Fine-structure interactions in the ground state of O2

Volume 5, number 8

CHEMXAL

FINE-STRUCTURE

PHYSlCS LETTERS

INTERACTIONS

Receiw33

IN

31

THE

‘hiarch

1 June 1970

GROUND

STATE

OF

O2

1970

‘1’0 explore the effects of electron correlation on fine-structure consta.nts, the spin-spin interaction Xss in the 3Xi grpund state of 03 is studied with single-configuration and configuration-interaction was@functions of varying quality. Ml integrals are computed accurately to 10-S au. It is faund that hs = = 0.7’7 ,L0.02 cm-1 for all the single configuration functions studied (including a close approach to the Wartree-Foci< limit), but that configurationd mixing lowers h,s to 0.5 em--I. This reduction is brought about primarily by the double excitntion iiu?&--17ig i!,. Comparing our results with previous calculations and with the experimental data of Tinkbam and Stran3 berg. we conclude that spin-orbit coupling plays a predominant

role

in determining

the observed

fine-structure

A& initio studies of the two-electron terms that appear in the Breit-Pauli hamiltonian have

been Iimited so far to atoms [l]** and tc a few diatomic hydrides [3] for which it is possible to constr~~ct accurate wavefunctions based on onecenter expansions. This has been true because of the difficulty involved in calculating the multicenter integrals X. Recently, however, general expressions [6] have been derived for all the oneand two-center matrix elements that occur for wavefunctions constructed’from Slater-type orbitals @TO’s) I thereby permitting systematic investigations of all the Breit-Pauli terms in diatomic molecules. A question of considerable interest in this regard concerns the effect of electron correiation on the various interaction

constants, particularly those that depend on the

interektronic coordinates such as are contained in the spin-spin and spin-other-orbit operators. We report here results of a prelimin_ary nature on this question for the ground 3Zg state of * present

address: University Computing Company, pal:, alto, California, USA. ** See Fraga and hlaili (21 for additional references, $ Other studies in which integrals were aPProxim3ted or neglected include refs.[lij. See also ref.l6].

splittings

of 0,.

03. In the absence of molecular rotation, the level separation cosresponding to states with S,=OandSz = * 1 is given by 2X. where X = X,, + &so is the sum of a spin-spin hss and a

Tabk! I

Spin-spm couplmg constant A,, for 32: ; of 02

X”

2.2810

-142.09228

f lb

2.2810

-149.13237

IIre

2.2610

-149.55366

wd

2.1775

-149.67166

V

2.2810

-149.14029

Vlf

2.2810

-1+s.r9393

vrrg

2.2810

“f49.20627

basis a. Slater’s rules single-zeta b. Minimal basis set, ref. f9].

set.

ref.

[Sl.

C. Double-zeta basis set, ref. 19). d. Near Hartree-Fock set, ref. 191. CI using basis set a (R3@g + B 3V%1). e. Two-term f. Four-term CI (6 spin-symmetry eige~fuRctio~s} using basis set a (340 plus the first three doi%bleWcitations in table 2). g* Fourteen-term CI (30 spin-symmetry eigenfunctions) using basis set a (table 2). 529

CHEMICAL

Vofume 5, number 8

PHYSICS

spin-orbit (including spin-other-orbit) &So term. Although bath of these contributions depend upon expectation values of two-electron operators,

ASSprovides,

in the present case, the simpler

of exploring corr&t.tion effects, since its first-order correction to the energy is nonvanishing. Table 1 shows the results of our calculations on Xss for 02(3X& All the energy and spindipob$ integrals that occur are computed accurately with errors no larger than IOv5 au. Wavefunctions I-IV are based on a single conmeans

in which the MO’s are determined self-consistently. These functions differ from each other in the Slater- type basis sets used to construct the MO’s (cf., footriotes to table I). Beginning in Function I with a Slater’s rules single-zeta set {S] and proceeding to a near Hartree-Fock set [9] in IV, we see that Ass = 0.77 & 0.02 cm-l over the whole range, in reasonable agreement with the results of Kayama [5], Furthermore, it is found with Function I that a variation in R about Re = 2.281 au off 1% changes Xss by T 2.5%, a rather negligible change (however, see below) compared to the one, now ta be described, of mixing additional configurations with 3*6. To first order, the value of hss should depend [lo] only on configurations involving changes in the unpaired electron system (z Zg). A configuration which one might expect f o have an important

effect

arises

If the configuration

from

the double

interaction

LETTERS

1 June 1970

Since the addition of the one configuration 3%1 had such a dramatic effect, a mure detailed study was undertaken with a wavefunction based on SIater’S rules single-zeta basis set. All the

configurations that could be constructed through

quadrupote excitations were examined and ordered on an energy criterion [ll]. The fourteen dominant configurations are shown in table 2. The spindipole interaction was computed for the full fourteen-term expansion and for an expansion using only the first four configuraticns, with the results in table 1. To two significant figures, the parameter X,, = 0.47 cm-l for bath expansions. The considerable increase from the two-term resuLt of 0.29 cm-1 can be attributed to the change (to - 0.1601) in the coefficient B. In fact, using this value of B in the two-term CI expansion gives Xss = 0.48 cm-l (f’lg. 1. ); tlz~s, the additional CML~~i~~tt~~s ckange hSS tkrot@ renormalizatitm. Moreover,

it seems

to us that although

the CI

energy lowering* is only a small fraction (= 15%) of the total correlation energy, our value of Xss may have converged reasonably well. This is plausible because (a) Xss is insensitive to the basis set in a single-configuration approximation, * A slightly tower energy has been obtained by Schaefer and Harris [12]. using 110 co~i~r~tio~s and a minimal basis set optimized for the atom.

excitation

(CI) wavefunction

3P = A39?o + B 3ql is constructed,

then

Fig. 1 illustrates

the strong

on the coefficient

3 using the Slater’s

Set (E

= 2.281

au).

The

dependence

single

of Xss

rule basis

configuration

value

cm-l) appears at B = 0. When the CE coefficients are variationally determined (A = 0.9669 and B= -0.2551), Xss = 0.29 cm-l. Thus, adding only the term 3Pl decreases hss by more than a factor of two. of X ss (0.16

f For a description of the computer programs, see Pritchard

530

[ 71.

0 -

I, a25

I

I -0.15

I 025

I

-0.05

oo!i

015

!

B Fig. 1. Plot of hss versus B for the two-term tion 39 =A

3\ko + B 391 described

in

CI func-

the text

Volume 5, number 8

CHEMICAL

PHYSICS

Table 2

Dominant configurations Spatial configuration

Excitation

levelb)I

d d d 9 s d d d d t d

t ‘1 The quantity A& for a given spat&i confi~ra~ion fi) is equal to IC~~~~~-~o*)J, averaged over all spin functions.

Here C; is the coefficient of each confirmeigenf*ktions) which has been normalized so that Co = 1. b)~ = single, d = double, t = triple, q = quadruple cxci-

ration in a 34-terr;l CI (78 spin-symmetry

tations.

(b) the valence-shell excitation ;;u?u -.+;ig?i is dominant, and (c) all of the configurations oQher than those considered have relatively small weights.

In other

words,

the non- relativistic

energy and the spin-dipole correction to it depend in complementary ways on the particle correlations. CI studies of the type described here, starting with the Hartree-Fock root function, would be very worth- while. Previous calculations of ~~~ for the 32; state of 03 by various methods of approximation were performed by Kayama 151. His values vary widely (from -0.2 to + 1.5 cm-l), but it is interesting to note that Kayama’s “o-MO, r-semilocalizationw function (similar to our function V) gives Xss = 0.26 cm-l; moreover, his “pelectron” nine-term CI function (similar to VII) yields & = 0.47 cm-l. Kayama recognized that these relatively small values of Xss obtained from his best wavefunctions were inconsistent with the results of Tinkham and Strandberg [13] (TS), in which the experimental value of hss was determined to be 1.97 cm-l, or essentially all of the spin coupling constant of X = Xss + hsO = 1.98 cm-l. As pointed out by Kayama and Baird [la],

LETTERS

1 June 1970

it appears that this value of h,, is in error because it is based upon incorrect theoretical assumptions. In particular, TS aSSumed that the second-order. spin-orbit contribu~on could be obtained from the phenomenologlcal operator A L-S (2 = constant) rather ulan from the true microscopic spin-orbit hamiltonian. Although TS were certainly aware of tklepossible dangers of this assumption (see ref. 21 of TS), it nevertheless led them to a spin-orbit contribution Xso of only 0.01 cm-l. The accuracy of the TS anaiysis was further confused by their theoretical calculations with a fitted form of Meckler’s [15] gau_s;ian wavefunction; they found A,, = 1.17 which they considered too small because ZL &me wavefunction eve a value of (l/F$ much less than that derived from a magnetic hyperfine measurement. Using the Brett-Pauff ham iltonian, Kayama and Baird fl4 J estimated Ass = 0.82 cm-l and h,, = 1.16 cm-l on the basis of approximate calculations. If we assume that our value of Xss x 0.5 cm-1 is correct and use the experimental value [I 3f * of x = x,, + xso = 2 cm-l. then we obtain X,, = 1.5 cm-l, implying that spin-orbit effects contribute about 75% to the observed spIittini;s. Spin-orbit coupling may also be crucial for f&e derivative of X with respect fo the internuclear distance. We find with Function E [R(dXssld&)~Re = -0.2 cm-l, [~2(dzxss/~2)jRe = 0.6 cm-l, to be compared with the experimental data [13] for the corresponding rates of change in the total splitting constant of t- 0.564 *0.005 and 3.3 L 1.3 AB inzitiostudies of Xso as a cm-l, respectively. function of R are needed to test these roncZusions and to identify the principal forces acting_ A number of associates have contributed to the computer programs used in this work, including Dr. R. R, Giiman, Professor R. L. Matcha, Professor I. Shavitt, and Dr. A. C. WahL. We also thank Professor P. E. Cade for sending us their 02 wavefunctions prior to publication. Comments on the manuscript by Professor M. Tinkham and by Professor J. IX.Van Vleck were also helpful. *

The value of X observed in the 02 micrasvzwe spectrum can be traced historically &rough references to the rcry recent paper by Wilhcit nnd Barrett [IS].

[I]

M. Blume and R. E. Watsan, Proc. Roy.Soc. [London) A270 (1962) 127; A271 (1963) 56%

531

volume

5. m.untn?r 8

CX-IEMICALPHYSICSLETXZRS

R.E.Watson and M.Blume, Phys.Rev.139 (1965) A1209; C.Froesq Can.Y.Phys.45 (1967) 1501. I21 S. Fraga and G,Malli, Many-electron systems: Properties and interactions (W. B. Saunders Company. Philadelphia, 1968). f3l J.B.Lounsbury. J.Chem.Phys.42 (1965) 1549; 46 (1967) 2193; M.Horani, J.Rostas and H. Lefebvre-Brion, Can. J.Phys.45 (1967) 3319.

141 L.-Y. C. Chiu. Ph~s.Rev. 137 (1965) A382; J.W.McIver &d ti.F.Hztmek& J.Chem.Phys.&S ww 767: ”

_

‘?.E.*H.Walker and W.G.Richards, Symp.Faraday Sot. 2 (1968) 64: Phys. Rev. 177 (l.969) 100; J. Cheni. Phys. 52 (1970) 1311; H.Lefebvre-Brion and h’.Bessis, Can.J.Phys.47 (1969) 2727. 151 K. Kayama, J. Chem. Phys.42 (1965) 622. [Gj R. L. Ma&ha, 6. W *Kern and D.&I. Scharder, 3. Chem. Phys. 51 (1969) 2152;

1 June1970

and C.W.Kern, J.Chem.Phys.51 (1969) 3434; R. L.Matcha, D.J.Kouri and C. W.Kem, .I. Chem. Phys., to be published, [7] R. H. Pritchard, M. S. Thesis, The Qhio State University, June 1969. [S] C.F.Bender, to be published. [9] P.E.Cade, G.Mallfand H.E.Popkie, to be published. [lo] H.M.McConnell. Proc.Natl.Aaad.Sci.~USA\ . , 45 (1953) 172. [ll] A.Pipano and I.Shavitt, Inrern_J.Quantum &em. 2 ff968) 741. [I21 8. F. Schaefer and F.E.Harris, J. Chem.Php.. 48 (1968) 4946. I131 M.Tintiam and M.W, P. Strandberg, Phys.Rev. 37 (195.5) 937, 951. [141 K.Kayama and J.C.Baird, J.Chem.Phys.43 (1965) 1082; 46 (1967) 2604. 1151 A.Meckier. J.Chem.Phvs.21 119531 1750. ilSj T.T. Wilheit Jr. and A.fi.Barx&t, khys.Rev.A157 (1970) 213.

R.L.Matcha