Received 3 October 1975

A procedure for analyzing charge distributions of FSCU wavefunctions, by partitioning bonding electrons according to the location of the orbital centers, is proposed. Electron populations are given for singlet states of eighteen diatornic species, and are shown to be in excellent agreement with results obtained from virial partitioning of Hartree-Fock wavefunctions.

. Since its inception by Frost El], the FSGO method, which empIoys floating orbit& not centered on nuclei, has been used to gencr&te crude but suggestive ab initio wavefunctions for a variety 0E systems (see, for example, refs, [2--4]). These wavefunctions are formed dire& from core, bonding, and lone pair orbitals, and may thus be interpreted in terms of classidal chemistry. Further, the FSGO method has been shown to yield both surprisingIy accurate moiecular geometries and reasonable molecular orbital structures 12-4 ] . Since time-comparisons with LCAO SCF procedures are W- vorabb [2,41, the FSGO method represents a useful stiuctural tool in chemistry. A significant drawback of the FSGO method, however, is that charge distribu- tions (which are certainly extensively used by chem- ists) are not readily defined. The conventional Mu&ken population analysis can&t be applied to wavefunctions which are not formulated according to LCAO MO pro- cedures [5,6{ *. in the course of a number of FSGO calculations, we have discovered a simple partitioning scheme for FSGO charge distributions which yields

electron populations of apparently rather high accu- racy. The impl$ation, therefore, is that there is addi-

tional chemically useful information which can and should be extracted from FSGO ~vavefunctions. ’

_ Our procedure assigns core and lone-pair electrons to their p,aent atoms, and partitions only the bonding eIectrons . Since FSGO bondins orbitak are generally

:_ * Altermtiv(? hroeadures for ana&i.ng FSGU charge densities

hove been propoxd, see refs. [7&l_

.I 422 . . . _,

.. - .

. - “_ .

on or nearly on bond axes, their locations may be spec- ified according to their fractional distance from a given atom [3]. Presumably, a fractional distance of0.5 in- dicates an evenly shared electron pair while a value of 1 .O denotes an ionic bond. By the Iogical extension, one could apportion each pair of bonding electrons be- tween its respective atoms according to distance and thereby define atomic electron populations in a given molecule. We have performed test FSGO calculations on I8 diatomic molecules? and present the computed electron popufations in tables i and 2. These may be compared to the electron populations recently goner- ated from Hartree-Fock wavefunctions by Bader and co-workers [IO--f7 J . These authors have developed a novel viriaf partitioning technique which characterizes molecular charge distributions in terms of zero-flux sur- faces of the first order density, and is thus basis-set in- dependent. One notes that there is a striking level of agreement between the two population analyses. In most cases, respective electron populations agree to within 0.10 or 0.15 electrons; t&s is within the detia- tion of the original [IO] and more refined fl I-141 values proposed by Rader. in contrast, a!ternative pro- cedures for computing atomic charges often differ from

** Except where lone iair electrons are centered well into the bonding region, irz which case they are also subdivided, Ex- amples of this are Be0 and C02+ in table I.

i Tlwse c&uIat.ions were made at the given bond !engths, for purposes of comparison with vi&l partitioning results_ Re- spfts on other species, namely certain third row hydrides, are given eisewheze 191. .

.‘. _’ _- : .

Volume~38, number 3 CHEMICAL PHYSICS LE-M%RS 15 March 1976

Table 1 Electron populations of AB ditomics”)

FsGO Wrial partitioning

AB % MAI N(B) ‘N(A) N(B)

LiF X’Z+ 2.9550

LIB X’r* 4.5000

BeF+ X ’ x3+ 2.5720

BFX*C+ 2.3910

LiO* A ’ SC 3.184

co x*c+ 2.132

Be0 x’Y+ 2.515 CN-’ )3+ 2.214 r-02+ ’ x* 2.132

2.0809 9.9191

2.3991 5.6009

2.1337 9.8663

4.1768 9.8232

2.0647 7.9353

49798 9.0202

2.3772’) 9.6228”

5.7390 8.2610

3.7652 8.2348

2.0630 (2.16)

2.2391 (2.48)

2.1389 (2.32)

4.0603 (4.26)

2.022b) (2;07)

(5.08) (2.79)

(5.64)

(3.78) -

9.9370 (9.84)

5.7609 (5.52) 9.8611 (9.68)

9.9397 (9.74)

7.Q78b) (7.93)

(8.92) (9.21)

(8.36)

(B.22)

3) Unless noted otherwise. refined viriil partitioning results are from ref. [ 121 and initial virial partkioning results, in parentheses, are from ref. [lo].

b, Fromref. 1111. ‘) A second minimum in the FSGO surface, with it slightly higher energy, yields populations of 2.53 and 9.47 electrons.

the virial partitioning results by 0.5 e or 1 .O e or more [ 18-22 J . Apparently the bond orbital location. as de-

termined by energy minimization in the FSGO proce- dure, monitors the location of the zero-flux surface.

The largest discrepancy among the non-hydrides is for BeO; in this case the virial partitioning results were computed using a p!anar separation surface. By com- parison to BeF* and BF, we anticipate that an improved

Table 2 Electron populations of AH diatomics

partitioning surface would yield results closer to the FSCO values. Trends in the charge of an atom such as Li in a series of compounds may be readily noted in table 1 and related to chemical properties. FSCO elec- tron populations for hydrides might be expected to be less accurate: given the lack of basis functions fcr the hydrogen 1s core, but with the exception of CH+, the agreement is rather good. Most molecules (HF and

FSGO Vi&l partitionin$)

AH Re WA) NW) NW MH)

LiH X’C+ 3.0!50 2.1937 1.8063

BH X’Z+ 2.3360 4.5717 1.4283

FH X1X+ 1.7328 9.6843 0.3157

CH+X’Z* 2.1370 4.9105 1.0894

NaH X1x+ 3.566 10.1512 1.8487

AlHX’Z+ 3.114 12.3819 1.6181

ClH X’ C* 2.4087 17.0534 0.9466

MgH+ X ’ X+ 3.116 10.5087 1.4913

SH- X’S+ 2.55 1 16.726 1.2744

2.0889 I 1.9111

4.2461 1.7539

9.7603 0.2397

5.2884 0.7116

10.190 1.810

12.175 1.825

17.241 0.759

10.402 1.598

16.728 1.272

a) Vi&l partitioning results from refs. [ 13,141.

423

Volume 38, numbs; 3 --. CHEhlICAL PHYSICS LETTERS 15 hfzrch 1976

He in particular) are computed to be slightly 1esS ion-

ic I$& prdposed’by Bader:The anomalous resuk [14] obtain&d &the &ial partitidning scheme for LLH as

izompgred to NaIj is not present in the FSGO calcula- tiork. FSGO electron populations reflect the chemical- ly anticipated r&x& that NaH is more ionic than LiH.

Since-the Bader procedure represents a welldocu- mented; unique, z&d chemically significant procedure for describing charge distributions; we therefbre be-

lieve t.hat the results displayed in tables 1 and 2 dem- onstrate the utility of the proposed partitioning scheme and ill&at< that FSGO wavefunctions may be useful- ly decomposed by a simple population analysis.

NV additional difficulties are anticipated for poly-

atomics with localized bonds. Indeed, we find quanti-

tative agreement between FSGO and virjal partition-

ing results for the charge-in hydrogen electron popula-

tion upon going from BH to BH3 [16] . We believe

that decompositicns of this nature may also prove

practical for systems with delocalized electrcns, and that related procedures could be applied in computa-

tional schemes which use additional FSGO basis func-

tions [i] _

We wish to thank Wichita State University for pro-

viding the nxessary computer time.

References

111 A.A. Frost. J. Chem. Phvs. 47 (1967) 3707.

[ 21 A.A. Frost, in: Modern theorztical chemistry, Vol. 2, ed. H.F. Schaefer, to be published.

[3] P-H_ Blustfn and J-W. Linnett, 1. Chem. Sot. Faraday II 70 (1974) 274.

[4] E.R. T$aty, AK. Schwartz and G. Simons, J. Am; Chem. sot. 97 (1975) 972.

[5] D.B. Boyd, discussion comment in: Energy structure and reactivity, eds. D.W. Smith and W.B. hfcRae.

[6] R.E. Christoffersen, discussion comment, in: Energy structure and reactivity, eds. D.W. Smith and W.B. McRae.

171 R.E. Christoffersen, Advan. Quantum Chem. 6 (1972) 333.

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G.R. Runtz and R.F.W. Bader, Mol. Phys. 30 (1975) 129. S. Srebrenik and R.F.W. Bader, J. Chem. Phys., to be published. P. Poiitzer, K.C. Leung, J.D. Elliott and S.K. Peters, Theoret. Chim. Acta 38 (1975) 101. W.L. Jolly and W.B. Perry, J. Am. Chem. Sot. 95 (1973) 5442. R.S. Evans and J.E. Huheey, Chem. Phys. Letters 19 (1973) 114. K. Jug, Theoret. Chim. Acta 31 (1973) 63. R.S. Mulliken, J. Chem. Phys. 23 (1955) 1833.

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