HartreeFock Wavefunctions, Potential Curves, and Molecular Properties forOH−(1Σ+) and SH−(1Σ+)Paul E. Cade Citation: The Journal of Chemical Physics 47, 2390 (1967); doi: 10.1063/1.1703322 View online: http://dx.doi.org/10.1063/1.1703322 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/47/7?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Variational timedependent Hartree–Fock calculations. III. Potential curves for twoelectron molecularsystems J. Chem. Phys. 65, 2104 (1976); 10.1063/1.433393 Complex and Unrestricted HartreeFock Wavefunctions J. Chem. Phys. 57, 2994 (1972); 10.1063/1.1678695 On the Unrestricted Hartree–Fock Wavefunction J. Chem. Phys. 51, 3175 (1969); 10.1063/1.1672491 Calculation of Extended HartreeFock Wavefunctions J. Chem. Phys. 48, 835 (1968); 10.1063/1.1668721 Force Calculation with Hartree—Fock Wavefunctions J. Chem. Phys. 39, 2397 (1963); 10.1063/1.1701470

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THE JOURNAL OF CHEMICAL PHYSICS VOLUME 47, NUMBER 7 1 OCTOBER 1967

Hartree-Fock Wavefunctions, Potential Curves, and Molecular Properties for OH-(t~+) and SH-(l~+)*

PAUL E. CADE

Laboratory of Molecular Structure and Spectra, Department of Physics, University of Chicago, Chicago, Illinois (Received 5 April 1967)

The potential curves for OH-(lu22u23u2br4, X 12;+) and SH-(1u22u23u2br440-25u221!'4, X 12;+) have ~een calculated using electronic wavefunctionswhich are believed ~o be very near theH.artree-Fock wa~:functlOns in accuracy. Using a Dunham analysis for these two potentIal curves and from mternal regulantIes of per­cent errors for the ground states of the first- and second-row hydrides, AH, it is predicted that for OH-19.1 <B. <19.4 cm-I, 3733~w.~3820 cm-I , and 1.81~R.~1.83 bohr and for SH- 9.36~B.~9.54 cm-I ,

2642<:w <2692 cm-I and 2.54<R <2.57 bohr. The potential curves of OH- and SH- are thus closely parallel ~;;-the potential curves ;{ OH(X 2ll.) and SH (X 2ll.) , respectively. Several ways of estimating the electron affinity (E.A.) of OH and SH are presented; the most reliable values are E.A. (OH) = 1.91 eVand E.A. (SH) =2.25 eV. These calculated results for OH- are in good agreement with recent p?otodetach~ent measurements by Branscomb. The electric dipole and quadrupole moments, th~ electnc field gradient at the nuclei the electronic force acting at the nuclei, and several molecular expectatIOn values are presented as a functio~ of internuclear separation for OH- and SH-. The relationships of these various properties to the analogous properties of the parent neutral systems [OH(X 2lli ) and SH(X 2lli ) J and the isoelectronic systems [HF(X 12;+) and HCI(X 12;+) J are discussed.

I. INTRODUCTION SH- "centers." It has been found that OH- and SH­

The experimental characterization of negative molec­ular ions has yielded relatively little information about the structural properties of this small group of mole­cules. The electron affinity of the parent molecule has been measured with good accuracy in a few cases, but such properties as the spectroscopic constants, Re,

electric multipole moments, and other quantities are usually completely unknown. For example, the Re value of such well-known molecular ions as O2- and CN­have not been experimentally determined. The fragility of molecular negative ions has almost always precluded the measurement of their properties by orthodox con­ventional experimental methods, although indirect information as to their presence and characteristics is readily available. We are concerned here with chemically stable molecular negative ions and not metastable or "resonance" states with ephemeral existence as found in electron scattering phenomena.

Two relatively new research activities have recently encouraged theoretical consideration of "alleged" stable negative molecular ions. The first group of studies is less directly related, although relevant, and concerns the nature and properties of (AH)- ions as imperfections in an alkali halide crystal lattice and as anions in MOH or MSH, alkali halide type crystals. As examples, particularly pertinent here, are the works of Luty and colleagues,la Shepherd and Feher,1b and Chau, ~lein, and Wedding'" on OH-, and Fischer and Griindlg2 on

* Research reported in this publication was supported by Advanced Research Projects Agency through the U.S. Army Research Office (Durham) under Contract No. DA-31-124-ARO­D-447 ARPA Order 368, ARO-D Project No. 3835-P, and by a grar:t from the National Science Foundation.

I (a) F. Liity, U. Kuhn, H. Paus, and H. Hartel, Solid State Comm. 2, 281 (1964); Phys. Status Solidi 12, 431. 347 (1965). (b) I. Shepherd, G. Feher, and H. B. Shore, Phys. Rev. Letters 15, 194 (1965); 16, 500 (1966). (c) C. K. Chau, M. V. KI~in, and B. Wedding. Phys. Rev. Letters 17,521 (1966); (also pnvate com­munication) .

2 F. Fischer and H. Griindig, Phys. Letters 13, 113 (1964); Z. Physik 184, 299 (1965).

ions apparently simulate halide ions X- in compounds such as LiOH, NaOH, or NaSH under certain condi­tions. For example, these compounds crystallize in cubic form at high temperature with the NaCI struc­ture, and several interesting studies have appeared by Coogan, Belford, and Gutowsky3 (for NaSH), and by Campbell and Coogan4 (for LiOH). These studies have, in addition to other aspects, sought to examine the nature of the OH- and SH- charge distributions in such crystals. The second new experimental evidence involves the continuous absorption (or photodetachment) spec­trum of OH- by Branscomb5 and of SH- by Steiner.6 The latter experimental studies involve the free molecu­lar ion, while the research on crystals deals with OH- or SH- ions, either trapped in KCI or KBr crystals as substitutes for Cl- or Br- ions, or as regular anions in MOH or MSH crystals. Therefore, the results obtained here for free OH- and SH- ions mayor may not be useful charscterizations of the properties of OH- and SH- "centers" in KCI or KBr, or as OH- and SH- ions in MOH and MSH crystals, but, in my opinion, caution would be wise in making such an association. It should be added that the wavefunctions obtained here for OH- and SH- can be used with earlier results for OH and SH to calculate photodetachment transition prob­abilities, but better than a crude estimate should not be expected.

The properties of OH- and SH- desired are Re, We,

other spectroscopic properties, the electric dipole moment and p,(R) , and the electron affinity of OH and SH. These quantities and others are considered here; in particular, reliable estimates of the electron affinity of OH and SH are possible from these calculations. In

3 C. K. Coogan, G. G. Belford, and H. S. Gutowsky, J. Chern. Phys. 39, 3061 (1963).

4 r. D. Campbell and C. K. Coogan, J. Chern. Phys. 42, 2738 (1965) .

6 L. M. Branscomb, Phys. Rev. 148, 11 (1966). . 6 B. Steiner (National Bureau of Standards, Washington, D.C.,

private communication).

2390

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ELECTRONIC STRUCTURE OF OH- AND SH- 2391

TABLE I. HFR wavefunction for OH-(1u22u23u21,r"XI2:+), R=1.781 bohr." E=-75.41754 hartrees, T=75.42619 hartrees, V=-150.84373 hartrees, V/T=-1.99989.

E1.= -20.17697, 02.= -0.90541, E3.= -0.25372, Eh"= -0.10679.

"'-C;~p Cl •• p Ca..P Ca.,p ~~P C1r•p

xP''''- Xpr

uiso (r=7.01681) 0.94287 -0.24772 0.07546 ,..2po(r=0.95041) 0.43052 u1so' (12.38502) 0.09316 0.00307 -0.00149 ,..2po'(2.06239) 0.48013 u2so (1. 57292) -0.00130 0.44289 -0.24103 ,..2po"(3.75295) 0.20873 u2so'(2.86331) 0.00397 0.57846 -0.18826 ,..2po'" (8.41140) 0.00967 u3so (8. 64649) -0.03824 -0.02924 0.00447 ,.. 3do (1. 66353) 0.02955 u2po(1.02266) -0.00046 0.03788 0.29294 ,..4/0(2.19941) 0.00876 u2po'(2.11724) -0.00049 0.08240 0.39746 ,..2PH(1. 76991) 0.02625 u2po" (3.75959) 0.00100 0.01943 0.16536 ,..3dH (3.32513) 0.00267 u2po"'(8.22819) 0.00055 0.00206 0.00908 u3do (1.18175) -0.00030 0.01470 0.04571 u3do' (2.82405) -0.00004 0.01374 0.02292 u4/0 (2.26641) -0.00005 0.00661 0.01247 ulsH (1.19859) 0.00175 -0.03064 0.23558 ulsH '(2.43850) -0.00032 0.09358 0.06121 u2sH (2. 30030) -0.00038 0.08217 0.05504 u2PH(2.80542) 0.00010 0.01306 0.00935

" Atomic units are employed, i.e .. one unit of energy"" 1 hartree =27.2097 eV and one unit of length"" 1 bohr =0.529172 1.

addition, we consider how certain physical properties of a system change with the addition of a surplus elec­tron to form a negative molecular ion.

Previous calculations on OH- include the one-center expansion results of Gaspar, Tamassy-Lentei, and Kruglyak,7 minimal SCF calculations by Rosenfeld,S limited valence-bond calculations by Grahn,9 and, finally, LCAO-MO-SCF calculations using a large Gaussian basis set by Moskowitz and Harrison.lo The only previous calculation involving SH- known to the author is due to Banyard and Hakell who used the one­center expansion approach. Only the more recent cal­culation of Moskowitz and HarrisonlO approaches the accuracy of the wavefunctions given here. (Moskowitz and Harrison obtain E = -75.3670 Hartrees versus -75.41751 Hartrees reported here.) The upper limit of accuracy of all calculated results given here is that defined by the Hartree--Fock approximation-the canonical independent-particle model.

II. CALCULATION OF WAVEFUNCTIONS, ENERGIES, AND CERTAIN MOLECULAR

PROPERTIES

The calculations reported in this paper are based on an independent-particle model, or more precisely on

7 R. Gaspar and I. Tamassy-Lentei, Ann. Physik. (Leipzig) 7, 208 (1958); Acta Phys. Acad. Sci. Hung. 10, 149 (1959); R. Gaspar, I. Tamassy-Lentei, and Y. Kruglyak, J. Chem. Phys. 36, 740 (1962); Zhur. Strukt. Khim. 3, 316 (1962). With regard to (OH) - results, these four papers are identical in content, but give one a choice of German. English, or Russian.

S J. L. J. Rosenfeld, J. Chem. Phys. 40, 384 (1964), OH-. 9 R. Grahn, Arkiv Fysik 28, 85 (1964), OH-. 10 J. W. Moskowitz and M. C. Harrison, J. Chem. Phys. 43,

3550 (1965), OH-. 11 K. E. Banyard and R. B. Hake, J. Chern. Phys. 41, 3221

(1964), SH-.

solutions of the Hartree--Fock-Roothaan matrix equa­tions for OH- and SH-. In this approximation the wavefunction is written as an antisymmetrized product of molecular orbitals, and applications of the variational method leads to the set of Hartree--Fock equations whose solution yields the usual canonical molecular orbitals. If, instead, one arbitrarily expands the molecu­lar orbitals in terms of a known set of expansion func­tions, that is,

N~A N~

IPi~",(r) = E xp~"(rA)C,,,/A)+ E xPA"(rH)C,,,p(H), po=l p=N~A+1

( 1)

then the solution of the Hartree--Fock equations to self-consistency in IP,,,,,(r) is replaced by the solution of the Hartree--Fock-Roothaan equations to self-con­sistency in the expansion coefficients CAP' The expan­sion basis functions xPA,,(r) are the usual Slater-type functions (STF's) which are not orthogonal, are node­less, and are defined by

xp)..,,(r) = (2~\p)"hP+i[(2n)..p) !]-irnhP"""l

Xexp( -tApr) YI~pm~a(8, cfJ). (2)

The summation of p in Eq. (1) is over the total number of basis functions of symmetry A, N A, and is written here as the sum of the NM functions centered on Nu­cleus A and the N)"H functions centered on the proton (N A = N M + NAH) • Only for adequately large and versatile expansions can the solutions of the Hartree-­Fock-Roothaan equations be expected to closely approximate the solutions of the Hartree--Fock equa­tions, the latter being the basis oj the canonical inde-

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2392 PAUL E. CADE

TABLE II. HFR wavefunction for SH-(1u22u23u'l~4u25u22,,-., X I2:+), R=2.551 bohr."

E=-398.1459 hartrees, T=398.1503 hartrees, V=-796.2961 hartrees, V/T=-1.99999. Elv= -91.65348, E2v= -8.65453, Eav= -6.33687, E4v= -0.63605, Es.r= -0.24670, Elr= -6.33476, E2.= -0.08878.

~AP CIV•P C,v.P Cav•p C4v.p Cs.r.P ~~p Cb •P C'r.p Xpv Xpr

ulss(t= 18.01116) 0.81379 -0.22471 0.00103 0.05768 -0.02190 1I'2ps (t= 4. 82431) 0.68144 -0.16613 u2ss (5.30794) 0.00062 0.90110 -0.00311 -0.37338 0.10471 1I'2ps' (7.84493) 0.32819 -0.07177 u2ss'(15.83751) 0.21931 -0.11868 0.00048 0.04115 -0.01488 1I'2ps" (13. 58306) 0.02848 -0.00630 u3ss (1. 49328) 0.00010 -0.00008 0.00014 0.36793 -0.32499 1I'3ps(1.l0061) -0.00094 0.55823 u3ss' (2.59027) -0.00020 0.00597 0.00101 0.60708 -0.24620 1I'3ps'(2.27212) 0.00732 0.54986 u3ss" (9.40187) 0.00268 0.16781 -0.00170 0.00923 0.01054 1I'3ds (2 .15151) 0.00062 0.00610 u2ps(4.81311) 0.00001 0.00348 0.68011 -0.02824 -0.14462 1I'2PH(1.18717) -0.00036 0.04308 u2ps' (7.83720) -0.00002 0.00041 0.32942 -0.00976 -0.05348 1I'3dH (1. 87968) -0.00035 0.00796 u2ps" (13.57266) 0.00009 0.00016 0.02853 -0.00124 -0.00624 u3ps (1.12168) 0.00002 0.00018 -0.00095 0.00761 0.24145 u3ps'(2.30113) -0.00001 0.00049 0.00793 0.06897 0.42760 u3ds(2.3OO36) 0.00000 0.00059 0.00090 0.02413 0.05945 ulSH(1.38774) -0.00035 -0.00136 -0.00029 0.31008 1.34402 u1sH'(1.80604) 0.00021 0.00117 0.00072 -0.06385 -0.64727 u2sH (1. 92071) 0.00013 0.00000 -0.00055 0.00244 -0.17740 u2PH(1.65793) 0.00001 -0.00026 -0.00024 0.03067 0.03949

" Atomic units are employed. i.e .. one unit of energy"" 1 hartree =27.2097 eV and one unit of length = 1 bohr=0.529172 A.

pendent-particle model. In all the calculations reported here, an expansion of the type in Eq. (1) is employed. For a complete discussion of the definition of the complex spherical harmonics Y1m«J, cJ», the explicit form of the Hartree-Fock-Roothaan matrix equations,

and details of accomplishing their solution, reference to earlier publications12- 14 is encouraged.

These calculations are for the explicit states and configurations of OH-(lu22u23u217r\ X 11:+) and SH-( 1u22u23u217r44u25u2211'\ X 11:+), which are ISO-

TABLE III. Summary of energy quantities as a function of internuclear distance- OH-(1u22u23u'h., XI2:+).

R E T V V/T EI, (2. Ea, frr

Ob -99.45937 99.45896 -198.91833 -2.00000 -25.82946 -1.07442 -0.18083 -0.18083 1.40 -75.34460 76.03718 -151.38178 -1.99089 -20.13976 -0.98477 -0.29347 -0.10633 1.50 -75.38321 75.82880 -151.21201 -1. 99412 -20.15021 -0.95953 -0.28369 -0.10601 1.60 -75.40515 75.65846 -151.06361 -1.99665 -20.16023 -0.93750 -0.27338 -0.10598 1. 70 -75.41537 75.51950 -150.93487 -1.99862 -20.16970 -0.91858 -0.26265 -0.10628 1. 75 -75.41725 75.46005 -150.87730 -1.99943 -20.17422 -0.91023 -0.25715 -0.10657 1. 781 c -75.41754 75.42619 -150.84373 -1.99989 -20.17697 -0.90541 -0.25372 -0.10679 1.79Qc -75.41751 75.41676 -150.83427 -2.00001 -20.17775 -0.90406 -0.25272 -0.10687 1.795d -75.41748 75.41161 -150.82909 -2.00008 -20.17819 -0.90332 -0.25216 -0.10691 1.835 -75.41671 75.37225 -150.78896 -2.00059 -20.18162 -0.89764 -0.24769 -0.10729 1.90 -75.41385 75.31506 -150.72892 -2.00131 -20.18705 -0.88929 -0.24038 -0.10805 2.00 -75.40646 75.24156 -150.64803 -2.00219 -20.19506 -0.87846 -0.22909 -0.10960 2.10 -75.39658 75.18303 -150.57961 -2.00284 -20.20274 -0.86984 -0.21786 -0.11163 2.25 -75.37907 75.11802 -150.49709 -2.00348 -20.21381 -0.86051' -0.20134 -0.11564 2.40 -75.36016 75.07453 -150.43470 -2.00380 -20.22456 -0.85509 -0.18547 -0.12078 2.60 -75.33493 75.04127 -150.37620 -2.00391 -20.23864 -0.85253 -0.16566 -0.12928 2.80 -75.31107 75.02785 -150.33892 -2.00377 -20.25261 -0.85419 -0.14763 -0.13938 ",- -75.28948 75.29531 -150.58479 -1.99992 -20.19775 -0.81322 -0.12889 -0.12889

- Atomic units are employed. i.e .. one unit of energy. 1 hartree = 27.2097 d Calculated R, value for OH (X -II.). eV, and one unit of length. 1 bohr =0.529172 A. e Separated atoms result; F(ls'2s'2P'. 'P) +H(ls. 'S); E1V ..... E1 •• E •• --+E ...

b United atom result; F-(1s'2s'2p'. 15); Elv-El •• E,v-*'" "v = Elr-"p, Eaer = E111' --..£211-

C Calculated R, values (R = 1. 781 gives lowest energy. R = 1. 790 gives best virial).

12 W. M. Huo, J. Chern. Phys. 43, 624 (1965). 13 P. E. Cade and W. M. Huo,47,'614 (1967). 14 P. E. Cade and W. M. Huo, 47, 649 (1967).

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ELECTRONIC STRUCTURE OF OH- AND SH- 2393

~~~~~~~~~;;b8~~~:sg; \J')OO~t-~O\tn......t~OO\oNOr-lI'}

~~;;:l;;:l~~$:g:g:8~~::8~~~ ~~~...;.....;.,....;,....;.,....;....;....;.,....;.,....;....;.....;.,....;,....; 80-0-0-0-0-0-0-0-0-0-0-0-0-0-0-

I I I I I I I I I I I I I I I I

§~~I~I~ii~~~~~~~ N""";""';'-;""';""';""';"";""";NNNNNNN I I I I I I I I I I I I I I I I

"". g g

"'-~a58S~~~~~i2~~~~~1D ,....;NNNNNNNNNM~~~8

o ..., -

electronic with HF and Hel, respectively. The ground states of OH- and SH- are thus assumed to be X 12;+ states; no reasonable alternatives, or even any stable electronic excited states, seem likely. In these cal­culations, considerable use was made of the OH(X 2IIi) and SH (X 2II.;) wavefunctions to obtain a starting basis-set composition for OH-(X 12:+) and SH-(X 12:+) calculations.13 .14

The wavefunctions and -energetic quantities obtained are given for R=R.(calc) for OR- in Table I, and for SH- in Table II. The variation of E, T, V, V IT, and the £i with internuclear separation R are summarized in Table III for OH-, and Table IV for SH-. Only a few remarks indicating explicitly how these particular results were obtained are presented here. The starting basis-set composition, that is, the particular distribution of Slater-type functions between Nucleus A and H, the specific type of STF's, and the orbital exponents, are taken from the basis sets previously obtained for OH(X 2II.) and SH(X 2IIi ) at R=Re(exptl) .15 Starting with the AH(X 2II.) basis set instead of, for example, with an atomic basis set of A-, seems a more reasonable and efficient procedure. The same set of molecular orbitals (not the same molecular orbitals) are involved in AH and AH-, and the expansion of the outer molecu­lar orbitals in AH- relative to AH is accomplished here by the new set of linear variational parameters, the set of C'AP coefficients, and reoptimization of selected nonlinear variational parameters s\p. If a basis set for AH- had instead been constructed from scratch from atomic STF's from A- results plus the needed supple­mentary basis functions of higher symmetry on A and on the H nucleus, considerably more calculations (i.e., experimentation with basis-set composition, optimization of many orbital exponents, etc.) would have been required to achieve the same quality molecu­lar orbitals for AH-. This perspective is entirely con­sistent with calculations on A- atomic ions starting from neutral A atomic results.

Thus, for OH-(X 12;+) the final basis set for OH(X 2II i ) was used as the starting basis set, and for R= 1.835 bohr the orbital exponents of the 1r2po, 1r2po', 1r3do, u2so, u2po, u3do, u1sH , and u2po' basis functions were individually optimized (see Table I for final optimal rxp values). The energy commencing this series of optimizations was -75.40346 hartrees and it was lowered to -75.41671 hartrees after these eight single optimizations. The starting basis set for SH-(X 12:+) was the final basis set for the parent neutral SH(X 2IIi)' The starting energy for SH- was -398.1317 hartrees and after optimizing orbital ex­ponents of the five key basis functions (apparent from the OH- results) this value was lowered to -398.1459

,. It should be realized that the molecular orbitals of AH- and AH are different inasmuch as the linear expansion coefficients, CiAp, are different, although the starting set of basis functions representing analogous ~iAa in AH- and AH are the same.

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2394 PAUL E. CADE

+15%

(b)

+10% +10% +10% +10%

+5% +5% +5% +5%

%Error 0 I--~~-------' O%Error %ErrorO I----'~~:;:-----"'-'-- 0 % Error

FIG. 1. A summary of the percent error of calculated values of R" Be, We,

and 01, for (a) first- and (b) second-row hydrides. Calculated results from Cade and Huo (Refs. 13 and 14) are to Hartree-Fock accuracy. i }.

~) -10% :.::t \ j -,:: t '," !

-5%

-10%

-15 % "-:-L-':-iH:-:s:-':eH-:-::':SH-:--::cLH -:-N"--'H""O:<-H:-H:C:F:--" - 15 %

DIATOMIC HYDRIDE

-15 % L.f ~. ~--,-,---.,---,---,-_I -15 % NoH MgH AIH SiH PH SH HCI

DIATOMIC HYDRIDE

hartrees.16 These final basis sets were adopted as the final results given in Tables I and II for OH- and SH-, respectively.

A brief recapitulation of the argument behind the quality of the various molecular orbitals, <Pi>.,,(r), of OH- and SH- is as follows. First, and foremost, the quality of these results for OH- and SH- depend on high-quality results for OH and SH as previously discussed.13 •14 Secondly, the relatively large and versatile basis set is;believed to be flexible enough to absorb most of the changes, mostly spatial expansion, of the valence-shell molecular orbitals in going from AH to AH-j finally, the reoptimization of the orbital ex­ponents of the key basis functions is perhaps adequate to preclude any further necessary changes in the va­lence-shell molecular orbitals,l7 Any remaining doubts the author has about claiming that the molecular orbitals of OH- and SH- are to Hartree-Fock ac­curacy, comparable to the OH and SH results, are associated with the significant spreading out of the AH­charge density, and orbitals, relative to the AH mole­cule. This problem of very long significant tails in the atomic or molecular negative ions is a difficult problem,

16 Characteristic behavior in optimizing the orbital exponents was a decrease in r~p values, that is, an expansion of the molecular orbitals is indicated. This is in addition to the expansion already effected through the Ci~p coefficients in the SCF process. The single most important optimization was for the 7r2po or 7r3ps for OH- and SH-, respectively, which accounted for a large percentage of the total energy gain. For example, in the optimization of the 7r3ps STF, r~p went from 1.29270-->1.10061 and lowered the energy by 0.01162 hartrees (out of the total of 0.01419 hartrees).

17 It is necessary to check the final basis set obtained for AH­(which differs from that of AH only in the reoptimized orbital exponents) in a calculation on AH, the parent neutral system. This will insure that the energy gained for AH- upon optimization of the orbital exponents, is a real improvement for AH- and not just a general improvement which could also lower the result for AH. This is indeed verified as the SH- basis set gives an energy -398.0947 hartrees for SH(X 211.) compared to the lower value of -398.1015 hartrees reported earlier (Ref. 14) for SH (X 211.). This is a more critical test than a recheck of OH(X 211.) would provide.

not limited to the expansion approach, but it should not be overlooked.

The basis sets obtained for Re(exptl) of AH were then used to obtain Hartree-Fock-Roothaan wave­functions for 14 or 15 R values each (that is, the set of STF's and ~},p's in Tables I and II, not, of course, the set of CAP, which change with internuclear separation). These give the energetic results found in Tables III and IV, i.e., E(R), T(R), V(R), V(R)/T(R), and the orbital energies, EfA • Once again, on the strength of the study of the AH neutral sequence, the orbital expo­nents were not reoptimized at the various R values. To repeat our previous refrain, in light of the very con­siderable computing time which would be required, the intrinsic shortcomings of a Hartree-Fock potential curve, and the strong likelihood that the characteristics of E(R), V(R), T(R), and E.A(R) would be almost indiscernibly affected, the expansion basis sets obtained at '"'-'Re(calc) were used for the various R values. The large and versatile basis-set composition is believed sufficient to absorb the changes in the (Na with R almost completely via the linear expansion coefficients C'AP' A full discussion of these energetic results is given beJow.

The wavefunctions obtained above were used to calculate a modest number of expectation values and molecular properties of OH- and SH-. In each case the quantity as a function of internuclear separation is of prime consideration. Thus, in Tables V and VI the electric dipole and quadrupole moments, the electric field gradient at A and H nuclei, the electric force on both nuclei, and several other useful expectation values are given for OH- and SH-. The explicit form of the expectation values, or operators, employed are given in previous publications.18 At best, these quantities are

18 See, for example, P. E. Cade, K. D. Sales. and A. C. Wahl, "The Electronic Structure of Diatomic Molecules. IILB. Certain Ex,pectation Values and Molecular Properties of N 2 (XI'1:,.+) and N. (X 22:.+, A 'lIu, B 2'1:,+u) ," J. Chern. Phys. (to be published) .

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ELECTRONIC STRUCTURE OF OH- AND SH- 2395

TABLE V. Summary of expectation value and molecular property functions, M(R)."

OH-(1.r20'23.r1~, X 12;+).

R(b) ( I ~ I )b ( I TA21 ) ( I TA -1 I ) ( I TH-1 I ) J.I (eao) Q Q(eao2)d FA (e2ao-2) FH(e2ao-2) qA(eaO-S) qH(eaO-S)

1.40 2.1120 2.1660 2.3476 0.7087 1.1703 1. 7491 -0.5111 -0.4914 0.7805 2.1701 1.50 2.0296 2.2086 2.3416 0.6721 1. 2142 1.8062 -0.2965 -0.2924 0.9849 1.5465 1.60 1.9579 2.2517 2.3364 0.6392 1.2554 1.8586 -0.1437 -0.1525 1. 2029 1.1081 1. 70 1.8949 2.2954 2.3318 0.6094 1.2927 1.9035 -0.0354 -0.0546 1.4303 0.7970 1. 75 1.8661 2.3176 2.3296 0.5956 1.3096 1.9221 0.0060 -0.0174 1.5467 0.6766 1. 781 1.8491 2.3315 2.3284 0.5874 1.3194 1.9321 0.0282 0.0024 1.6196 0.6113 1.790 1.8443 2.3355 2.3280 0.5851 1.3221 1.9348 0.0342 0.0078 1.6409 0.5936 1.795 1.8416 2.3378 2.3278 0.5838 1.3237 1.9362 0.0375 0.0107 1.6527 0.5840 1.835 1.8209 2.3560 2.3263 0.5737 1.3352 1.9464 0.0614 0.0319 1. 7477 0.5124 1.90 1.7891 2.3861 2.3240 0.5582 1.3519 1.9578 0.0933 0.0601 1.9034 0.4143 2.00 1.7444 2.4339 2.3206 0.5361 1.3724 1.9609 0.1289 0.0914 2.1451 0.2983 2.10 1.7039 2.4838 2.3176 0.5160 1.3860 1.9440 0.1521 0.1115 2.3880 0.2139 2.25 1.6502 2.5635 2.3135 0.4893 1.3929 1.8750 0.1710 0.1274 2.7506 0.1279 2.40 1.6033 2.6501 2.3098 0.4661 1.3836 1.7472 0.1774 0.1325 3.1061 0.0739 2.60 1.5494 2.7785 2.3055 0.4396 1.3477 1.4772 0.1743 0.1299 3.5597 0.0313 2.80 1.5036 2.9229 2.3016 0.4173 1.2886 1.0901 01.640 0.1220 3.9805 0.0079

a Atomic units are employed throughout; 1 unit of length. 1 bohr= Q To convert to the dipole moment in debyes. multiply by 2.54154. 0.529172 A. and 1 unit of charge is e-=4.80286 X 10-10 esu. Origin chosen at geometric center of the molecule; a =!.

b The expectation values indicated by (I M I) are the sum of d To convert to the Quadrupole moment in buckinghams. multiply by NiA (l"iAI M.ll"iA) values divided by the total number of electrons. Thus. 1.34491. Origin chosen at A nucleus; a =0. tbese Quantities represent the average value of M. averaged over all occu-pied molecular orbitals.

no better than Hartree-Fock quality for these closed- III. DISCUSSION OF ENERGETIC RESULTS AND shell systems. Contrasted to neutral systems, those MOLECULAR PROPERTIES expectation values of AH- which depend importantly on the details of the molecular orbitals at large r values, The potential curves calculated here, together with e.g., rn , will probably be less reliable. empirical corrections based on the neutral-hydride

TABLE VI. Summary of expectation value and molecular property functions, M (R)."

SH-(10'22.r3.rl~4.r5.r2~, X 12;+).

R(b) ( II; I )b ( I TA21 ) (ITA-II) ( I TH-l I ) p.(eao) Q Q(eao2)d FA (e2ao-2) FH(e2aO-2) qA (eao-S) qH(eaO-S)

1.80 1.8321 2.5160 3.3403 0.5634 1.1459 2.9621 -0.6250 -0.6351 0.1010 1.6374 2.00 1. 7375 2.5757 3.3365 0.5166 1.2265 3.0085 -0.3654 -0.3307 0.7446 1.0053 2.10 1.6976 2.6071 3.3348 0.4959 1.2639 3.0302 -0.2921 -0.2276 1.0442 0.7904 2.20 1.6615 2.6393 3.3332 0.4769 1.2996 3.0496 -0.2433 -0.1475 1.3302 0.6240 2.30 1.6285 2.6726 3.3318 0.4592 1.3329 3.0653 -0.2123 -0.0855 1.6035 0.4935 2.40 1.5989 2.7069 3.3305 0.4429 1.3638 3.0757 -0.1939 -0.0374 1.8655 0.3913 2.512 1.5686 2.7466 3.3292 0.4261 1.3952 3.0788 -0.1837 0.0039 2.1466 0.3030 2.551 1.5587 2.7607 3.3287 0.4205 1.4051 3.0772 -0.1820 0.0150 2.2418 0.2775 2.70 1.5236 2.8164 3.3271 0.4007 1.4394 3.0568 -0.1813 0.0487 2.5923 0.1984 2.90 1.4825 2.8956 3.3252 0.3771 1.4738 2.9841 -0.1880 0.0756 3.0337 0.1278 3.05 1.4554 2.9586 3.3239 0.3615 1.4909 2.8888 -0.1944 0.0856 3.3436 0.0914 3.20 1.4310 3.0249 3.3228 0.3474 1.5007 2.7538 -0.1998 0.0933 3.6355 0.0663 3.40 1.4019 3.1186 3.3213 0.3305 1.5025 2.5060 -0.2042 0.0938 3.9965 0.0409 3.60 1.3762 3.2188 3.3200 0.3156 1.4929 2.1763 -0.2050 0.0914 4.3246 0.0243 4.00 1.3329 3.4403 3.3178 0.2907 1.4447 1.2618 -0.1973 0.0792 4.8847 0.0035

"Atomic units are employed throughout; 1 unit of length. 1 bohr= • To convert to the dipole moment in debyes. multiply by 2.54154. Q.529172 A. and 1 unit of charge is r=4.80286X10-10 esu. Origin chosen at geometric center of molecule; a =!.

b The expectation values indIcated by (I M I> are the sum of d To convert to the Quadrupole moment in buckinghams. multiply by NiA (l"iAI M.ll"iA) values divided by the total number of electrons. Thus. 1.34491. Origin chosen at A nucleus; a =0. these quantities represent the average value of Mil averaged over all occu-pied molecular orbitals.

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TABLE VII. Comparison of calculated and experimental spectroscopic constants for HF(XJ2:+) and OH(X2JI.); predictions for OH-(I2:+) ion.

Molecule Source B,(cm-I ) a.(Cm-l ) ",.(cm-I) ",.x.(cm-I ) R.(bohr) k.XlO-6

(dyn/cm)

Experiment 18.871 0.714 3735.2 82.81 1.8342 7.791 OH (1cr22cr23cr2l,r3, X 2JI.) Calculated 19.712 0.6502 4062.0 74.54 1.795 9.216

% Error +4.46% -8.94% +8.75% -9.99% -2.14% +18.29%

Experiment 20.949 0.797 4139.0 90.44 1.7328 9.657 HF(1cr'2cr'3cr'1,.-4, X 12:+) Calculated 21.868 0.7693 4469.0 80.34 1.696 11.261

% Error +4.39% -3.48% +7.97% -11.17% -2.12% +16.61%

Experimenta 18.87 3735 1.834

OH-(1cr'2cr23cr'1,.-4, X 12:+) Calculated 20.023 0.722 4087.9 87.82 1.781 9.334 % Errorb (+3% to +5%) (rv-10%) (+7% to +9.5%) (rv-l0%) (-1.5% to -2.5%) (,-+20%) Predictedb (19.4 to 19.1) (rvO.80) (3820 to 3733) (rv97.6) (1.81 to 1.83) (rv7.8)

• Lewis M. Branscomb. Phys. Rev 148, 11 (1966). The photodetachment spectrum of OH- was the experimental means employed to obtain these results.

b The percentage errors and "predicted" experimental values for OH- are based on systematic

internal error trends exhibited by the first-row hydrides. as exemplifiied by OH(X'II;) and HF(X l~+) given here. The signs of the % error are probably reliable. but the magnitude may be considerably off for at. WeX~, and ke.

TABLE VIII. Comparison of calculated and experimental spectroscopic constants for HCI(XJ2:+) and SH(X2JI,); predictions for SH-(I2:+) ion.

Moleculea Source B.(cm-I) a.(cm-I) ",.(cm-I) ""x. (em-I) R.(bohr) k.XIQ-V (dyn/cm)

Experiment [9.461J [2702J [60J 2.551 [4.201J SH(KL4u25cr'2r', X'II.) Calculated 9.756 0.2496 2860 42.32 2.513 4.707

% Error +3.12% +5.85% -1.49% +12.04%

Experiment 10.591 0.302 2989.7 52.05 2.4087 5.157 HCl(KL4u25cr'2 .... , X 12;+) Calculated 10.767 0.2682 3181 47.76 2.389 5.838

% Error +1.66% -11.19% +6.39% -8.24% -0.82% +13.21%

Experiment

SH-(KL4cr'5cr'2 ... ., X 12;+) Calculated 9.734 0.2471 2827 43.84 2.516 4.600 % Errorb (+2% to +4%) (rv-10%) (+5% to +7%) (rv-l0%) (-1% to -2%) (rv+15%) Predictedc (9.54 to 9.36) (rvO.27) 12692 to 2642) (rv48.7) (2.54 to 2.57) (rv4.0)

• The shorthand notation KSlcr' and Ls 2cr'3cr'1.r< is used. C These "predicted" results are values based on rough internal regnlarities and estimated % error b These are rough estimates of the actual error. The sign is probably correct but the magnitude above.

may be off considerably for ae, WeXe. and ke.

N W '0 0\

'"d ;.-C1 t"'

t=J

(")

;.­I;j

t=J

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ELECTRONIC STRUCTURE OF OH- AND SH- 2397

results, permit what should be reliable estimates of spectroscopic constants for OH- and SH-. Also, several ways of estimating the electron affinity of OH and SH are considered here although only one such procedure seems reliable to ::;0.2 eV, which is not small for elec­t~on affinities. Finally, the manifestations of the expan­SIOn of the charge distribution of AH- relative to AH on certain molecular properties are examined; that is, comparable results for (OH; OH-; HF) and (SH; SH-; HCI) triples are considered.

A. Calculation of Spectroscopic Constants and the Estimation of the True Values

The earlier calculations on the first- and second-row neutral hydrides AH gave potential curves which in turn gave reasonably good R. values and spectroscopic constants,l3·14 These calculated spectroscopic constants were the result of a Dunham analysis of the E(R) curves using results for nine or ten R values. In Fig. 1 the percent errors of the calculated R., w., B. and lXe values of LiH~HF and NaH~HCI, relative ~o the experi­mental values are shown. The characteristic sign and the roughly systematic magnitude of the error from CH and SiH onward for Re, B., and w. suggests that for hydrides (e.g., hydride ions and excited states) whose spectroscopic constants are unknown, that the calcu­lat~d values can be "corrected" to obtain very good estImates of the actual experimental values.

Using the potential curves calculated for OH-(X 11":+) and SH-(X 11":+), that is, nine or ten E(R) values from Tables III and IV, a Dunham analysis was performed and the resulting spectroscopic constants are given in Tables VII and VIII under the heading "calculated" values. From the discussion earlier and Fig. 1, one would expect that these calculated values for OH- and SH- would be somewhat in error. Furthermore, the systematic trends, at least for R e, We, and Be, observed for ~he neutral AH systems can be used to apply "cor­rectIOns" to these calculated values in order to arrive at rather good predictions of R" We, and Be for OH­and SH-. In light of the difficulty in obtaining such quantities for negative ions, or as a complement to photodetachment measurements of Branscomb5 and Steiner,S such predictions are given below. A conserva­tive .attitude is adopted here and, instead, ranges of ~red~cted values are given.19 Thus, for example, con­sidenng the general trend of errors in R. for the first-row hydrides (specific values are given for OH and HF in Table VII), it appears reasonable that for OH- the calculated R. value is probably between 1.5% to 2.5% too small, and hence the "predicted" value of Re is 1.81::; R.::; 1.83 for OH-. Analogous considerations are used to obtain predicted ranges of Be, W" and R. for OH- and SH-. Predicted values for lX" W.x., and k. are

19 In t~e earlier report of results for OH- (see Ref. 5, Table I, p. 17) speCific valu~s, n~t ranges were given. The specific values are, of course, .contamed In the ranges given, but specific predicted values belie the confidence behind the predictions given.

-75,30,--.-----,-----,-------,

'75,35

EChartree)

I ·7540

-IT j

-"~'75,45

FIG.2. Theoretical potential curves for OH(X IIIl) and OH-(12':+) (both on a common scale).

probably reliable with respect to the direction (or sign) of the correction, but the magnitude is less reliable. This approach to predicting R., W" and B. should not be viewed as preferable to an elaborate ab initio calculation of E~R), for OH- or SH-, but the latter is not easily obtamed for molecules with this many electrons.

Comparison of the predicted B., w., and R. values for OH-(X 11":+) with those obtained from photodetach­ment spectra of OH- and OD- by Branscomb6 is quite g?od. The uncertainty in w. is greatest having a pre­dIcted spread of almost 90 cm-I which leaves much to be desired. The photodetachment measurements on SH­and SD- are in progress by Steiner,6 but the indications again are that the potential curves of SH(X 2IIi) and SH-(X 11":+) are very nearly parallel. That is one of the major con~lusions of Branscomb's work; namely, that the potential curves of AH- and AH are nearly identical except for a constant vertical lowering of the AH­curve by electron affinity (E.A.) relative to the AH curve. Thus in Table I of Branscomb5 (repeated in Table VII here), R., Be, and w. of OH(X 2II;) and OH-(X 11":+) are essentially identical. Figures 2 and 3, based on the numbers given in Tables III and IV give the Hartree-Fock potential curves for OH- versu~ OH and SH- versus SH, respectively. These calculated results illustrate convincingly the fact that the potential curves are very nearly the same for AH and AH- except for a vertical displacement by (E. A.).

The p~otodetachn:ent measurements are apparently very delIcate, espeCIally as relates to comparing OH and OH- (or OD and OD-). However, Branscomb gives argument indicating how the photodetachment cross section versus wavelength for OH- and OD- can be used to determine if we(OH-) >we(OH), w.(OH-) < w.(OH), or w.(OH-)~w.(OH). As already indicated, Branscomb found the latter, i.e., w.(OH-)~w.(OH), and the calculations given here predict w.(OH-) > w.(OH) by 0 to 85 cm-I • However, for SH-, the calcula-

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2398 PAUL E. CADE

2.0 2.5 3.0

R (boh,)

tions predict that we(SH-) <w.(SH) by 10 to 60 em-I, an opposite shift of We for SH- relative to SH. This mayor may not be a real distinction between OH- and SH-, and it is near the limits of reliability of the pre­dicted w. values.

Finally, the characteristics of the OH- potential curve relative to the OH(X 2II.) potential curve is no cause for alarm (see Fig. 2). This would indicate that OH- is not stable relative to OH(X 2IId +e-, as indeed is the case to Hartree-Fock accuracy. The reverse is true of SH- relative to SH (X 2IIi). The difference in behavior shown by OH- and SH- relative to OH and SH, respectively, is due to the decrease in the change in correlation energy for SH(X 2IIi)---7SH-(X l~+). This is discussed in detail below.

B. Predictions of the Electron Affinity of OH and SH

As emphasized earlier, the results given here are at best the results characteristic of the Hartree-Fock approximation, or an independent-particle model. Hartree-Fock calculations alone will permit a direct calculation of tbe electron affinity (E.A.) for AH---7AH­which is reliable only if

(E.A.)>>I E""rr(AH)(~) -Eoor/AH-) (R.) I. (3)

Extensive calculations by Clementi and McLean20 for first-row atoms and by Clementi, McLean, Raimondi,

iO E, Clementi and A, D. McL~Il, Phys. Rev.l33, A419 (1964).

-397.95

-398.00

-398.05

E(ha,I,ee)

-398.10

FIG. 3. Theoretical potential curves for SH(X 211.) and SH-(X 1:t+) (both.on a common scale).

and Yoshimine21 for second-row atoms has demonstrated in detail that Eq. (3) is not valid and more usually

(E.A.)~I Eoorr(AH) (R.) -Eoor/AH-) (Re) I, (4)

holds.22 That is, the electron affinity, usually 0---74 eV, is about the same magnitude as the change in the correla­tion energy, also usually a few electron volts. The relative magnitudes of (E.A.), Eoor/A) - Ecorr(rl, and Erel(Al-Erel(A-), taken from the work of Clementi and McLean2il are summarized in Fig. 4. In summary, Hartree-Fock calculations alone cannot give reliable estimates of the electron affinities of atoms and, without doubt, the same is true for the diatomic hydrides (in fact, the same is true for almost all molecules).

Several alternative approaches to estimating the electron affinity for OH and SH using the calculated energy results given above are considered. Generally, the estimates must be good to 0.1 to 0.3 eV or less to be useful since the electron affinities of molecules are usually less than "'2.0 eV and this imposes a serious constraint as accurate estimates are required. It should be mentioned here that extrapolation procedures based on several isoelectronic members as has been repeatedly done for atoms are much less useful for molecules due

21 E. Clementi, A. D. McLean, D. L. Raimondi, and M. Yoshimine, Phys Rev. 133, A1274 (1964).

22 It can be recalled that in the case of ionization potentials, the change in correlation energy in going from AH->AH+ (a few electron volts) is usually smaller thall I.P. by a factor from 5 to 10,

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ELECTRONIC STRUCTURE OF OH- AND SH- 2399

TABLE IX. Estimates of the electron affinity of OH and SH (electron volts).

System" (E.A.)HFb

F (1s22s22p6, 2 P) 1.36

OH(lu22u23u21r, X2JI,) -0.10

CI(KL3s23p6,2P) 2.57

SH(KL4u25u22r, X2JI,) 1.21

a The F+.-_F- and Cl+r-Cl- calculated or estimated electron affinities are from the results of Clementi, McLean, Raimondi, and Yoshl­mine [phys. Rev. 133, A1274 (1964) l.

b These results employ directly the Hartree-Fock energies for A and A- or AH- and AH pairs.

• These results employ the outermost orbital energy of A-or AH-. The orbital involved is 2p, 3P, 1,.., and 2,.., respectively. for F-, Cl-, OH-, and SH-.

d These results employ (EA)HF plus a correction for the change in the

to the lack of such series; e.g., OH-, HF, NeH+, (NaH+ +, MgH+3, etc.), do not form such a series since the highly ionized members do not exist as stable entities. This will also inhibit the extrapolations of Eerm such as Clementi and McLean20 employ for first­row atoms. The various estimates of (E.A.) for OH and SH, together with analogous results for the iso­electronic and united atoms F and CI for comparison, are collected in Table IX.

The electron affinity is given by

(E.A')exptl = Eexact(AH) (Re) - Eexact(AH-) (Re') 2::0, (5)

= (E.A')HF+oEoorr(AH) (Re) +oEre,(AH) (Re),

(6)

in which the change in the correlation energy and the

• 3.0 3.0

2.0 2.0

E(eV) E(eV)

1.0 1.0

o o

.2 (2 ) 4 (2 ) LI( S) Bet'S) B P C(3p) N( S) O(3p) F p

ATOM

FIG. 4. Comparison of the magnitude of (E.A.), lIE.orr, and lIEr• 1 for the first-row atoms associated with the electron-attach­ment process, A+e--.A-. Values from Clementi and McLean (Ref. 20).

- .... (E.A.hd (E.A.)ue (E.A·)"Pt\f

4.92 3.08 3.448

2.90 1.40 1.91 1.83

4.08 3.56 3.613

2.42 2.11 2.25 (2.3)

correlation energy for an isoelectronic process (ionization of the lsoelec­tronlc neutral. i.e., Ne, Ar, HF, and HCl).

e These results employ (EA)HF plus a correction for the change in the correlation energies between UA and AH- systems, and the relative ZA independence of this value (ZA is the nuclear charge of A).

I The experimental values for F and Cl are from R. S. Berry and C. W. Reimann, J. Chern. Phys. 38, 1540 (1963); for OH the result is from L. M. Branscomb, Phys. Rev. 148, 11 (1966); for SH the experimental value Is a preliminary unpublished value of B. Steiner (private communication).

change in the relativistic energy are defined as

oEoorr(AH) (Re) = Eoorr(AH) (R.) - Eoorr(AH} (Re') 2:: 0, (7)

and

OErel(AH) (R.) = Ere,(AH) (R.) -Ere,(AH-)(Re')~O. (8)

The electron affinity (involving !lE between Re' of OH- and R. of OH) is usually different from the elec­tron detachment energy (involving the vertical !lE between OH- and OH at R=R.' of OH-). But in the cases involved here R,~Re' as we have already seen so that the electron affinity and the electron detach­ment energy are practically identical. In Eq. (5) we have indicated a positive electron affinity, or a stable negative ion, and that oEoorr(AH)(R.) is usually positive also. The Hartree-Fock electron affinity, (E.A.)HF, is not necessarily positive and often is actually even negative.23 The change in the correlation energy, o Ecorr'AH) (R.) , can, of course, be negative, but it is difficult to visualize such a situation involving a stable negative ion. As also indicated by Eq. (8), the oEr.l(AH) (R.) can be considered unimportant.24 To use the Hartree-Fock energy values to predict (E.A')exptl accurately, we can employ

(E.A').xptl = (E.A.)HF+oEoorr(AH) (R.) , (9)

so that the problem is to know oEoorr(AH) (R.) to within 0.1 to 0.3 eV at worse.

In Table IX the (E.A.)HF values are given (including the results of Clementi and colleagues20 ,21 for F and Cl. Two avenues of estimating oEoorr(AH) (R.) leading to estimates (E.A.h and (E.A.)u are given in Columns 4 and 5. The first approach obtains oEoorr(AH)(R.) from

23 Although it cannot be proved formally, a positive value for (E.A.) HF almost guarantees a stable negative ion, i.e., a positive (E.A').'Pt\. No exceptions are known.

M In addition, IB..I(AH) (14) - B..\(AH-) (R.) I must be very small. As Clementi and McLean (Ref. 20) show, this is indeed the case for atoms as one would expect since only differences in outer shells are involved as these make a small contribution to the relativistic energy. See Fig. 4 also for atomic values.

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2400 PAUL E. CADE

3.5 3.5 3.5

3.0 3.0 3.0

2.5 2.5 2.5 25 r " (OebyeJ ~ (Oebye)

20 OH(X'n) l2.0 2.0

I

J:: -6~4~"·""""'0!--,--'--'c0='-.4~--'-="0.8 1.2

1.5 .

1.0

-.0.8

SH(X'n;l

1.5 1.5

1.0 1.0

-0.8 -0.4 o 0.4 0.8 1.2 1.6

R-R,CBohr) R-R,(Bohr)

FIG. 5. Electric-dipole-moment function p.(p) as a function of p= (R-R.); Coordinate system with origin at geometrical center of molecule, a=t. (a) Comparison of OH(X 211;), OH-(X 1~+), and HF(X 1~+). (b) Comparison of SH(X 211;), SH-(X 1~+), and HCI(X I~+).

an isoelectronic pr.ocess (I.P.) involving different nuclear charges and is in the spirit of the extrapolation methods ordinarily used for atoms; specifically we consider the isoelectronic processes [(LP.)exptl and (E.A')exptl are in electronvoltsJ

HF(X l~+)---tHF+(X 211;) +e; (I.P')exptl = 16.04,

OH-(X l~+)-tOH(X 211i) +e-; [(E.A')exptl = 1.83J;

and

HCI(X l~+)---tHCI+(X 211;) +e; (I.P')exptl = 12.74,

SH-(X l~+)-tSH(X 211;) +e-; [(E.A')exptl"-'2.3J.

Using (I.P.hF values for HF and HCI reported earlier13.14 and the experimental ionization potentials, the change in the correlation energy in the HF-tHF++ e- process is 1.50 eV and the change in the correlation energy in the HCI-tHCI++e- process is 0.90 eV [neg­lecting OErel(AH)(R.) in both cases]. Adding and sub­tracting this quantity leads to the expression used to give (E.A.)r:

(E.A')exptl= (E.A')HF+oEcor/A/H) (R.'; ZA+1)

+ [OEeorr(AH) (R.; ZA) -OEeorr(A'H) (R/; ZA+1)J,

(E.A')exptl= (E.A.h+[oEcor/AH) (R.; ZA)

In these equations the atomic number ZA refers to the AH----tAH+e process while the primes and (ZA+l) refer to the isoelectronic ionization process AH-t AH++e-. Therefore, (E.A.) I will be reliable if the ZA

dependence of oEcorr is small. This does not require oEcorr be small or negligible (as is the case for oEr.I) , but that the difference of the differences be small. The resulting (E.A.)r values are given in column four of Table IX. This procedure is used by Clementi, Mc­Lean, Raimondi, and Y oshimine21 for the electron affinity of F and CI and their results are also included for comparison and perhaps as a guide.

The second procedure employed here is a windfall resulting from observations of the neutral hydride calculations previously reported.t3,l4 It was found that the decrease in the correlation energy of AH at R=R. relative to its appropriate united atom is small (",0.3-0.5 eV) and appears to change very little for different hydrides. This can be expressed

Eoor.'AH) (ZA; R.) -Eoor.'UA) (ZA+1; R=O)~K, (11)

in which K is ",0.3-0.5 eV and, very important for this procedure, K is roughly independent of the nuclear charge ZA or hydride involved. This observation can be used to predict (E.A.)u by adding and subtracting

oEoorr (UA) = Eoorr (A) - Eeorr (A -), (12)

for the corresponding united atom process. Straight­forwardly then,

-rE (UA)] u oorr ,

= (E.A.) u+[oEcor/AH) (R.) -OEcorr(UA)].

(13)

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ELECTROXIC STRUCTURE OF OH- A;.;rD SH- 2401

(al (b) ~~'~~/7-' . 7.0 7.0 7.0 7.0

6.0 6.0 6.0

/,/HCl(X'r')

6.0

HF(X'r') . ,'l ,

5.0, ,/-

5.0 5.0 ,," 5.0

I ,I

,. , ,/'

, " ,.' I

, ,-,.tt>-·-.A .... "'6..

SH(X'n,) 4.0 c /" . 4.0 4.0 4.0

, '" Q(Buckinghaml /i Q(Buckinghaml "

Q(Buckingham) '6

'. 3.0 ,./" OH(X'n,) 3.0 3.0

", 'b 3.0 ,

'.

-" , ..... 4 .. \ ,

".

/ \ 2.0 2.0 '''\ 2.0 2.0 \

". \ \, SH-(X'r')

OH1X'r') 1.0 .~LO 1.0 LO

, , "-' -'--'---'-

-O.B -0.4 0 0.4 O.B 1.2 -O.B -0.4 0 0.4 0.8 1.2 1.6

R-R,(Bohr) R·R,(Bahr)

FIG. 6. Electric-quadrupole-moment functions Q(p) as a function of p= (R-R.); Coordinate system with origin at A nucleus; a=O. (a) Comparison of OH(X 2fI.), OH-(X 12:+), and HF(X 12;+). (b) Comparison of SH(X 2IT.) , SH-(X 12;+), and HCl(X 12;+). Units in buckinghams ( ... 1~ esu·cmS) as suggested earlier (Ref. 18).

Superficially, this appears to be nothing more than the preceding approach except using the corresponding isoelectronic atomic process. This is, however, only the weakest aspect of this procedure as we can see by re­writing Eq. (13) in the form

(E.A')exptl = (E.A.) n+[Eoor/AH) (R.) - Eoorr(UA)]

= (E.A.)n+K(AH~UA) -K(AH-~UA-).

(15)

Therefore, the accuracy of (E.A.)n depends directly on the relative independence of the decrease in the cor­relation energy of AH relative to its united atom. It is important to note that the accuracy of (E.A.)n does not primarily depend on the K value being small, but on the alleged ZA independence of K. The resulting values of (E.A.)n are given in Column 5 of Table IX for OH and SH. The oEoor/UA) values used to obtain (E.A.)n are taken from Clementi and colleagues20 •21 [OEoorr(UA)(F~F-) =0.074 hartrees=2.01 eV] and/or (E.A').xpt![oEoor/UA) (CI~CI-) =0.038 hartrees = 1.04 eV].

The final estimates of (E.A.) for OH and SH come simply from Koopmans' Theorem which is just as valid for electron affinities to closed-shell systems as it is for ionization potentials for closed-shell systems. The resulting -EIlp, -Eap, -El.-, and -Ell". values in eV are given in column three for F-, Cl-, OH-, and SH-, respectively.

The various estimates indicate the well-known diffi-

culties associated with obtaining good estimates of (E.A.) values. The -E values are only rough estimates unless the change in the correlation energy just balances the energy gain accompanying orbital reorganization. The (E.A.)I values, as pointed out by Clementi and colleagues,21 should become more reliable for systems with more electrons and indeed this seems also the case for hydrides. The (E.A.)n values are surprisingly reliable, although we only have evidence that this approach should be useful for diatomic hydrides. The success of the (E.A')I for second-row hydride SH (needing only the change in correlation energy for AH~AH++e-, available using experimental ionization potentials) and (E.A.)n for both OH and SH (needing the change in correlation energy for A-~A+e, which is less available) has prompted an investigation of the electron affinity of CH, NH, SiR, PH, and other hydrides.25 At worst, it should be possible to predict whether (E.A.).xpt! is positive or negative for AH+ e-~AH- processes unless (E.A')exptl is ~0.1.

C. Comparison of Molecular Properties of OH- and SH- with (OH and HF) and (SH and HCI),

Respectively

The various expectation values and molecular prop­erties calculated here for OH-(X 12;+) and SH-(X 12;+) are given in Tables V and VI, respectively. The cor­responding quantities were given for OH(X 2IT.) , HF(X 12;+), SH(X 2IT.) , and HCI(X 12;+) in earlier

2Ii P. E. Cade "The Electron Affinities of the Diatomic Hydrides, CH, NH, SiH, and PH," Proc. Phys. Soc. (London) 91, 842 (1967) .

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2402 PAUL R. CADE

5.0 5.0 5.0 lb) ,,.,..,..

5.0 (a) II

,./' .' " ,e' .'.

.' ,

a"~" , ,. 4.0 4.0 4.0 -,' ,.

4.0 ./' Stf(X'r') ~

~/

3.0 3.0 ti"".

3.0 ,. ,,/"

/

2.0 2.0 2.0 1'/1' 2.0

QA( eo;) QA(eci!) QA(eci!) / I'

/

" 1.0 1.0 1.0 ,," 1.0

I

J 2 i R'

0 0 0 'O' ... -a_ .. o ..... oo._ •• oQ:o~_ •••••• 0

-1.0 -1.0 -1.0 -1.0

-0.8 -0.4 0 0.4 0.8 1.2 -0.4 0 0.4 0.8 1.2 1.6

R-R, (Bohr) R-R,(Bohrl

FIG. 7. Electric field gradient at the heavy nucleus, qA (p), as a function of p= (R- R.). (a) Comparison of OH (X ill;) ,OH-(X 12;+) and HF(X 12;+). (b) Comparison of SH(X ill;), SH-(X 12;+), and HCI(X 12;+).

publications26 •27 and are summarized in Figs. 5-8. One would expect the molecular charge distribution of OH- (or SH-) to be considerably expanded relative to its parent OH (or SH) and also even more expanded relative to the isoelectronic molecules HF or NeH+ (HCI or ArH+). In another publication26 this is indeed seen to be the case, but here the change in the charge distributions of OH- relative to OH and HF on the one hand, and of SH- relative to SH and HCI on the other hand, is gauged in terms of certain molecular properties. As mentioned earlier, more attention is directed to the molecular-property functions M (R) rather than on the calculated value at R=R •.

Four molecular properties, which are in principle measurable, are chosen to illustrate how electron at­tachment, AH+e~AH-, affects the molecular charge distribution. Thus Fig. 5 contrasts dipole-moment functions p(R); Fig. 6 contrasts quadrupole-moment functions Q(R); Fig. 7 contrasts electric-field-gradient functions at the heavy nucleus qA(R); and, finally, Fig. 8 contrasts electric-field-gradient functions at the proton (or deuteron) qH(R). The electric multipole moments provide a useful "external" gauge of the character of the molecular charge ... distribution and

liP. E. Cade and W. M. Huo, "The Electronic Structure of Di­atomic Molecules. VI.B. Certain Expectation Values and Molec­ular Properties of First Row Hydrides," J. Chern. Phys. (to be published) .

27 P. E. Cade and W. M. Huo, "The Electronic Structure of Di­atomic Molecules. VII.B. Certain Expectation Values and Molec­ular Properties of Second-Row Hydrides," J. Chern. Phys. (to be published).

Ji P. E. Cade, "The Rearrangement of the Molecular Charge Distribution of OH (X III;) upon Excitation, Ionization, and Elec­tron Attachment", (to be published).

reciprocally the electric field gradients at the nuclei offer a useful "internal" gauge.

The general characteristics of the electric dipole and quadrupole moments were discussed at length earlier26 •27

for the neutral series, AH. These results for OH(X 2II,) and HF(X 1~+), and for SH(X 2IIi) and HCI(X l~+) are here reproduced for comparison with OH- and SH-. A glance at Figs. 5 and 6 indicates a distinct change in the character of the p.(R) and Q(R) functions. In the Appendix the explicit dependence of p.(R) and Q(R) with coordinate system is summarized; for exam­ple, the dipole moment of AH- varies for two choices of coordinate systems by

p.{j(R) =p.a(R) +R(,s-a) , (16)

where a and ,s(,s>a) denote the position of the origin with respect to the A nucleus (see Fig. 9). Thus, any two dipole-moment curves for AH- are related by the straight line, R(,s-a); the point is that in comparing p.(R) for OH- (or SH-) with p.(R) for OH (or SH), p.(R) for OH- is uniquely determined to within addition of a straight line. In these comparisons the dipole moment is calculated using the geometrical center of the molecule (a=!), and the quadrupole moment is calculated using the coordinate system centered on the A nucleus in each case. Several other choices of the origin may actually be more useful for comparing the electric multipole moments of AH and AH-. For exam­ple, a could be chosen such that it corresponds to the center of negative charge or the center of mass, both cases giving a near zero (near Nucleus A). The data enclosed permit the results of any such alternatives to be obtained very easily.

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ELECTRONIC STRUCTURE OF OH- AND SH- 2403

3.0 3.0 '~3.0 3.0 HCl(X'r) \ \ (a)

~ · \ · \ I , • ~.5 2.5 • 2.5 2.5 • \ • . • \,

\ , • • \ \ ,

2.0 2.0 . 2.0 2.0 • \ \ • \. '0 . .. , \. 2ZA(Z 8) 1.5 1.5 \ '\ 2ZA(Z -16) 1.5 1.5 \R' ." · \ R3 A · · \"

qH{eci!) '\" q.(ea!) · q.(eo!) ~ "\', . \

1.0 " 1.0 1.0 \ '-., 1.0 .. '\ \ ',,-

'0 SH(X2ni~'\ "''n.

0.5 0.5 ................ !O

0.5 0.5 " "

'-. -. 0 0 0 0

-0.5 -0.5 -0.5 -0.8 -0.4 0 0.4 0.8 1.2

-0.5 1.6 -0.8 -0.4. 0 0.4 0.8 1.2

R·R.{Bohrl R-R.{Bohr)

FIG. 8. Electric field gradient at the proton (or deuteron), lJH(p), as a function of p= (R-R.). (a) Comparison of OH(X !fl.), OH-(X 12:+), and HF(X 12:+). (b) Comparison of SH(X2IIi), SH-(X 12:+), and HCI(X 12:+). (OH- and SH- designated by .6., and OH and SH designated by 0.)

The outstanding distinction in the p.(R) curves for OH- and SH- is that these curves show a maximum at large R, a maximum which will not be erased by using a new coordinate system. Generally, these curves behave with R as is expected with respect to the limiting values (especially as R-HX) ), while the curves for OH, HF, SH, and HCI are obviously incorrect as R-'>oo. Thus, as before, the dipole moment function for OH­and SH- can be represented by a relatively few terms of a polynomial expansion in (R - R.). Several such polynomial expansions were obtained, but a less com­plicated expression valid for a wide range of p is a simple parabola centered about the maximum, i.e., (in debyes) with p= (R-R.) ,

p.(OH-)(p) =3.S40-0.889(p-0.41)2 (17) and

p.(SH-)(p) =3.822-0.404(p-O.79) 2. (18)

Obviously these curves depart considerably from the near-linear behavior found earlier for OH, HF, SH, and HCI.

A point of further interest concerns the near-parallel behavior of the AH(X 2JI.) and AH-(X 12;+) potential curves and the significant difference in the p.(p) curves and how this would affect the line intensities in the infrared vibration-rotation bands of AH- versus AH. We are aware that the prospects of measuring the vibration-rotation bands of OH- or SH- do not seem encouraging, but the following remarks may yet be experimentally tested. To a good first approximation, the electronic potential acting on the nuclei in AH- is

just E(AH)(R) +(E.A.), the first term being the poten­tial for the parent AH molecule and (E.A.) a constant. [We are not distinguishing here between the electron affinity (E.A.) and the electron detachment energy.] Therefore, a common set of vibration-rotation wave­functions ~.J (p) are solutions to the SchrOdinger equations for nuclear motion for both AH and AH-. The key quantity measuring the intensities of the vibration-rotation bands is (suppressing rotation in­dices and valid for JI =J" =0 transitions only)

p.,.,,=1 Rvl•If 12=1 <~v, I pCp) 1 ~.,,) 12,

with (19)

Therefore, the quantity measuring the intensity dif­ference between AH- and AH for the same transition (v"---tv') is

l::.P"v" = I R",.1f j2(AJr) -I R",,,If 12(AH)o

or

flp.,v" = t l::.ak2/ <~., 1 pc I ~.,,) 12 + 1:; 2l::.(akal)

10-1 10<1-1

and l::.(akal)· 50' (aka,) (AIr) - (akaZ) (AR). (20)

The differences, flak' and l::.(akal), are from the poly­nomial representations of pep) for AH and AH- and a common integral over vibration-rotation wavefunctions is assumed for AH and AH-. The sum up to n, the degree of the polynomial to represent p. (p), involves

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2404 PAUL E. CADE

only a few terms and k, l=O is excluded unless J' or II! ;eo. The specific character of the shifting of intensity is not completely known unless some form for E(R) and hence '¥vJ(R) is assumed. However, the various !J,.ak2 and !J,.(aka/) values seem adequat~ to insure a substantial rearrangement, especially for transitions involving v' or v" greater than 2 or 3.

The electric quadrupole moments for OH- and SH­are also indicative of a pronounced change relative to OH and HF or SH and HCI, respectively. It was pointed out earlier that similar to the dipole-moment functions, the quadrupole-moment functions for OH, HF, SH, and HCI are almost straight lines (see Fig. 6). The analytical representation for Q(R) of OH- and SH- is again less simple and is adequately represented over a wide range of p by a parabola. Therefore, one may write (in buckinghams)

Q(OH-)(p) =2.6178+0.4735p-1.6937p2 (21)

and Q(SH) (p) =4.1386+0.0793p-1.1767 p2. (22)

[Several extended polynomial forms were also ob­tained.] As is well known the quadrupole moment depends on the coordinate system, or origin, used. All the results given here employ a coordinate system centered on the heavy nucleus. To convert these values to another coordinate system with origin along R, the relationship in the Appendix should be employed.

These two "external" probes corroborate one another and reveal a considerable change in the external per­spective of the AH- ion relative to the parent hydride AH. It is possible that the unusual character of p.(R) and QCR) for AH- is due to a difference in the Hartree­Fock approximation relative to AH and the isoelec­tronic system. This is, however, not a priori apparent and no indication that this is the case is available from other calculated quantities.

The second set of properties singled out for compari­son with the parent molecule and an isoelectric homolog involves the electric field gradients at the proton (or deuteron) and the A nucleus. These useful "internal" gauges are plotted in Figs. 7 and 8 in which the nuclear contribution, ,....,constX R-8, is also shown [using the (R-Re) scale for (OH; OH-) or (SH; SH-)]. The explicit values of qA(R) and qH(R) are given in Tables V and VI in atomic units e{l;J-3. It is possible that the quadrupole coupling constant for the deuteron in OD- or SD- ions trapped in alkali halide crystals can be measured, and using the well-known electric quad­rupole moment of the deuteron, the electric field gradi­ent at the deuteron can be obtained from experimental data.29

29 Campbell and Coogan (Ref. 4) have considered the electric field gradients qLI,qO, and I/H in LIOR crystals. These are calcula­tions based on a simple point-charge model and have absolutely no relationship to the values cited here •• In my opinion, ~e simple point-charge model for such a system 15 a useless and mIsleadmg characterization of the situation.

The qA(R) values for OH- and SH- behave in a regular manner. Thus the qA(R) curve for OH- (or SH-) lies intermediate between the qA(R) curves for OH(X 2ll.) and HF(X l~+) [or SH(X 2lli) and HCI(X l~+)]. These qA(R) curves for OH- and SH­are well represented by a simple polynomial form; i.e., withp=(R-R.),

qoCOW)(p) =1.7477+2.3403p+O.1692p2, (23) and

qS(SH-)(p) =2.2418+2.4365p-0.4577p2. (24)

(More extended polynomial forms were also obtained.) Thus, as for most hydrides, the quadrupole coupling constant (eqQh increases for increasing vibrational quantum number. As the nuclear contribution due to the proton, 2/ R3, indicates, most of the actual field gradient at A is due to the electronic charge density which is concentrated nearer the A nucleus. The united atoms involved, F-(IS) and CI-(lS) , have a zero electric field gradient as is approached by both OH- and SH- at small R values. Thus the increasing field gradi­ent with R is a direct measure of the distortion of the UA spherical charge density (as was the case for HF and HCI also).

The electric field gradients at the proton (or deu­teron) qH(R) for OH- and SH-, are almost indistin­guishable from those of OH(X 2lli) and SH(X 2II.), respectively. No significant approach towards the isoelectronic systems HF or HCI is evident. In sum­mary, qH (R), for AH- is slightly greater than qH(R) for the parent molecule, AH. In fact, I qH;(AH-) (R) -qfl.(AH) (R) I is a nearly constant quantity for both OH-; OH and SH-; SH pairs. The exponential form of qH(R) advanced earlier26 ,27 is also a good representation for OH- and SH-; therefore, a simple accurate repre­sentation of qH;(R) is

InqH(or)(p) = -O.669-3.283p (25)

and InqH(sr)(p) =-1.280-2.281p. (26)

As Fig. 8 indicates, the net electric field gradient at the proton is the result of nuclear and electronic parts opposite in sign and the parts are again (as with the neutral AH series) an order of magnitude greater than the resulting qH{R) values.

In summary, the electric field gradients, rather effective "internal" probes, detect relatively little noteworthy changes when an electron is attached to OH or SH. The resulting slight change in qH(R) and the reasonable shift in the qA (R) curve are both con­sistent with expectations as the major change in the charge distribution (which follows·· ·lT1+e--t·· ·11r4

or" • 27r3+e--t27r4 for OH- or SH-, respectively) is characterized by expansion and a net increase of charge more or less localized on Nucleus A. In particular, the attached electron does not go into a molecular orbital of key significance in the hydride bond.

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ELECTRONIC STRUCTURE OF OH- AND SH- 2405

The remaining expectation values and properties are consistent with the "external" and "internal" probes just discussed. Several trends between first- and second­row hydride properties are again evident; for example, the consistently smaller (w I ~ I w) value for SH­compared to OH- is again due to the tightly bound K and L shell contributions for SH-.

Finally, a note about the forces on the nuclei is necessary. The basic data for comparison are found in Table V and VI, and in two earlier studies.26 .27 It is evident that the FA (R) and F H (R) curves show the usual behavior and these force curves are almost coin­cidental for OH- and OH(X 2lli) [and for SH- and SH (X 2lli)]. This is a necessary and supporting fact for the close similarity of the potential curves of AH and AH- as discussed earlier. In fact, these curves, for OH- and SH- could also be used to obtain certain spectroscopic constants for further comparison. The force on the S nucleus is again27 disturbingly large.

IV. CONCLUSIONS

In general, the quality of results obtained here seems to parallel that previously obtained for the neutral hydride sequence, AH. Certain general conclusions stated earlier26 •27 are therefore also applicable here. Specific conclusions relevant for the AH- series are:

(1) Once a Hartree-Fock wavefunction is obtained for an AH system, it is usually a straightforward matter to obtain a wavefunction for AH- which is comparable in quality. This is especially true if no new molecular orbitals are involved in AH +e-~AH-, but the attached electron goes into an already partially jilled molecular orbital (e.g., CH, NH, and OH, but not BH and HF) . The same basis set composition may be used and ap­parently only one or two key orbital exponents need reoptimization. The major modification of the molecu­lar orbitals in AH- is thus accomplished via the linear expansion coefficients C,)..p'

(2) As suggested from photodetachment measure­ments of Branscomb5 for OH- and SteinerO for SH-, these calculations indicate that the potential curves of OH(X 2ll.) and OH-(X l~+) or SH(X 2lli) and SH-(X l~+) are very near parallel over a wide range of R values around Re. In particular, using the results of a Dunham analysis of EHF(R) and a correction from the neutral AH studies enables one to give very accurate estimates of Re, B., a., and w. to within a narrow range although the EHF(R) results are no better than Hartree­Fock accuracy.

(3) Despite initial pessimism about the calculated results being adequate to permit reliable prediction of the electron affinities of OH and SH, I am now con­vinced that reliable estimates for the energy of the AH +C---4AH- process are possible from only Hartree­Fock calculations and relevant atomic data. The most promising approach seems to be using the apparently small ZA dependence of the change in the correlation

energy of AH or AH- relative to UA or UA-. In par­ticular, these estimates are no worse than (E.A.) values given by Clementi, McLean, Raimondi, and Yoshi­mine21 for second-row atoms, and the second scheme given above (involving K) seems even more accurate.

(4) The attachment of an electron has dramatically modified the dipole- and quadrupole-moment functions from their behavior for the parent molecules OH and SH. No significant relationship to the isoelectronic molecules HF and HCI seems apparent either. As indicated, if this is actually the case, the molecular­vibrational-transition intensity distribution (v"<-?v') should be markedly different for OH(X 2lli) versus OH-(X 1~+) or SH(X 2IIi) versus SH-(X l~+) even though the shape of the potential curves and vibra­tional wavefunctions are very similar. This is consistent with the pronounced expansion of AH- relative to AH. It appears doubtful that this difference between jJ.(R) for OH and OH- (or SH and SH-) is due to any sharp difference in the Hartree-Fock approximation for AH versus AH-.

(5) The "internal" indices reveal that the proton (or deuteron) in AH- sees only a slightly different electric field gradient compared to the case in AH. The addition of an electron changes qA (R) in magnitude but only slightly in R dependence; thus, qA(R) for OH­(or SH-) now lies intermediate between qA(R) for OH and HF (or SH and HCI). As indicated above, this is consistent with the expectation that the extra electron and most of the e:h.'Pansion of the charge distribution is localized near the heavy nucleus in OH- or SH-.

ACKNOWLEDGMENTS

I am grateful to Professor L. M. Branscomb for call­ing to my attention the desirability of calculating good potential curves for OH- and SH- and comparing them to OH and SH, respectively. Professor Branscomb and Dr. B. Steiner have been most helpful in providing useful unpublished data and conversations about AH­negative ions. These calculations have made heavy use of programs written largely by Dr. Winifred M. Huo and Dr. Arnold C. Wahl, and I am pleased to acknowl­edge their cooperation. The continuing support, en­couragement, and advice from Professor R. S. Mulliken and Professor Clemens C. J. Roothaan is, as usual, very generous.

APPENDIX:"COORDINATE SYSTEMS AND DEPENDENCE OF PROPERTmS

Several of the properties discussed in this paper depend on the choice of coordinate systems and this appendix summarizes the general interrelationships for different coordinate systems. In Fig. 9, four relevant coordinate systems are indicated which differ only in the location of the origin along the internuclear separa­tion, and for the (XB , VB, ZB) coordinate system the direction of the Z axis is reversed. The arbitrary a-co-

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2406 PA UL E. CADE

Z.. za

v.

Z. ~_-lB

i 1 1 1

i I i

------ R --------<.,;

coordinate system differing only in location of the origin along R. The dipole-moment functions given in Tables V and VI are based on a=t, the geometrical center (gc) of the nuclei. A particularly valuable other choice is use of the c.m. coordinate system, a=MB(MA+MB)-\ so that in summary

FIG. 9. Coordinate systems; systems on Nuclei A and B are (as in Tables V and VI), and mirror images; the a and (J coordinate systems are arbitrary systems defined by the location of the origin aJongRwithZ" or Z~ p.o.m.(R) =p.g.(R) --21 R[MA - MB)/(MA+MB)]. in the same direction as ZA.

ordinate system (X .. , Va, Za) and p-coordinate system (Xp, Yp, Zp) may, for example, be a geometrically centered coordinate system (a or p=!) or a coordinate system located at the center of mass [a or p= MB(MA+MB)-l for a nuclear c.m. system]. For con­venience p> a is assumed and I a I or I p I ~ 1.

The electric dipole moment in an arbitrary a-co­ordinate system and in atomic units is

p.a(R) = -ZA(aR) +ZBR(l-a)

- L: N.A(~.A I z" I ~'A>' (AI) iC),

in which ZA and Zn are nuclear charges and N'A is the number of electrons in the ith molecular orbital of A symmetry. The dipole moment function employing the p-coordinate system is related to p.,,(R) by the equation

p.p(R) =p.a(R) -R(p-a) (ZA+ZB-N) , (A2)

in which N is the total number of electrons. Clearly, for a neutral system (ZA+Zn=N) the .choice of the coordinate system does not matter. For a singly charged diatomic molecule, the relationship is simply

p.p(R) =p.a(R) ±R(a-p), (A3)

[ +sign for (AB)+-sign for (AB)-] remembering that p>a. Obviously, one need calculate the (~'AI z I ~'A) matrix elements only once since Eq. (A3) can be used to obtain the dipole-moment function for any other

(AS)

The comparison of the R-dependence of p.g.(R) between OH and OH- (or SH and SH-) is therefore dependent on the coordinate system used for OH- and SH-. The curves in Figs. Sand 6 are based on the same geo­metrical center for both OH and OH- (and SH and SH-).

The electric quadrupole moment in an arbitrary a-coordinate system and in atomic units is

Q,,(R) =ZA(aR)2+ZB(1-a)2R2

- -! L: N'A<~'AI3z,,2-r,,21 ~iA>. (A6) iC),

The quadrupole moment employing another arbitrary p-coordinate system is related to Q,,(R) by the equation

Qf3(R) =Q,,(R) -2R(p-a)p.,,(R)

+R2(P2_a2) (ZA+ZB-N) , (A7)

which also involves the dipole moment p.,,(R) [not p.p(R) J. Once again the relevant expectation value over the electronic wavefunction need be calculated only once, as Qf3(R) for any other coordinate system is easily obtained.

The formula (A7), of course, satisfies the expected relationships, e.g., if p.,,(R) =0 and ZA+ZB=N, Q,,(R) is independent of the coordinate system. However, contrary to some suggestions, Q",(R) does depend on coordinate system if only p.a(R) =0 or ZA+ZB~N, that is, the quadrupole moment of A2+ or A2- molecules does depend on the coordinate system.

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