aufderheide and Alice ChungPhillips Citation: The Journal of Chemical Physics 73, 1789 (1980); doi: 10.1063/1.440315 View online: http://dx.doi.org/10.1063/1.440315 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/73/4?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Localized atomic orbitals for atoms in molecules. III. Polyatomic molecules J. Chem. Phys. 76, 1885 (1982); 10.1063/1.443162 Localized atomic orbitals for atoms in molecules. I. Methodology J. Chem. Phys. 73, 1777 (1980); 10.1063/1.440314 Uniform Localization of Atomic and Molecular Orbitals. II J. Chem. Phys. 48, 800 (1968); 10.1063/1.1668714 Localized Atomic and Molecular Orbitals. II J. Chem. Phys. 43, S97 (1965); 10.1063/1.1701520 Approximation of Molecular Orbitals in Diatomic Molecules by Diatomic Orbitals J. Chem. Phys. 22, 774 (1954); 10.1063/1.1740191

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Localized atomic orbitals for atoms in molecules. II. Diatomic molecules

Keith H. Aufderheidea)b)C) and Alice Chung-Phillips

Department of Chemistry. Miami University. Oxford. Ohio 45056 (Received 6 February 1980; accepted 9 May 1980)

Using a previously described method. localized atomic orbitals (LAOs) for atoms in molecules are found for the atoms Li. B. C. N. O. and F in the diatomic molecules LiH. Li,. LiF, DH. B" DF, C,. CO. NH. N2 • and F2 •

In all instances LAOs partition into sets of core, lone pair. and bonding orbitals which, except in C2• are arranged according to the hybridization schemes suggested by the localized molecular orbital (LMO) analyses of Edmiston and Ruedenberg. C, is of special importance as the LMO result for this molecule is unorthodox and difficult to interpret whereas the LAO description is more lucid and enlightening. Generally. core LAOs are doubly occupied and lone pair LAOs are either unoccupied (Li) or doubly occupied. Bonding is usually described principally as the interaction of bonding LAOs on the adjacent. bonded atoms. Exceptions to these generalities are found for C2 (not a Lewis compound) and NH (considered in an excited state). The composition of core LAOs for atoms in molecules is found to be a function primarily of the atom itself and is rather insensitive to chemical environment of the atom. On the other hand, composition of the valence LAOs is found to vary in a meaningful manner reflecting changes in both the number and orbital arrangement of valence electrons.

I. INTRODUCTION

A wave function taken as a single Slater determinant of spin-orbitals will yield a probability density which is invariant under a unitary transformation among the constituent molecular orbitals (MOs). j Because of their valuable properties, 2 canonical MOs (CMOs) are generally employed in an electronic structure calculation. However, CMOs do not purvey the localized description of valence Which has been so conceptually and historically useful. A number of schemes3 exist for designing unitary transformations to change CMOs into localized MOs (LMOs) which by their spatial arrangements, illustrate the con~ept of directed valence. Among these, the procedure suggested by Lennard-Jones and pople4 and implemented by Edmiston and Ruedenberg5 is perhaps the best known and most useful for closed shell system(!l. Hereafter the term "LMO" shall refer exclUSively to an MO localized by the Edmiston-Ruedenberg approach. We shall refer to Refs. 5(a), 5(b), and 5(c) as ER1, ER2, and ER3, respectively. LMOs have enjoyed considerable success for diatomic molecules 5 (b) as well as for larger closed shell systems within both the ab initio 5

(C) and semiempirical6 frameworks and have been used in conformational studies. 7

If some set of MOs are expanded as linear combinations of atomic orbitals (AOs) then we may choose the MOs to be invariant under a unitary transformation of the AOs.2 This permits one to construct sets of hybrid AOs (HAOs) to emphasize certain aspects of particular problems. We are here concerned with construction of HAOs for.atoms in molecules (as opposed to free atoms), for only in this fashion may one ascertain variations in

aJpresented, in part, at the Twelfth Annual Midwest Theoretical Chemistry Conference, Purdue University, May, 1979.

bJPresented in partial fulfillment of the requirements for the Ph.D. degree in PhYSical Chemistry from Miami University.

cJpresent address: Oglethorpe University, 4484 Peachtree Rd., N. E., Atlanta, Georgia 30319.

hybridization as a function of chemical environment. Because LMOs are localized largely on or between atoms, the forms of the LMOs localized on or around some atom A, in a molecule suggests the hybridization of A. Let us define localized AOs (LAOs) as a set of HAOs which by their arrangement and functional partitioning into sets of core, lone pair, and bonding orbitals match this suggested hybridization. A large number of methods exist for forming HAOss of one kind or another but none before has attempted to yield LAOs.

In a previous publication, one of us9 has proposed such a method. Beginning with a minimal basis of AOs, here taken as Slater-type orbitals (S'I'Os), a procedure was developed to construct a unitary transformation relating STOs to LAOs on some atom A within the ab initio MO framework. In the event of a free atom, the LMO and LAO methods converge as they should. Additionally, it was shown that the LAO scheme yields the correct hybridization for the HF molecule: we find one core, one bonding and thre~ trigonally equivalent lone pair LAOs on F and the bond in HF is described principallY by a linear combination of the His orbital and the bonding LAO on F. The present work considers all stable diatomics treated in ER2 and ER3. In every instance save one (see below) LMOs and LAOs predict the same type of atomic hybridization. In addition, the Mulliken populations of LAOs correspond quite closely to classical thought, each core LAO being essentially doubly occupied and lone pair LAOs tending to be either unoccupied (Li atoms) or doubly occupied (B, C, N, 0, and F atoms). Interatomic overlap populations tend to be quite small in magnitude except for the distributions between two bonding LAOs which are appreCiably populated. This is to be expected since these distributions are primarily responsible for bonding and should therefore contain most of the internuclear bonding charge. Exceptions to one or more of the foregoing generalities occur in C2 and NH, the former of which is not a Lewis structure compound and the latter of which is here considered in an excited state. We shall find that the forms and populations

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1790 K. H. Aufderheide and A. Chung-Phillips: Localized atomic orbitals. II

of LAOs and their overlap distributions offer a perfectly adequate and lucid localized picture of chemical bonding even in the absence of LMOs.

We therefore have two formalisms, each independent of the other which yield essentially equivalent pictures of atomic hybridization and bonding, one in the form of LAOs on every atom in the molecule and the other in the form of LMOs which tend to be localized on an atom or between two bonded atoms. The complementarity is aesthetically pleasing. However, on a more practical level there are several factors which may make LAOs more attractive than their LMO counterparts, as discussed previously. 9 To summarize:

(1) The LAO result for C2, as discussed in this publication, is far preferable to the LMO description of this molecule. 5

(C)

(2) For all systems studied except C2, the LAO and LMO methods provide complementary viewpOints. Preliminary results to prove this correlation extends to larger systems, as well. As the system size increases, the LAO method becomes extraordinarily more computationally efficient than the LMO approach because the dimensionality of an isolated atomic calculation is independent of molecule size. Further, the number of atomic calculations which must be performed is frequently reduced because H atoms may be ignored and LAOs need be calculated explicitly only on one member of a set of equivalent atoms.

(3) In ER2 it was suggested that a set of HAOs (here called ERAOs) be formed by normalizing the intraatomic portions of LMOs. Of course, LAOs and ERAOs are quite similar in form and arrangement except in C2. gRAOs were shown to vary in composition in a meaning~ ful fashion and other workers have employed this concept to some avail. 11 We show later in this publication that the composition of LAOs also varies in a meaningful fashion similar to ERAOs. Yet, ERAOs possess none of the advantages of LAOs as ERAOs are defined entirely arbitrarily and in terms oj the LMOs. That is, since one must know the LMOs to form ERAOs, the ERAOs cannot claim the advantages listed for LAOs in (1) and (2) above. Also, ERAOs on a given atom are not unitary which greatly complicates the Mulliken population analyses; LAOs on a given atom are unitary, greatlyenhancing the simplicity of the population analyses we later show to be so valuable.

II. REMARKS AND CONSIDERATIONS

All molecules treated herein are closed shell systems and all were considered in their ground states except B2 and NH which were treated in excited t2::; and t2::+ states, respectively. The experimental equilibrium geometries 12 for the appropriate states were employed and a coordinate system was consistently chosen so that the atoms on the left in Fig. 1 were placed at the origin and the atoms on the right were placed along the positive x axis. Electronic structures were calculated using mini~ mal bases of real STO-3G orbitals via GAUSSIAN 70. 13

Because we treat only atoms in the first two rows of the periodic table any atom may have, at most, five STOs

OLIO 0 OLIOC)LI'O

OBO 0 OOC)B'O gFC) 0 gFOC)F'© FIG. 1. Schematic representations of LAOs discussed herein. No attempt has been made to indicate fraction of s character, differences in polarities or any other small variations in directional properties of the orbitals. These refinements are, however, discussed fully in the text.

associated with it. Following gR2 we label these

Ak= 1s STO on atom A

As = 2s STO on atom A, made orthogonal (unitary) toAk

Apa = 2px(2pa) STO on atom A

Aprr =::: 2py(2prr) STO on atom A

Ap1i =::: 2pz(2p1i) STO on atom A •

LAOs were found on all atoms according to the scheme described in detail elsewhere. 9 For a given atom A, the procedure is as follows: If A has an initial set of unitary STO basis functions (ao, ao • ..• ) then these AOs give rise to the elements L(Aao I Aao)' L(Aao I Aao) is a measure of the "MO~label" exchange energy associated with the charge distribution between AOs (Aao) and (Aao) on A. The STOs give rise to the following functions:

Lo(A) = 2::. oL(Aao !Aao)

Ao(A) = 2::.o¢ioL (Aao !Aao) .

(1)

(2)

Obviously, Lo(A) is a measure of the total MO-label exchange experienced by the orbital (Aao =Aao) distributions on A where Ao(A) represents the total MO-label exchange associated with the overlap (Aao * Aao) distributions on A. STOs are then combined to form LAOs on A via a unitary transformation. If the LAOs on A are represented as (at> ato ... ) then these give rise to the elements L(AatIAat), the MO-Iabel exchange elements in the LAO basis. As before, we may define the functions

Lt(A) = 2::. tL(Aat !Aat)

At(A) = 2::Gt¢itL(AatIAat) •

The LAO basis is characterized by having Lt(A) be a maximum. Since the invariance

(3)

(4)

(5)

exists, it must follow that, because Lt(A) is a maximum,

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K. H. Aufderheide and A. Chung-Phillips: Localized atomic orbitals. II 1791

At(A) must be a minimum for LAOs.

LAOs thus formed are shown in Fig. 1 (core orbitals are not included). Depending upon the particular valence situation of A in the molecule the following labels may be ascribed to the LAOs on this atom (modified from ER2):

iA = core LAO on atom A.

ba-A(B) := a- bonding LAO on A polarized toward atom B along the internuclear axis.

la-A = a- lone pair LAO on A polarized away from B along the internuclear axis.

btjA(B) (j = I, 2, 3) = three trigonally equivalent bonding LAOs on A polarized toward B. They have the internuclear axis as a C 3 symmetry axis.

ltjA (j = 1,2,3) = three trigonally equivalent lone pair LAOs on A polarized away from B. They have the internuclear axis as a C3 symmetry axis.

Each molecule in Fig. 1 has a data table associated with it consisting of the following matrices:

(1) The elements 4L(AaoIAao) in the STO basiS, one matrix per each unique nonhydrogen atom A in the molecule. Lo(A) is the trace of this matrix divided by 4 and Ao(A) is the sum of the off-diagonal elements divided by 4. Units are hartrees (a. u.).

(2) The elements 4L(AatIAat) in the LAO basis, one matrix for each unique nonhydrogen atom A in the molecule. A comparison of the 4L(AaoIAao) and the 4L(AalIAal) will show to what extent L 1(A) has been maximized and A1(A) minimized. Also note that the invariance of Eq. (5) holds in all instances. Units=a.u.

(3) The unitary matrices relating STOs to LAOs, one matrix per each unique nonhydrogen atom in the molecule. The phases are consistent throughout: all core LAOs are written so that the Ak coefficient is positive and all valence LAOs are expressed with a positive As coefficient. Trigonally equivalent orbitals deserve special attention. Because LAOs are formed by a series of 2 x 2 rotations it usually follows that trigonally equivalent LAOs are shoum equivalent to only three or four significant figures. To correct for this, these "almost equivalent" orbitals were arbitrarily reorthogonalized and equivalenced and then used as initial "guesses" to form the final set of LAOs which are indeed equivalent to five Significant figures. This procedure is, rigorously, unnecessary except that it does lead to some very pleasing cosmetic advantages and was therefore considered worthwhile. Also, any set of trigonally equivalent LAOs on atom A in a diatomic mOlecule A-B have no preferred orientation with respect to the LAOs on atom B and vice versa. Put another way, trigonallyequivalent orbitals may be rotated by any angle about the internuclear axis in a diatomic without altering Lt(A). For the sake of consistency, all trigonally equivalent orbitals have been rotated to the same reference position: btlA(B), bt2A(B), and bt3A(B) are rotated until the Apn component in btlA(B) vanishes. The orbitals UlA,

lt2A, and lt3A are treated similarly.

(4) The LAO Mulliken populations, q(Aall Bbt ). Units = a. u. When Aal = Bb1 we have the population of the (Aat)th LAO. When Aat '* Bbl , 2q(Aatl Bbt ) yields the population of the overlap distribution between LAOs Aal and Bbl' Insofar as Aal and Bbl are on different atoms, A,* B, q(Aatl Bbt ) may be nonvanishing and its value reflects the amount of charge shared between A and B due to the interaction of Aat and Bbt . If Aat and Bb1 are bonding LAOs pointing toward one another then q(Aat \ Bb1)

is expected to be large and positive, indicative of the fact that the bond between A and B heavily involves interaction of Aal and Bbl' On the other hand, if either Aat or Bb1 (or both) is a core or lone pair LAO then q(Aatl Bbt) is expected to be small in magnitude, as core and lone pair LAOs are not ordinarily extensively involved in bonding. We define the PrinCipal Bonding Overlap population (PBOP) as the total population of LAO overlap distributions between bonding LAOs on atoms A and B. Usually the PBOP accounts for almost all of the charge shared between A and B. Finally, we expect core LAOs to be doubly occupied and each lone pair LAO to be either unoccupied (Li) or doubly occupied. As we shall see, except for very unusual molecules (C2 and NH) the above predictions are verified lucidly by LAOs.

III. LAOs IN DIATOMIC MOLECULES

LiH (Table I)

Li in LiH shows a core LAO essentially doubly occupied and a- bonding and lone pair LAOs. Both the bonding and lone pair orbitals contain approximately equal Lis and Lipa- contributions and as expected for any Li atom the lone pair LAO is essentially unoccupied. The only significantly populated overlap distribution is between the ba-Li(H) and Hk orbitals, which ~s quite reasonable since it is the interaction of these two orbitals which

TABLE!. LAOs for LiH.

Lik Lis Lipa

Lik 6.72302 0.00910 0.00473 Lis 0.00910 0.22083 0.05252 Lipa 0.00473 0.05252 0.15150

iLi baLi(H) 1 aLi

iLi 6.75832 0.00573 - O. 01253 buLi(H) 0.00573 0.48163 - O. 00271 laLi - 0.01253 -0.00271 0.00715

iLi buLi(H) loU

Uk 0.99802 0.05652 0.02763 Lis - O. 05978 0.71495 0.69661 Lipu -0.01962 0.69689 - O. 71692

iLi baLi(H) luLi Hk

iLi 1.99804 0.00000 0.00000 - 0.00304 buLi(H) 0.00000 0.62407 0.00000 0.39049 luLi 0.00000 0.00000 0.01256 -0.00598 Hk -0.00304 0.39049 -0.00598 0.60242

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TABLE II. LAOs for Li2•

Lik Lis Lipu

Lik 6.75425 0.01677 0.00045 Lis 0.01677 0.29743 0.00994 Lipu 0.00045 0.00994 0.01064

iLi buLi(Li ') 1uLi

iLi 6.78902 - O. 00094 - O. 002 87 buLi(Li') -0.00094 0.32579 -0.01097 1uLi -0.00287 -0.01097 0.03144

iLi buLi(Li ') 1uLi

Lik 0.99798 0.05650 0.02911 Lis -0.06350 0.86630 0.49547 Lipu -0.00278 0.49632 -0.86814

iLi buLi(Li ') 1uLi iLi' buLi '(Li) 1uLi'

iLi 2.00332 0.00000 0.00000 - O. 000 06 - O. 006 48 0.00139 buLi(Li ') 0.00000 0.56654 0.00000 - O. 00648 0.43388 -0.03127 1uLi 0.00000 0.00000 0.07078 0.00139 -0.03127 -0.00171 iLi' -0.00006 -0.00648 0.00139 2.00332 0.00000 0.00000 buLi '(Li) -0.00648 0.43388 -0.03127 0.00000 0.56654 0.00000 1uLi' 0.00139 -0.03127 -0.00171 0.00000 0.00000 0.07078

primarily causes bonding. The PBOP = q(21 4) + q(412) = 0.78098, larger than the population of either the baLi(H) or Hk orbitals.

li2 (Table II)

LAOs for Li in Li2 are similar to those found for Li in LiH. The core orbital is doubly occupied and the lone pair LAO contains very little charge. As before, the only significantly occupied overlap distribution is between the two bonding LAOs, baLi(Li') and baLi'(Li), so that bonding in Li2 may be considered as principally an interaction between these two orbitals. There are also

TABLE III. LAOs for BH.

13k Bs Bpu

Bk 11.71335 0.15360 0.03295 Bs 0.15360 1. 868 38 0.22609 Bpu 0.03295 0.22609 0.80800

iB buB(H) 1uB

iB 11.95205 0.01111 0.04056 buB(H) 0.01111 0.65264 -0.00727 luB 0.04056 - O. 007 27 2.52155

iB buB(H) luB

Bk 0.99206 0.05803 0.11159 Bs -0.12336 0.62190 0.77332 Bpu 0.02452 0.78095 -0.62412

iB buB(H) luB Hk

iB 2.00084 0.00000 0.00000 -0.00087 buB(H) 0.00000 0.55305 0.00000 0.34057 luB 0.00000 0.00000 1. 998 73 -0.02676 Hk -0.00087 0.34057 -0.02676 0.82154

some interesting differences between LiH and Li2 reflecting changes in LAOs as chemical environment is altered. The PBOP=0.86776, larger than in LiH. The increase in the PBOP results in a concomitant decrease in the population of the Li bonding orbital from its LiH value of 0.62407 to 0.56654 in Li2.

BH (Table lit)

As expected, the major difference between the LAOs for B in BH and those for Li in LiH and Li2 is that the laB orbital is doubly occupied whereas the laLi orbitals are unoccupied. The B-H bond is seen to arise primarily because of interaction between the two bonding LAOs, baB(H) and Hk, and the PBOP= 0.68114, smaller than the PBOP for either LiH or Li2•

B2 (Table IV)

The ground state of B2 is 3~;KK(2a,i(2au)2 (hi (tiTY so we here consider the excited 1~;KK(2ag)2('2a.)2(3ag)2 state. The LAOs are remarkably similar in composition to those on B in BH in spite of the fact that B2 is in an excited state. This is to be contrasted to the excited state LAO result for NH which is presented later. Both core and lone pair LAOs are doubly occupied and the B-B' bond is due essentially to interaction of the baB(B') and baB'(B) bonding orbitals with a resulting PBOP =0.67850.

FH (Ref. 9)

This example was treated in detail in our previous publication. LAOs consist of one doubly occupied core LAO, three trigonally equivalent lone pair LAOs, also doubly occupied, and a a bonding LAO. Neither core nor lone pair LAOs are Significantly involved in bonding and PBOP == O. 63908. Let us define B as the angle of separation between any trigonally equivalent LAO and the ac-



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TABLE IV. LAOs for B2(1:E;>'

13k Bs Bpu

13k 11.73357 0.14977 0.04152

Bs 0.14977 1. 840 28 0.26782

Bpu 0.04152 0.26782 1. 045 60

iB buB(B') luB

iB 11. 96201 0.01579 0.04294 buB(B') 0.01579 0.83481 0.00194 luB 0.04294 0.00194 2.61959

iB buB(B') luB

13k 0.99223 0.06089 0.10850

Bs -0.12223 0.64001 0.75859

Bpu 0.02325 0.76596 - O. 642 47

iB buB(B') luB iB' buB '(B) luB'

iB 2.00286 0.00000 0.00000 -0.00002 -0.00185 -0.00031 buB(B') 0.00000 0.69787 0.00000 - 0.00185 0.33925 -0.03797 luB 0.00000 0.00000 2.04123 -0.00031 -0.03797 -0.00089

iB' - O. 00002 -0.00185 -0.00031 2.00286 0.00000 0.00000 buB'(B) - O. 00185 0.33925 -0.03797 0.00000 0.69787 0.00000

1uB' -0.00031 -0.03797 - O. 000 89 0.00000 0.00000 2.04123

companying a-type valence LAO. In FH 0 represents the angle between any ltjF LAO and the baF(H) orbital. If F were pure Sp3 hybridized this angle would be 109.5°. In fact 0 = 103. 9°, less than the tetrahedral value. Hence, each lone pair is closer to the bonding LAO than for Sp3 hybrids and at the same time any two lone pairs are more separated from one another than for Sp3 orbitals. This may be understood using classical arguments. Any given lone pair LAO with a population of about 2 is repelled more by another lone pair LAO (also doubly occupied) than by the bonding LAO which has a population of only O. 844 02. To correct for this lone pair LAOs tend to move away from one another and toward the bonding orbital which causes 0 to be less than the tetrahedral value. In the ensuing discussions, orbital populations, PBOPs, and Os will be compared and contrasted liberally for atoms containing trigonally equivalent LAOs. Table XU, appearing at the end of this section should be consulted frequently as it summarizes the data to be compared.

F2 (Table V)

Like F in FH, F in F2 shows doubly occupied core and trigonally equivalent lone pair LAOs and a a bonding orbital and PBOP = O. 405 82. 0 = 101. 6°, again less than the tetrahedral value and reflective of the differential lone pair-lone pair and lone pair-bonding LAO repulsions. The fact that 0 for F in F 2 is slightly less than 0 for F in FH is a bit puzzling since the baF(H) and baF(F') LAOs are essentially equally populated, and lone pair LAOs are essentially doubly occupied in both molecules, also. However, the PBOP which accounts for most of the internuclear charge, is less in F2 than in FH. As a result, lone pairs should experience somewhat less repulSion from the internuclear region in F2 allowing them to move slightly further away from each other than in FH. Hence, 0 is somewhat smaller in F2•

N2 (Table VI)

Until now we have examined systems classically considered to be singly bonded and indeed each atom has shown a single bonding LAO. N2, however, is classified as a triply bonded species. Accordingly, we find the presence of three "banana" bonding LAOs on each N atom which may be considered to be the major constituents of three "banana" bond LMOs. The PBOP = 6 q(31 8) + 12 q(31 9) = 1. 35510, significantly larger than the PBOPs of the previously encountered singly bonded systems as would befit a triply bonded molecule. In addition, there exist doubly occupied core and a lone pair LAOs on each N atom. Based upon the small magnitudes of populations of overlap distributions involving core and lone pair LAOs it is surmised that these orbitals do not partiCipate to any great extent in bonding. For N2, 0 would be the angle between a btjN(N') LAO and the laN orbital. In light of our previous discussions it is interesting to note that 0 = 115.4°, Significantly larger than the tetrahedral angle. This is just the reverse of what was found in FH and F2 where 0 was found to be smaller than the tetrahedral value. The reversal is easily explained. Whereas in FH and F2 each trigonally equivalent LAO was a doubly occupied lone pair orbital and 0 decreased to compensate for the large interlone pair repulSion, in N2 each trigonally equivalent LAO is a bonding orbital with a population of only 0.78921. The 0.78921 electron in a bonding LAO is repelled more by the two electrons in the ZaN LAO than by the 0.78921 electron in each of the remaining bonding orbitals causing 0 to increase in response. In summary, when trigonally equivalent LAOs serve as lone pair orbitals they tend to separate from one another as compared to Sp3 hybrids whereas just the reverse is true if the orbitals serve as bonding LAOs, which tend to be closer to one another thansp3 hybrids. Obviously, this is very advantageous: as 0 on the Nand N' atoms in-



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TABLE V. LAOs for F2•

Fk Fs Fpu FP~

Fp1i

iF buF(F') ItIF lt2F It3F

Fk Fs Fpu FP~

Fp'ii

iF buF(F')

ltlF lt2F lt3F iF' buF'(F) 1tlF' lt2F' lt3F'

Fk

21. 61194 0.25932 0.04936 0.11239 0.11239

iF

22.00634 0.03496 0.07589 0.07589 0.07589

iF

O. 99~ 65 -0.12068

0.00885 0.00000 0.00000

iF

1. 999 94 0.00000 0.00000 0.00000 0.00000 0..00000 0.00000

-0.00002 -0.00002 -0.00002

Fs

0.25932 3.59387 0.37832 0.78470 0.78470

buF(F')

0.03496 I. 316 22 D.16511 0.16511 0.16511

buF(F')

0.02648 0.28800 0.95727 0.00000 0.00000

buF(F')

0.00000 0.83834 0.00000 0.00000 0.00000 0.00000 0.20291

-0.01124 -0.01124 -0.01l~4

Fpu

0.04936 0.37832 1. 253 71 0.10087 0.10087

ltlF

0.07589 0.16511 4.53447 0.30346 0.30346

ltlF

0.06817 0.54848

- 0.166 90 -0.81650

0.00000

ltlF

0.00000 0.00000 2.01352 0.00000 0.00000

-0.00002 - O. 011-24 -0.00461 -0.00004 - O. 000 04

FP"

O. li2 39 0.78470 0.10087 4.00277 0.21588

Jt2F

0.07589 0.16511 0.30346 4.53447 0.30346

1t2F

0.06817 0.54848

-0.16690 0.40825 0.70711

1t2F

0.00000 0.00000 0.00000 2.01352 0.00000

- O. 00002 -0.01124 -0.00004 -0.00461 -0.00004

Fp1i

0.11239 0.78470 0.10087 0.21588 4.00277

!t3F

0.07589 0.16511 0.30346 0.30346 4.53447

l/3F

0.06817 0.54848

-0.16690 0.40825

- O. 70711

lt3F

0.00000 0.00000 0.00000 0.00000 2.01352

- O. 00002 -0.01124 -0.00004 -0.00004 - O. 00461

creases, the btjN(N') and btjN'(N) LAOs have greater overlap which should increase the bond strength.

CO (Table VII)

Like N2, CO is a triply bonded system with doubly occupied core and (J lone pair LAOs on both the C and 0

TABLE VI. LAOs for N2•

Nk Ns NPu Np" Np1i

iN luN btlN(N') bt2N(W) bI3N(W)

Nk Ns Npa NP" Np1i

iN luN btlN(N') bI2N(N') bI3N(N') iN' luN' btlN'(N) bt2N'(N) bt3N'(N)

Nk

16.74140 0.16249 0.04423 0.03330 0.03330

iN

16.97943 0.04905 0.02409 0.02409 0.02409

iN

0.99441 -0.10411

0.01761 0.00000 0.00000

iN

2.00542 0.00000 0.00000 O. 000 00 0.00000

-0.00003 -0.00087 - O. 000 95 -0.00095 -0.00095

Ns

0.16249 2.47561 0.35801 0.23525 0.23525

luN

0.04905 3.35121 0.11537 0.11537 0.11537

luN

0.08289 0.66633

-0.74105 0.00000 0.00000

luN

0.00000 I. 99126 0.00000 0.00000 Q. 000 00

-0.00087 -0.00821 -0.00435 -0.00435 -0.00435

Npu

0.04423 0.35801 I. 407 44 0.05530 0.05530

btlN(W)

0.02409 0.11537 1.12127 0.05499 0.05499

btlN(N')

0.03777 0.42629 0.38753

-0.81650 0.00000

btIN(N')

0.00000 0.00000 0.78921 0.00000 0.00000

-0.00095 -0.00435

0.28631 -0.03023 -0.03023

Np"

0.03330 0.23525 0.05530 0.91796 0.03688

bI2N(N')

0.02409 0.11537 0.05499 1.12127 0.05499

bI2N(N')

Np1i

0.03330 0.23525 O. 055 30 0.03688 0.91796

bI3N(W)

0.02409 0.11537 0.05499 0.05499 1.12127

bI3N(N')

0.037770.03777 0.42629 0.42629 0.38753 0.38753 0.40825 0.40825 0.70711 - O. 70711

bt2N(N') bt3N(N')

0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.78921 0.00000 0.00000 0.78921

-0.00095 -0.00095 -0.00435 -·0.00435 - O. 03023 -0.03023

0.28631 -0.03023 - O. 030 23 0.28631

iF'

0.00000 0.00000

- O. 00002 - O. 00002 - O. 00002

1. 999 94 0.00000 0.00000 0.00000 0.00000

buF'(F)

0.00000 0.20291

-0.01124 -0.01124 -0.01124

0.00000 0.83834 0.00000 0.00000 0.00000

ltlF'

- O. 00002 -0.01124 -0.00461 -0.00004 - O. 000 04

0.00000 0.00000 2.01352 0.00000 0.00000

112F'

-0.00002 -0.01124 -0.00004 -0.00461 -0.00004

0.00000 0.00000 0.00000 2.01352 0.00000

lI3F'

- O. 00002 -0.01124 - O. 00004 -0.00004 -0.00461

0.00000 0.00000 0.00000 0.00000 2.01352

atoms. However, CO shows polarized triple bonds, a feature obviously not possible in N2• The C atom bonding LAOs contain a total population of 1.31115, considerably smaller than the corresponding value of 3.69846 for the 0 atom bonding LAOs. The PBOP = 1. 097 70, smaller than the PBOP for N2, probably because the

iN'

- O. 00003 - O. 000 87 -0.00095 -0.00095 - O. 000 95

2.00542 0.00000 0.00000 0.00000 0.00000

lUN'

- O. 000 87 -0.00821 -0.00435 -0.00435 -0.00435

0.00000 I. 99126 0.00000 0.00000 0.00000

btlN'(N)

-0.00095 -0.00435

0.28631 -0.03023 -0.03023

0.00000 0.00000 0.78921 0.00000 0.00000

bI2N'(N)

-0.00095 -0.00435 -0.03023

0.28631 -0.03023

0.00000 0.00000 0.00000 0.78921 0.00000

bI3N'(N)

-0.00095 -0.00435 -0.03023 -0.03023

0.28631 0.00000 O. 000 00 0.00000 0.00000 0.78921



205.208.120.231 On: Sun, 30 Nov 2014 10:31:52

TABLE VIT. LAOs for CO.

Ck Cs Cpu cprr

Ck 14.23515 0.15211 0.03199 0.01662 Cs 0.15211 2.10197 0.25490 0.10501 Cpu 0.03199 0.25490 0.90768 0.02311 cprr 0.01662 0.10501 0.02311 0.38204 Cp1i 0.01662 0.10501 0.02311 0.01120

iC luG btlC(O) bt2C(0)

!- iC 14.46611 0.04342 0.01194 0.01194 0 luG 0.04342 2.86337 0.05188 0.05188 :T

btlC(O) <D 0.01194 0.05188 0.52884 0.01711 ~ bt2C(0) 0.01194 0.05188 0.01711 0.52884 'll :T bt3C(0) 0.01194 0.05188 0.01711 0.01711 -< r < iC luG btlC(O) bt2C(0) ~ ..... Ck 0.99371 0.09442 0.034 77 0.03477 .w

Cs - 0.10980 0.71578 0.39817 0.39817 z !:l Cpu 0.02201 -0.69193 0.41665 0.41665 .~ cprr 0.00000 0.00000 -0.81650 0.40825 - Cp1i 0.00000 0.00000 (11 0.00000 0.70711 » c:

<C c: iC luG btlC(O) bt2C(0) '" ... - iC 2.00633 0.00000 0.00000 0.00000 CD 00 luG 0.00000 1. 95147 0.00000 0.00000 0

bt1C(0) 0.00000 0.00000 0.43705 0.00000 bt2C(0) 0.00000 0.00000 0.00000 0.43705 bt3C(0) 0.00000 0.00000 0.00000 0.00000 iO -0.00003 -0.00037 -0.00032 - O. 00032 1uO -0.00143 -0.00666 -0.00247 -0.00247 btIO(C) -0.00111 0.00059 0.23729 -0.02717 bt20(C) -0.00111 0.00059 -0.02717 0.23729 bt30(C) -0.00111 0.00059 -0.02717 -0.02717

Cp1i Ok Os

0.01662 Ok 19.18702 0.20236 0.10501 Os 0.20236 3.00387 0.02311 Opu 0.06418 ,0.49181 0.01120 oprr 0.06197 0.43721 0.38204 op1i 0.06197 0.43721

bt3C(0) iO 1uO

0.01194 iO 19.48290 0.05903 0.05188 1uO 0.05903 3.82383 0.01711 btlO(C) 0.04370 0.20127 0.01711 bt20(C) 0.04370 0.20127 0.52884 bt30(C) 0.04370 0.20127

bt3C(0) iO 1uO

0.03477 Ok 0.99385 0.07686 0.39817 Os -0.10992 0.59572 0.41665 opu -0.01364 0.79951 0.40825 Oprr 0.00000 0.00000

-0.70711 op1i 0.00000 0.00000

bt3C(0) iO 1uO btlO(C)

0.00000 -0.00003 -0.00143 -0.00111 0.00000 -0.00037 - O. 00666 0.00059 0.00000 - O. 00032 -0.00247 0.23729 0.00000 - O. 00032 -0.00247 -0.02717 0.43705 - O. 000 32 -0.00247 -0.02717

-0.00032 2.00202 0.00000 0.00000 -0.00247 0.00000 1. 970 00 0.00000 -0.02717 0.00000 0.00000 1.23282 -0.02717 0.00000 0.00000 0.00000

0.23729 0.00000 0.00000 0.00000

opu oprr

0.06418 0.06197 0.49181 0.43721 2.05614 0.10143 0.10143 1. 859 88 0.10143 0.08853

btlO(C) bt20(C)

0.04370 0.04370 0.20127 0.20127 2.12791 0.13084 0.13084 2.12791 0.13084 0.13084

btlO(C) bt20(C)

0.04605 0.04605 0.45936 0.45936

-0.34670 -0.34670 -0.81650 0.40825

0.00000 0.70711

bt20(C) bt30(C)

-0.00111 -0.00111 0.00059 0.00059

-0.02717 -0.02717 0.23729 -0.02717

-0.02717 0.23729 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 1. 232 82 0.00000 0.00000 1. 232 82

Oprr

0.06197 0.43721 0.10143 0.08853 1. 859 88

bt30(C)

0.04370 0.20127 0.13084 0.13084 2.12791

bt30(C)

0.04605 0.45936

-0.34670 0.40825

-0.70711

?' I

l> c: -c. (D ... ';j (D

c. (D

III ::l c.

~ () ';j c: ::l

<C

~ ';j

'0 en

r 0 0 III

N' (D

c. III ... 0 3 o· 0 ... C" ;:t. III Ui"

-..... CD (11

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 205.208.120.231 On: Sun, 30 Nov 2014 10:31:52

TABLE vm. LAOs for BF.

Fk Fs Fpu Fprr Fp1i

Fk 21. 63124 0.24613 0.08432 0.09684 0.09684

Fs 0.24613 3.50871 0.62264 0.67419 0.67419 Fpu 0.08432 0.62264 2.71474 0.15515 0.15515 Fprr 0.09684 0.67419 0.15515 3.21576 0.16762 Fp7i 0.09684 0.67419 0.15515 0.16762 3.21576

iF luF btlF(B) bt2F(B) bt3F(B)

~ iF 21.99620 0.06902 0.06584 0.06584 0.06584 (") luF 0.06902 4.14953 0.29102 0.29102 0.29102 :::r CI> 3.46736 ? btlF(B) 0.06584 0.29102 0.23424 0.23424

" bt2F(B) 0.06584 0.29102 0.23424 3.46736 0.23424 :::r

bt3F(B) 0.06584 0.29102 0.23424 0.23424 3.46736 < !' < £. iF 1uF btlF(B) bt2F(B) bt3F(B) ......

Fk 0.99318 0.06842 0.05450 0.05450 0.05450 ,c.J

Z Fs - 0.116 21 0.51651 0.48980 0.48980 0.48980 0

,'" Fpu 0.00929 - O. 853 55 0.30076 0.30076 0.30076

- Fprr 0.00000 0.00000 -0.81650 0.40825 0.40825 01 Fp'if 0.00000 0.00000 0.00000 0.70711 - O. 70711 » <:

<C <: iF luF btlF(B) bt2F(B) bt3F(B) U) ... -U) iF 2.00008 0.00000 0.00000 0.00000 0.00000 00 0 1uF 0.00000 1. 91616 0.00000 0.00000 0.00000

bt1F(B) 0.00000 0.00000 1. 655 78 0.00000 0.00000 bt2F(B) 0.00000 0.00000 0.00000 1. 655 78 0.00000 bt3F(B) 0.00000 0.00000 0.00000 0.00000 1.65578 iB -0.00001 - O. 00128 -0.00034 - O. 00034 -0.00034 luB -0.00010 -0.00303 0.00341 0.00341 0.00341 btlB(F) -0.00002 0.00293 0.12280 -0.01440 -0.01440 bt2B(F) -0.00002 0.00293 -0.01440 0.12280 -0.01440 bt3B(F) - 0.00002 0.00293 - O. 01440 - O. 01440 0.12280

Bk Bs

Bk 11. 718 86 0.15313 Bs 0.15313 1. 85929 Bpu 0.02260 0.17291 Bprr 0.00517 0.02923 Bp'if 0.00517 0.02923

iB 1uB

iB 11. 960 00 0.03804 luB 0.03804 2.37079

btlB(F) 0.00416 0.00952 bt2B(F) 0.00416 0.00952 bt3B(F) 0.00416 0.00952

iB 1uB

Bk 0.9921l 0.11055 Bs -0.12252 0.75511 Bpu - O. 026 55 0.64623 Bprr 0.00000 0.00000 BpTi 0.00000 0.00000

iB 1uB btlB(F)

- O. 00001 -0.00010 - 0.00002 -0.00128 -0.00303 0.00293 -0.00034 0.00341 0.12280 -0.00034 0.00341 - O. 01440 - O. 00034 0.00341 -0.01440

2.00235 0.00000 0.00000 0.00000 1. 895 91 0.00000 0.00000 0.00000 0.20913 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

Bpu Bprr

0.02260 0.00517 0.17291 0.02923 0.54918 0.00563 0.00563 0.09320 0.00563 0.00148

btlB(F) bt2B(F)

0.00416 0.00416 0.00952 0.00952 0.21634 0.00605 0.00605 0.21634 0.00605 0.00605

btlB(F) bt2B(F)

0.03414 0.034 14 0.37186 0.37186

- 0.44035 - 0.44035 -0.81650 0.40825

0.00000 0.70711

bt2B(F) bt3B(F)

-0.00002 -0.00002 0.00293 0.00293

-0.01440 - O. 01440 0.12280 - O. 01440

-0.01440 0.12280 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.20913 0.00000 0.00000 0.20913

BpTi

0.00517 0.02923 0.00563 0.00148 0.09320

bt3B(F)

0.00416 0.00952 0.00605 0.00605 0.21634

bt3B(F)

0.03414 0.37186

- 0.44035 0.40825

-0.70711

...... U) (l)

?" :I

» c ..... 0-CD .... :;;T CD

0.: CD Q) ;j Co

?> () :;;T c ;j

'-9 "tI =.

"0 !'! r 0 (") Q)

N' CD 0-Q)

g 3 o· 0 .... CJ ;::+. Q)

!i'


TABLE IX. LAOs for LIF.

Fk Fs Fpu FP1T

Fk 21. 63192 0.24230 0.09987 0.09286 Fs 0.24230 3.38046 0.69502 0.64364 Fpu 0.09987 0.69502 3.38248 0.16765 FP1T 0.09286 0.64364 0.16765 2.98128 Fpi 0.09286 0.64364 0.16765 0.15515

iF 1uF btlF(Li) bt2F(Li) c.... (")

iF 21. 99177 0.07600 0.06561 0.06561 ::r 1uF 0.07600 4.57183 0.28554 0.28554 .. ? btlF(Li) 0.06561 0.28554 3.36496 0.24021 "tI bt2F(Li) 0.06561 0.28554 0.24021 3.36496 ::r < bt3F(LI) 0.06561 0.28554 0.24021 0.24021 ~III

< 0

bt1F(Li) bt2F(Li)· :- iF 1uF "-J ,w Fk 0.99326 0.06865 0.05391 0.05391 z

Fs -0.11565 0.53474 0.48328 0.48328 ? ,~ Fpu 0.00753 - O. 84222 0.31124 0.31124 - Fp1T 0.00000 0.00000 - O. 816 50 0.40825 C11

» FpTi 0.00000 0.00000 0.00000 0.70711 <: co <: III iF 1uF bt1F(Li) bt2F(Li) ... <D

1. 999 71 (I) iF 0.00000 0.00000 0.00000 0

1uF 0.00000 2.05681 0.00000 0.00000 btlF(Li) 0.00000 0.00000 1. 64135 0.00000 bt2F(Li) 0.00000 0.00000 0.00000 1. 64135 bt3F(Li) 0.00000 0.00000 0.00000 0.00000 iLl -0.00002 -0.00207 -0.00133 -0.00133 luLl 0.00000 0.00093 - O. 00025 -0.00025 btl Li(F) 0.00001 -0.01714 0.12313 -0.01088 bt2L1(F) 0.00001 -0.01714 -0.01088 0.12313 bt3Ll(F) 0.00001 -0.01714 -0.01088 -0.01088

FpTi Lik Lis

0.09286 Lik 6.71905 -0.00912 0.64364 Lis -0.00912 0.04255 0.16765 Llpu 0.00171 0.01493 0.15515 Llp1T 0.00374 0.00352 2.98128 Lipi 0.00374 0.00352

bt3F(Li) iLi luLl

0.06561 iLl 6.73699 -0.01517 0.28554 1uLi -0.01517 0.00484 0.24021 btl Li(F) 0.00098 0.00008 0.24021 bt2Li(F) 0.00098 0.00008 3.36496 bt3Li(F) 0.00098 0.00008

bt3F(Li) iLl 1uLi

0.05391 Lik 0.99884 0.02733 0.48328 Lis -0.04690 0.73915 0.31124 Lipu 0.01095 0.67299 0.40825 Llp1T 0.00000 0.00000

-0.70711 LlpTi 0.00000 0.00000

bt3F(Li) iLi 1uLi bt1LI(F)

0.00000 -0.00002 0.00000 0.00001 0.00000 - O. 002 07 0.00093 -0.01714 0.00000 -0.00133 -0.00025 0.12313 0.00000 - O. 00133 - O. 00025 -0.01088 1. 64135 -0.00133 -0.00025 -0.01088

-0.00133 1. 998 63 0.00000 0.00000 - 0.00025 0.00000 0.00654 0.00000 -0.01088 0.00000 0.00000 0.17358 -0.01088 0.00000 0.00000 0.00000

0.12313 0.00000 0.00000 0.00000

Lipu Llp1T

0.00171 0.00374 0.01493 0.00352 0.04623 0.00200 0.00200 0.10464 0.00200 0.00159

btlLi(F) bt2Li(F)

0.00098 0.00098 0.00008 0.00008 0.11740 0.00038 0.00038 0.11740 0.00038 0.00038

btlLi(F) bt2Li(F)

0.02290 0.02290 0.38793 0.38793

- 0.426 99 -0.42699 -0.81650 0.40825

0.00000 0.70711

bt2Li(F) bt3LI(F)

0.00001 0.00001 -0.01714 -0.01714 -0.01088 -0.01088

0.12313 -0.01088 -0.01088 0.12313

0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.17358 0.00000 0.00000 0.17358

Lip1T

0.00374 0.00352 0.00200 0.00159 0.10464

bt3LI(F)

0.00098 0.00008 0.00038 0.00038 0.11740

bt3LI(F)

0.02290 0.38793

-0.42699 0.40825

-0.70711

A

;r: » c -a. CD ..., ::r CD c: CD Q)

::J a. » (") ::r c ::J "? ." ::r

"0 II)

r 0 n I»

N' CD a. Q) r+ 0

~. n 0 ..., 0-~: I»

ii'

-"-J <D "-J


1798 K. H. Aufderheide and A. Chung·Phillips: Localized atomic orbitals. II

polar situation tends to concentrate charge more on the o atom than in the internuclear region. 9 = 117.0° for carbon, larger than 9 for N in N2 because the btjC(O) orbitals contain less charge (0.43705 electron each) than the btjN(N') LAOs, allowing the former to approach one another more closely than the latter. Similar ly, each btjO(C) LAO contains more charge (1.23282 each) than does a btjN(N') orbital so it is not surprising that 9 = 113. 0° for 0 in CO, smaller than 9 for N in N2 but still larger than the tetrahedral value. It should be noted that the experimental CO dipole is found to be C-O· whereas the single-determinental ab initio result yields C·O-. Hence, the polarity (or, at least, the degree of polarization) of the banana bonds as suggested by LAOs is questionable since LAOs depend upon and reflect the calculated electronic structure.

SF (Table VIII)

Until now LAOs have predicted classical hybridizations: the functional partitioning of LAOs into core, lone pair, and bonding sets has followed Lewis structure guidelines. BF is an exception. The Lewis structure of this system suggests a valence arrangement of a (J lone pair on B, three equivalent lone pairs on F, and a single bond between the two atoms. As did LMOs, the LAO procedure offers an alternative description. LAOs consist of doubly occupied core and (J lone pair orbitals on both Band F with a triple bond connecting the two atoms. The nature of the triple bond is quite important. First, the bonds are much more polarized toward F in BF than they are toward 0 in CO. For example, each btjB(F) LAO contains 0.20913 electron whereas the btjC(O) orbitals each have a population of 0.43705. Similarly, each btjF(B) orbital contains 1. 655 78 electrons while a btjO(C) LAO has a population of only 1. 232 82. The extensive polarity necessarily leads to a small PBOP of 0.56400 which is much less than the PBOPs of the typical triply bonded systems N2 and CO, and much closer in magnitude to the PBOPs of singly bonded molecules like FH and F2• Finally 9 for B in BF is 118.3° while 9 for Fin BF is 110.2°, the latter being quite close to the tetrahedral value of 109.5°. Obviously, the great disparity in 9 for Band F in BF may be attributed to the fact that btjB(F) LAOs are only slightly populated whereas btjF(B) orbitals are quite considerably populated. The disparity also implies that the bonding LAOs on B and F do not overlap strongly, causing the B-F triple bond to be weaker than either the N-N' or the C -0 triple bonds. From these considerations14 we might surmise bonding in BF to be less strong than that in typical triply bonded systems. The same conclusion was arrived at in ER2 where it was pointed out that the binding energy of BF was greater than that of similar singly bonded systems but less than that of either N2 or CO. It was also suggested in ER2 that the l(JF orbital is significantly involved in bonding. This is not verified for LAOs, as the l(JF LAO is seen to form no well-populated overlap distributions.

LiF (Table IX)

LiF, like BF, is hybridized in an unusual fashion. Both Li and F possess core and (J lone pair LAOs, all of

which are approximately doubly occupied except the l(JLi orbital whi«h is essentially unoccupied as it should be. Once more we find a triple bond where we would expect a single bond. In some ways the LiF triple bonds are even more polar than the BF ones: while the F bonding LAOs are about equally populated in both molecules, the PBOP = 0.60822 in LiF larger than that of BF while at the same time the Li bonding LAOs are less populated than the B bonding orbitals. This indicates more charge has been transferred from Li to the bonding region in LiF than was transferred from B in BF. Again, the PBOP is more similar in magnitude to a singly bonded system than to a triply bonded one. Finally, 9 for Li is 117.6°, large like B in BF while 9 for F equals 110.9°, close to the tetrahedral value just like F in BF. Once more, the disparity in 9 between F and Li decreases overlap of the btjLi(F) and btjF(Li) orbitals which should result in weak triple bonds. This triply bonding character of LiF is even more striking than it was for BF which is at least isoelectronic with the typical triply bonded systems N2 and CO.

C2 (Table X)

C2 is an especially important result because LMOs for this molecule are quite peculiar. Indeed, the LMO valence arrangement is almost impossible to visualize. On the other hand, LAOs yield a very lucid and understandable valence picture for C2• Lewis structure considerations suggest C2 to be a triply bonded system with a singly occupied "lone pair" LAO on each C atom. In fact, we find each C atom to possess a doubly occupied core LAO, a (J "lone pair" LAO containing only 0.37586 electron and three trigonally equivalent bonding LAOs each with a population of about one electron. This is consistent with Lewis structure arguments except as regards the lone pair electrons which are "smeared out, " as opposed to being localized, one per atom. Upon further examination certain other peculiarities emerge. The PBOP = 0.55728, more comparable to singly bonded systems than to triply bonded molecules. Furthermore, 9 = 99. 9°. This is extremely striking, as it represents the only example of a triply bonded molecule in which 9 is less than the tetrahedral value! The small 9 indicates that the bonding LAOs are being strongly repelled from some sort of additional charge in the internuclear region beyond that found in the bonding LAOs and their distributions. This extra charge is seen to arise from the bonding character of the "lone pair" LAOs. The "lone pair" LAO on one C atom interacts strongly with the bonding LAOs on the other C atom giving rise to a total lone pair-bonding LAO overlap population of 12 q(218) = O. 836 76, which is larger than the PBOP! Apparently, the tendency to share a lone pair between two atoms allows for strong bonding interaction between the lone pair LAO on one atom and the bonding orbitals on the other. This causes appreciable bonding charge to build up in the internuclear region which, in turn, repels electrons in the actual "bonding LAOs" causing 9 to decrease dramatically. The small 9 creates little overlap between the btjC(C') and btjC'(C) orbitals so that the PBOP is quite small. However, C2 should probably be considered to be more strongly bonded than weak triply bonded sys-



205.208.120.231 On: Sun, 30 Nov 2014 10:31:52


TABLE X. LAOs for C2•

Ck Cs Cpa cprr cprr

CI, 14.2504ti 0.14314 0.00854 0.03202 0.03202

Cs 0.14314 1. 799 17 0.05288 0.19494 0.19494 Cpa 0.00854 0.05288 0.171 ti7 0.01797 0.01797

CP" 0.03202 0.19494 0.01797 0.80989 0.03253 CpTf 0.03202 0.19494 0.01797 0.03253 0.80989

iC laC btlC(C ') bt2C(C ') bt3C(C ')

iC 14.46278 0.01034 0.02614 0.02614 0.02614 laC 0.01034 0.35393 0.03544 0.03544 0.03544 htlC(C ') 0.02614 0.03544 1. 261 00 0.05058 0.05058 ht2C(C ') 0.02614 0.03544 0.05058 1. 261 00 0.05058 bt3C(C ') 0.026 J.l 0.03544 0.05058 0.05058 1. 261 00

iC laC btlC(C ') bt2C(C ') bt3C(C')

CII 0.99441 0.03252 0.05800 0.05800 0.05800 Cs -0.10538 0.24532 0.55639 0.55639 0.55639 Cpa 0.00669 -0.96890 0.14282 0.14282 0.14282 cprr 0.00000 0.00000 -0.81650 0.40825 0.40825 cp'ii 0.00000 0.00000 0.00000 0.70711 -0.70711

iC IrrC btlC(C ') bt2C(C ') bt3C(C ') ie' laC ' btlC '(C) bt2C'(C) bt3C '(C)

iC 2.00594 0.00000 0.00000 0.00000 0.00000 - O. 000 06 - O. 00337 - O. 000 36 - O. 00036 - O. 00036 laC 0.00000 0.37586 0.00000 0.00000 0.00000 - O. 00337 -0.04218 0.06973 0.06973 0.06973 btlC(C ') 0.00000 0.00000 0.99078 0.00000 0.00000 - O. 00036 0.06973 0.17868 - O. 042 90 - 0.042 90 bt2C(C ') 0.00000 0.00000 0.00000 0.99078 0.00000 - O. 00036 0.06973 - O. 042 90 0.17868 -0.04290 bt3C(C ') 0.00000 0.00000 0.00000 0.00000 0.99078 - O. 00036 0.06973 - O. 042 90 - O. 042 90 0.17868 ie' -0.00001i - O. 003 37 -0.00036 - O. 00036 -0.00036 2.00594 0.00000 0.00000 0.00000 0.00000 IrrC ' - O. 003 37 -0.04218 0.06973 0.06973 0.06973 0.00000 0.37586 0.00000 0.00000 0.00000 btlC '(C) -0.00036 0.06973 0.17868 -0.04290 -0.04290 0.00000 0.00000 0.99078 0.00000 0.00000 bt2C '(C) -0.00036 0.06973 - O. 042 90 0.17868 - O. 042 90 0.00000 0.00000 0.00000 0.99078 0.00000 bt3C '(C) - O. 000 36 0.06973 -0.04290 - O. 042 90 0.17868 0.00000 0.00000 0.00000 0.00000 0.99078

tem" like BF and LiF which like C2 have a small PBOP but possess no "extra" bonding charge due to bonding contributions from "lone pair" electrons. Experimentally, it is found that the C-C force constant for C2 (9. 5X 105 dynes/cm)15(&) is far closer in magnitude to the C -C force constant in a doubly bonded system like ethylene (9. 6X105 dynes/cm)15(b) than to the value for a singly bonded molecule such as ethane (4. SOX 105 dynes/ cm).15(b) Interestingly, the C-C force constant for acetylene (15. 6x 105 dynes/cm)15(b) suggests the bonding in C2 to be less strong than for a typical C -C triple bond. Further, some recent calculations16 of a fairly sophisticated nature suggested the bond order of C2 to be about 2.6_ Apparently, the augmentation of the weak triple bonds in C2 by "lone pair" bonding interactions results in a system having C -C bond strength somewhere between that of typical C -C double apd triple bonds.

NH (Table XI)

Since NH has a 3:EXK'(3a)2(h)1(17T)1 ground state we consider the excited l:E+ KK'(h)2(17T)2 state. Like C2,

NH presents an unusual valence arrangement. On the N atom there is a doubly occupied core LAO, a a lone pair LAO which is essentially unoccupied and three trigonally equivalent bonding LAOs. Since the H atom does not possess three trigonally equivalent bonding LAOs, we cannot really claSSify NH as a triply bonded system_ Rather, the bonding seems to represent a compromise between single and triple bonds. The PBOP = O. 56448, similar to most singly bonded molecules. 8=97.1°, very much smaller than the tetrahedral value and indicative of the extremely small btjN(H) and Hk overlap so that the "pseudo" triple bond should be weak compared

to N2 or CO. Most likely, it seems that NH being a hydride like LiH, BH, and FH would prefer to be a singly bonded species. However, because of the peculiar electronic structure of the given excited l:E + state this is impossible so the next best alternative is selected: a set of "triple bonds" emerge from N which gradually coalesce into essentially a single bond in the region around the H atom.

IV. COMPOSITION OF LAOs

Whenever trigonally equivalent LAOs were evidenced on an atom we found it useful to define the angle 8 which was shown to vary in a meaningful manner from system to system. To interpret 8 we simply assumed that the trigonally equivalent LAOs and the accompanying a valence LAO were pure Sp3 hybridized. Next, we populated these orbitals with the charges which they are known to hold in the molecule under consideration. Finally, the orbitals were unfrozen so that they could adjust to the differential repulsions from one another which causes 8 to either increase or decrease from 109.5°. This method which was the implicit basis for our interpretation of f) variations proved to be quite useful. It is obvious that f) depends on the relative contributions of the Apa, Aprr, and Ap1i orbitals to the LAOs under consideration so that f) is a particular manifestation of the more general concept of LAO composition which is subsequently discussed in detail.

We shall find it instructive to employ the population ratio (PR) parameter, as introduced in ER2. For any neutral atom A in a particular valence arrangement pictured in Fig. 1, the PR is the ratio of the number of bonding electrons to the number of lone pair electrons.



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TABLE Xl. LAOs for NH(lL;+l.

Nk Ns Npa Nprr

Nk 16.70619 0.11021 0.00175 0.08299 Ns 0.11021 1. 473 52 0.03506 0.39936 Npa 0.00175 0.03506 0.05153 0.00521 Nprr 0.08299 0.39936 0.00521 3.05122 Np7i 0.08299 0.39936 0.00521 0.16447

iN IaN btlN(H) bt2N(H)

iN 16.86776 - O. 00143 0.05381 0.05381 IaN -0.00143 0.00046 0.00032 0.00032 btlN(H) 0.05381 0.00032 2.86940 0.18494 bt2N(H) 0.05381 0.00032 0.18494 2.86940 bt3N(H) 0.05381 0.00032 0.18494 0.18494


Nk 0.99574 0.00810 0.05311 0.05311 Ns -0.09199 0.17495 0.56596 0.56596 Npa -0.00815 -0.98455 0.10101 0.10101 Nprr 0.00000 0.00000 -0.81650 0.40825 Np7i 0.00000 0.00000 0.00000 0.70711


iN 1. 997 57 0.00000 0.00000 0.00000 IaN 0.00000 0.00025 O. 00000 0.000 00 btlN(H) 0.00000 0.00000 1.73609 0.000 00 bt2N(H) 0.00000 0.00000 0.00000 1. 736 09 bt3N(H) 0.00000 0.00000 O. 000 00 0.00000 Hk -0.00057 0.00119 0.09408 0.094 08

PRs for all the atoms in all valence configurations studied here are listed in Table XIII.

Composition of core LAOs

Any core LAO is of the form

iA = chAk + csAs + caApa (6)

where Ck' Cs> and ca are determined by the LAO procedure. These coefficients are compiled in Table XIII along with the coefficients for free atom core LAOs. The following features are evident:

Np7i

0.08299 0.39936 0.00521 0.16447 3.05122

bt3N(H)

0.05381 0.00032 0.18494 0.18494 2.86940

bt3N(H)

0.05311 0.56596 0.10101 0.40825

-0.70711

bt3N(H) Hk

O. 00000 -0.00057 0.000 00 0.00119 0.00000 0.09408 0.00000 0.09408 1. 736 09 0.09408 0.09408 0.22839

(1) For a given atom A in a series of molecules, the ck and Cs are quite similar in both sign and magnitude, suggesting that core localization is rather insensitive to chemical environment for a particular atom. The ca values are less similar but because of their small magnitudes these affect the analysis almost negligibly.

(2) There is not necessarily a great similarity between the values of the coefficients for a free atom and those for the same atom in a series of molecules. For example, the ck and Cs values for Li in LiH, Li2, and LiF are all quite similar to each other but markedly

TABLE XII. Summary of important comparative data for atoms possessing trigonally equivalent LAOs. pt=population of one trigonally equivalent LAO; pa=population of the accompanying a valence LAO. Above the blank line the trigonally equivalent LAOs are lone pair orbitals and the a LAO is a bonding orbital. Below the blank line the reverse is true.

Constituent Molecule PBOP non-H atom(s) fit pa 0(°)

FH 0.63908 F 2.05743 0.84402 103.9

F2 0.40582 F 2.01352 0.83834 101. 6

N2 1. 355 10 N 0.78921 1. 99126 115.4

CO 1. 09770 C 0.43705 1. 95147 117.0 0 1. 23282 1. 970 00 113.0

BF 0.56400 F 1. 655 78 1. 91616 110.2

B 0.20913 1. 895 91 118.3

LiF 0.60822 F 1.64135 2.05681 110.9 Li 0.17358 0.00654 117.6

C2 0.55728a C 0.990 78 0.37586 99.9a

NH 0.56448 N 1. 73609 0.00025 97.1

"see discussion in text.



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K. H. Aufderheide and A. Chung-Phillips: Localized atomic orbitals. II laOl

TABLE XID. Composition of core LAOs. For all the free atoms A, the isolated Ak and As orbit-als were localized. Under "polarization" the symbols are: A indicates the core LAO is polarized away from the other atom in the molecule and T indicates the core LAO is polarized toward the other atom in the diatomic species. The polarization is determined by comparing the sign of the Apu coefficient with the position of A in Fig. 1.

AtOm Species PR Ak As

Li Li 0.99312 -0.11707 LiH 00 0.99802 -0.05978 Li2 00 0.99798 -0.06350 LiF 00 0.99884 -0.04690

B B 0.99107 -0.13335 BH 1/2 0.99206 -0.12336 B2(12:;) 1/2 0.99223 -0.12223 BF 1/2 0.99211 -0.12252

C C 0.99263 -0.12120 C2 3 0.99441 -0.10538 CO 1 0.99371 -0.10980

N N 0.99353 - 0.113 59 NH(12:1 0.99574 -0.09199 N2 3/2 0.99441 - 0.10411

0 0 0.99342 -0.11454 CO 2 0.99385 -0.10992

F F 0.99333 -0.11527 FH 1/6 0.99279 -0.11918 LiF 5/2 0.99326 -0.11565 BF 5/2 0.99318 -0.11621 F2 1/6 0.99265 - 0.12068

different than the coefficients for a free Li atom. This implies that hybridization promotion of core LAOs is non-negligible but is approximately constant for a given atom in any molecule. Finally, the discrepancy between the free atom coefficients and the coefficients for atoms in mOlecu.les seems to diminish in gOing from Li to F, indicative that hybridization promotion effects decrease as the atomic number increases. This might be expected based upon shielding effects; that is, as the number of valence electrons of a neutral atom increases, the less perturbed should be the core electrons upon bond formation.

(3) The PR which is defined in terms of the valence arrangement is not expected to correlate with the coefficients of core LAOs and indeed, the coefficients are seen to be rather insensitive to this parameter. For example, c.- is quite comparable for Fin FR, LiF, BF, and F 2 even though F in FH and F 2 has PR =! while F in LiF and BF has PR = ~. This would be expected based upon (1) above. Interestingly, there does seem to be some correlation between the (extremely tiny) polarization of iA and PR: when PR = "", iA is polarized away from the other atom in the molecule, whereas just the reverse is true when PR remains finite. Again, because of the quite small magnitudes of the c" the effect is minute.

Composition of Valence LAOs

Valence LAOs are of two kinds: (a) a-type LAOs, va, which have the form

va = vkAk + V8As + v"Apa; v~ + v; + v; = 1 • (7)

Apu Polarization

-0.01962 A - O. 002 78 A

0.01095 A

0.02452 T 0.02325 T

-0.02655 T

0.00669 T 0.02201 T

-0.00815 A 0.01761 T

-0.01364 T

0.01312 T 0.00753 T 0.00929 T 0.00885 T

Included are baA(B) and laA LAOs. There are also (b) trigonally equivalent valence LAOs, vtj (j = 1, 2, 3), which have the form

~ - ..f'll'J AP11; j = 1

vtj=a+ 1/~AP11+1/$AP:; j=2

1/~ AP11 -1/$ AP11; j = 3 where

a= tkAk + t&As + t"Apa; t~ + t; + t~= t .

(8)

(9)

Relevant LAOs would be ltjA and btjA(B) (j = 1,2,3).

In discussing the composition of valence LAOs it is obvious that the AP11 and Ap1i components are of no interest, since their contributions are completely determined once the valence situation is known: any a-type LAO has no 1r contribution while any trigonally equivalent set of valence LAOs has a well-defined, fixed 1r composition. Thus, the problem is reduced to a study of Ak, As, and Apa contributions. Following ER2 let us define the normalized sigma, Ncr composition of LAOs as follows:

~ vkAk + vsAs + v"Apa for va LAOs

Na= {..ff (t.-Ak + tsAs + t"Apa) for vtj LAOs' (10)

The factor of ..ff for vtj LAOs is a normalization constant which assures that (Nal Na) = 1 in all instances. The normalized fraction of s character of an LAO, fs is defined directly from Na as

lv~ + v; = 1 - v! for va LAOs

fs = 3(~ + t;) = 1 - 3t! for vtj LAOs . (11)



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1.00~--------------------------------------~

0.90

0.80

0.70

0.80

0.50

0.40

0.30

0.20

0.10

1/6 1/2 1 3/2 2 5/2 POPULATION RATIO

FIG. 2. Plot of PH versusf~ (e) andf!(6j, C2 is not included because it is uncertain as to the correct value of PR: recall that almost 2/3 electron is shared between a "lone pair" LAO on one C atom and the bonding LAOs on the other C atom in C2. It is not clear how much of this 2/3 electron should be assigned to a bonding LAO and how much should be allocated to the "lone pair" LAO in the neutral atom. Conceivably, PH could fall anywhere between 3 and 00; however, the fact that r. = O. 938 and f!= 0.061 does suggest that PH is larger than 3.

Whenf. refers specifically to a bonding LAO we shall use the symbolf! while f! will indicate specifically the fraction of s character of a lone pair LAO. When fs = 1 the N(J portion of the LAO has all s character and no p(J character and when fs = 0 the N(J part of the LAO has no s character and all p(J character. In Fig. 2 we plot as did ER2 PR versus f~ and f!. Several features are noticeable.

(1) When an atom has the same PR in a series of molecules the compositions of the valence LAOs are remarkably similar in all systems. For example, B in BF shows trigonally equivalent bonding LAOs and a (J lone pair LAO whereas B in BH and B2 evidence both (J bonding and lone pair LAOs. While the bonding picture is different in BF than in BH and B2, B has the same PR in all molecules and indeedf!andf! are very similar for B in all three species. The same phenomenon is noted for F in HF and F 2 (PR =~) and again for F in LiF and BF (PR=~). It was shown in ER2 that ERAOs also exhibit this tendency; however, it is much more pronounced for LAOs than for ERAOs. For example, f~ for Bin BH, B2, and BF runs from 0.390 to 0.418 (range = 0.028) for LAOs but spans the region 0.135 to 0.257 (range = O. 122) for ERAOs. Quite evidently the compositions of LAOs are tremendously more similar than those of ERAOs for a given value of PR. This indicates that LAOs are far more transferable than ERAOs and this posit shall be further explored in a forthcoming publication.

(2) When PR = 0 the valence arrangement consists of a single lone pair and no bonding electrons. It is to be expected in this event that the lone pair LAO contains only s character U! = 1) which in turn requires the unoccupied bonding LAO to be pure p(J in composition U!

=0). This is shown to be the case in Fig. 2. As PR increases from zero the bonding LAOs show an increase in s character while the lone pair LAOs evidence a similar decrease in s character. The same gross trends were found for ERAOs in ER2 and have been anticipated and explained elsewhere! 7 in terms of s - p hybridization promotion and we shall not reiterate these arguments here. However, there are several features of this plot which are unique to LAOs and need to be mentioned further. First, the plots of f! and f! are essentially mirror images of one another. That is, to an excellent approximation,

(12)

While the unitary nature of LAOs on a given atom is certainly involved in causing Eq. (12) to hold to some degree, there is still no a priori reason to expect such extraordinary correlation and in fact, no such relationship is evidenced for ERAOs. The fact that LAOs do behave in such a manner is clearly an advantage, since one may readily understand that whatever s character is lost by a lone pair LAO(s) as PR increases is immediately gained, essentially in its entirety, by the bonding LAO(s). Second, Fig. 2 implies that as PR-oo, f! becomes large and f! becomes correspondingly small. In fact, in ER2 it was suggested that as PR - 00, f! should fall between O. 8 and O. 9 with f! appropriately tiny. LAOs do not, in fact, bear out these predictions. Of the four atoms we have studied which have PR = 00 (Table XIV), only N in NH and to a lesser extent Li in Li2 show a largef~ and a smallf!. Li in both LiF and LiH diverge drastically from the anticipated result. The wide range of the f! and f! values when PR = 00 is all the more striking considering the ranges are typically quite small (see the discussion in Sec. I above). The answer to this dilemma is actually contained in ER2, although these authors arrived at a different conclusion. When PR- 00

there are no lone pair electrons and a finite number of bonding electrons. Thus, the bonding LAO(s) is (are) allowed to adjust optimally to a particular bonding environment without making any compromises for the adjustment of the lone pair LAO(s). Thus, when PR = 00

the bonding LAO(s) is (are) maximally sensitive to chemical environment which allows for extreme variations in the values of f! (and hence f!). This would appear to be a more reasonable conclusion than that of ER2 which postulated that the f~ should fall within a relatively small range (0.8-0.9) at PR=oo. This conclusion could not account for the great sensitivity of the bonding LAOs to chemical environment when no lone pair electrons are present.

TABLE XIV. The fraction of s character for bonding and lone pair LAOs for atoms which have PH = 00.

Atom Molecule r. f~ Li LiH 0.514 0.486

Li2 0.754 0.246 LiF 0.453 0.547

N NH(I~+) 0.939 0.061



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K. H. Aufderheide and A. Chung·Phillips: Localized atomic orbitals. II 1803

(3) Whenever the valence arrangement of an atom contains trigonally equivalent LAOs the angle 8 was found to be a useful parameter. We have heretofore interpreted 8 in terms of Valence Shell Electron Pair Repulsion (VSEPR)-type arguments. In terms of the fraction of s character, 8 is expressible as

[ (1 _x)1/2]

8 =cos·1 - 3 -x

where

~! for a valence arrangement of

berA(B) and ltjA LAOs x-

- : for a valence arrangement of lerA and btjA(B) LAOs

(13)

As x increases from zero to one, 8 decreases from 125.26° to 90°, according to Eq. (13). We may distinguish two mechanisms whereby 8 may vary for our systems:

(a) Variations in PR: For triply bonded systems we found e to vary as

B in BF(1l8. 3°) > C in CO(1l7. 0°» N in N2(1l5. 4°)

>Oin CO(1l3.00»F in BF(1l0.2°) .

Reference to Fig. 2 will show that this series represents an increase in PR in going left to right which from the discussion in (2) above is known to correspond to an increase in f! due to variations in requirements for s - p hybridization promotion. As mentioned above, increasing f! must correspond to decreasing 8 toward 90°, which is just the trend observed. This interpretation of 8 variation in triply bonded systems in terms of PR differences is really quite equivalent to the earlier VSEPRlike tenets: since each of the above systems contains a doubly occupied er lone pair LAO, the increase in PR in going from left to right is due to an increase in the number of bonding electrons in this direction. As the number of bonding electrons in the trigonally equivalent bonding LAOs increases the interbonding LAO repulsions increase so 8 decreases.

(b) Variations for a constant PR: F in BF and LiF both have PR = ~, yet

F in LiF(llO. 9°) > F in BF(llO. 2°) •

9 is not the same in the two systems even though both molecules show a similar bonding picture and PR is the same for both F atoms. Thus the change in 9 is entirely due to the difference in the counteratoms (Li versus B) in the two molecules rather than to differences in the number and type of valence electrons on the F atoms. In this instance, changing the counteratom from B to Li causes f! to decrease somewhat. Note that these variations in 9 are much smaller than alterations induced by varying PRo

V. CONCLUSION

In discussing atomic contributions to localized phenomena it is evident that LAOs compare favorably to ERAOs. For a given PR (except 00), f! andf! are less sensitive to alterations in chemical environment for

LAOs than for ERAOs indicating that for a particular PR, LAOs are far more transferable than ERAOs. The alteration of valence LAO composition versus PR is much smoother and more understandable for LAOs than for ERAOs as f! varies as 1 - f! for the former but not the latter. Mulliken population analysis is greatly simplified for LAOs, as they are unitary on a given atomic center while ERAOs are not. Finally, ERAOs because they depend upon the specification of LMOs for their very definition are necessarily inadequate when the LMO method fails as it does for C2•

In addition, it has been shown that the populations of LAOs and their overlap distributions offer an excellent picture of localized chemical bonding even in the absence of LMOs as was the case in C2• For larger systems where the LMO process becomes impractical the LAO method may be the only efficient way to study localized properties within the ab initio MO framework. We intend to shortly publish results of LAO computations for a number of such large closed shell molecules to prove the suitability of the process for such systems. As in C2, LAOs occasionally offer a better picture of bonding in these larger systems than do LMOs. For example, the two bonding LAOs in water are separated by an angle of about 101°, quite close to the equilibrium bond angle of 104.5°. On the other hand, the two bonding LMOs are separated by about 900

• 5(0)

Finally, the LAO method is formally applicable to open shell systems even in the Unrestricted Hartree-Fock framework. This is an area in which the LMO procedure experiences a number of fundamental difficulties. Although tentative results are only now being gathered. LAOs may possibly allow for a study of localization in this class of compounds.

ACKNOWLEDGMENT

The authors would like to express their gratitude to Professor David B. Phillips for many inspired and insightful discussions and to Dr. Joseph Simpson for computational aids at the outset of this work. One of us (A.C.P. ) would like to thank the Miami University Faculty Research Committee for partial support of this work.

IV. Fock, Z. Phys. 61, 126 (1930). 2C. c. J. Roothaan, Rev. Mod. Phys. 23, 69 (1951). 3(a} Equivalent Orbitals: J. E. Lennard-Jones, Proe. Roy.

Soc. (London) Ser. A 198, 1 (1949); 198, 14 (1949); G. G. Hall and J. E. Lennard-Jones, ibid. 202, 155 (1950); Ref. 4; (b) Extremization Methods: S. F. Boys, Rev. Mod. Phys. 32, 296 (1960); J. M. Foster and S. F. Boys, ibid. 32, 300 (1960); S. F. BOyS in Quantum Theory of Atoms, Molecules and the Solid State, edited by P. O. wwdin (Academic, New York, 1966) p. 253; V. Magnasco and A. Perico, J. Chern. Phys. 47, 971 (1967); 48, 800 (1968).

4J. E. Lennard-Jones and J. A. Pople, Proc. Roy. Soc. (London) Ser. A 202, 166 (1950).

5(a) C. Edmiston and K. Ruedenberg, Rev. Mod. Phys. 35, 457 (1963); (b) C. Edmiston and K. Ruedenberg. J. Chern. Phys. 43, S97 (1965); (c) C. Edmiston and K. Ruedenberg in Quantum Theory of Atoms, Molecules and the Solid State , edited by P. O. Lowdin (Academic, New York, 1966), p. 263.

6C. Trindle and O. Sinan~lu, J. Chern. Phys. 49, 65 (1968); J. A. Hunter, QCPE 11, 355 (1979).

7W. England and M. Gordon, J. Am. Chern. Soc. 93, 4649

J. Chem. Phys., Vol. 73, No.4, 15 August 1980


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(1971); ibid. 94, 4818 (1972). 8(a) sp" Hybrids: L. Pauling. J_ Am. Chern. Soc. 53, 1367

(1931); J. C. Slater, Phys. Rev. 37, 481 (1931); (b) Overlap Extremization Criteria: M. Randic and Z. B. Maksic, Chern. Rev. 72, 43 (1972); (c) Valence Activity Extremization: K. Ruedenberg, Rev. Mod. Phys. 34, 326 (1962); (d) Symmetric Orthogonallzation Criteria: R. McWeeny and K. A. Ohno, Proc. Roy. Soc. (London) Ser. A 255, 367 (1960); (e) Symmetry Orbitals: Ref. 2. ~. H. Aufderheide, J. Chern. Phys. 73, 1769 (1980); pre

ceding paper. 1<1<. H. Aufderheide and A. Chung-Phillips (to be published). l1C. Trindle and O. Sinano~lu, J. Am. Chern. Soc. 91, 853

(1969). lzAll geometries taken from B. J. Ransil, Rev. Mod. Phys.

32, 245 (1960) except that of Bz(l:!;;), which was taken from J. A. Pople and D. L. Beveridge, Approximate Molecular Orbital Theory (McGraw-Hill, New York, 1970), p. 89.

13W. J. Hehre, W. A. Lathan, R. Ditchfield, M. D. Newton, and J. A. Pople, QCPE 11, 236 (1979).

14It must be pointed out that the PBOP is generally not a good measure of bond strength within a class of compounds. That is, the relative bond strengths of singly bonded systems like LiH, Liz, BH, Bz, FH, and Fz are not well reflected by the PBOPs of these molecules. However, it does appear that the PBOP can suggest whether bond strength is more similar to typical singly or typical triply bonded molecules.

15(a) G. Herzberg, SPectra of Diatomic Molecules (Van Nostrand, New York, 1950>; (b) G. Herzberg, Infrared and Raman Spectra (Van Nostrand, New York, 1945), p. 192.

16K. Jug and B. M. Bussian, Theor. Chim. Acta (Berlin) 52, 341 (1979).

17E . M. Layton, Jr., and K. Ruedenberg, J. Chern. Phys. 68, 1654 (1964); R. R. Rue and K. Ruedenberg, ibid. 68, 1676 (1964).



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Localized atomic orbitals for atoms in molecules. II. Diatomic molecules