Theoretical and experimental determination of the polarizabilities of the zinc 1S0 state

Dirk Goebel,1 Uwe Hohm,1,† and George Maroulis2,†1Institut fur Physikalische und Theoretische Chemie der Technischen Universita¨t Braunschweig, Hans-Sommer-Straße 10,

D-38106 Braunschweig, Federal Republic of Germany2Department of Chemistry, University of Patras, GR-26110 Patras, Greece*

and Department of Mathematics, University of the Aegean, GR-83200 Karlovassi, Samos, Greece~Received 12 March 1996!

A combined experimental and theoretical study of the polarizabilities of the ground-state zinc atom ispresented. In our interferometric measurements we have obtained the frequency dependence of the lineardipole polarizabilitya(v) betweenv50.05EH\21 and 0.14EH\21 ~hereEH is the Hartree energy!. A de-tailed theoretical CCSD~T! study~coupled-cluster with single and double excitations and perturbational treat-ment of triple excitations! gives the static~zero-frequency! values of the linear dipole polarizabilitya, thesecond dipole hyperpolarizabilityg, the quadrupole-quadrupole polarizabilitya2, and the dipole-dipole-quadrupole polarizabilityB. From these studies we conclude with the static values ofa538.8(8)e2a0

2EH21

~from experiment!, a539.2(8)e2a02EH

21 ~from theory!, g531(6)3103e4a04EH

23 , a25324.8(16.2)e2a04EH

21 ,andB522.37(24)3103e3a0

4EH22 . @S1050-2947~96!05009-3#

PACS number~s!: 31.15.Ar, 07.60.Ly, 31.25.2v, 32.10.Dk

INTRODUCTION

At present there is a renewed interest in the dipole polar-izabilities of the group IIb metals zinc, cadmium, and mer-cury, from theory@1–3# as well as from an experimentalpoint of view @4–7#. However, most of theab initio resultsare hardly comparable with measurements, because of a lackof precise experimental data. Therefore, in this paper we willpresent a combined experimental and theoretical study of thepolarizabilities of the zinc atom in the1S0 state, in order toget a comprehensive and thoroughly evaluated set of polar-izabilities. These include the static and dynamic dipole po-larizability a, the second dipole-hyperpolarizabilityg, thequadrupole-quadrupole polarizabilitya2, and the dipole-dipole-quadrupole polarizabilityB. These properties are rel-evant to the interpretation of various phenomena in atomic,molecular, and cluster physics, ranging from intermolecularinteractions to optical processes@8–12#. Atomic units areused throughout this paper; see Table I.

THEORY AND COMPUTATIONAL STRATEGY

The energy of an unchargedS-state atom in a weak, gen-eral static electric field can be written as@8,13,14#

E5E02~1/2!aFz22~1/8!a2Fzz

2 2~1/4!BFzzFz2

2~1/24!gFz41•••, ~1!

whereE0 is the energy of the free atom,Fz the field, andFzz the field gradient.a is the dipole polarizability,a2 thequadrupole-quadrupole polarizability,B the dipole-dipolepolarizability, andg the second dipole hyperpolarizability.

When the general field is that of chargeQ placed at a dis-tanceR from the atom, Eq.~1! is written as

E5E02~1/2!a~Q2/R4!2~1/2!a2~Q2/R6!

1~1/2!B~Q3/R7!2~1/24!g~Q4/R8!1•••. ~2!

The extraction of the atomic polarizabilities from finite-fieldcalculations has been presented in some detail in previouswork @14,15#. Both self-consistent field~SCF! and electron

*Permanent address.†Authors to whom correspondence should be addressed.

TABLE I. Theoretical polarizability and hyperpolarizability ofZn calculated with the B1@18s14p10d5 f # basis set, 145 CGTF. Allvalues in atomic units. Energy, 1EH54.359 748 23 10218 J; length,1a050.529 177 249310210 m; a, 1e2a0

2EH21 5 1.648 778

3 10241 C2 m2 J21; g, 1e4a04EH

2356.235 378310265 C4 m4 J23 ;a2, 1e2a0

4EH2154.617 048310262 C2 m4 J21; B, 1e3a0

4EH22

51.696 733310264 C3 m4 J22.

Method a 1022g a2 1021B

SCF 53.88 607 472.54 2482D2 212.34 2288 2143.70 227D3 1.10 29 22.40 215S4 20.65 27 27.11 4D4 21.54 234 215.93 31T4 21.74 229 220.27 32QR4 0.25 24 21.67 23DCCSD 211.09 2230 2112.37 189T 21.10 221 28.92 15

MP2 41.54 319 328.84 2256MP3 42.64 349 351.24 2271DQ-MP4 41.35 311 333.64 2243SDQ-MP4 40.70 304 326.53 2239MP4 38.95 275 306.26 2207CCSD 42.79 378 360.16 2294CCSD~T! 41.69 357 351.24 2278

PHYSICAL REVIEW A SEPTEMBER 1996VOLUME 54, NUMBER 3

541050-2947/96/54~3!/1973~6!/$10.00 1973 © 1996 The American Physical Society

correlation calculations rely on the above expansions. In thiswork we use fourth-order many-body perturbation theory~MP4! and single and double excitation coupled-clustertheory ~CCSD! and its extension, which resorts to a pertur-bational treatment to estimate the connected triple excitations@CCSD~T!#. We refer to other authors for the definition ofthese theories@16,17#. We give here only the definitions ofthe various orders of MP, MP25 SCF1 D2, MP35 MP21 D3, DQ-MP45 MP3 1 D4 1 QR4, SDQ-MP45 DQ-MP4 1 S4, MP45 SDQ-MP41 T4, where the numeralsindicate the order of many-body perturbation theory; S, D, T,and Q, respectively, stand for contributions from single,double, triple, and quadruple substitutions with respect to thezeroth-order SCF wave function, and QR denotes the contri-bution from disconnected quadruple substitutions plus thenormalization term, and for coupled-cluster theory, CCSD5SCF1 D CCSD, CCSD~T! 5 CCSD1 T.

To ensure the quality of our SCF calculations we use avery large basis set consisting of (21s13p10d) Gaussian-type functions~GTF! contracted to@16s10p7d# or 81 CGTF@18#. Adding 2s4p3d diffuse GTF, with exponents forminga geometric progression with the two most diffuse GTF ofthe same symmetry in the primitive set, we obtained@18s14p10d#. The addition of fivef -GTF with exponentsequal to those of the five most diffused-GTF gave@18s14p10d5 f #, hereafter basis B1, consisting of 145CGTF. A larger basis B2 was obtained from B1 with theaddition of two tightf -GTF. B2 or@18s14p10d7 f # consistsof 159 CGTF. An even larger basis set B3, 166 CGTF, wasobtained from B2 with the addition of another tightf -GTFwith exponentnf 5 4.894 923. The calculation ofa andguses five values ofE(F) with field strengthsF50, 0.002,0.004, 0.006, and 0.008e21a0

21EH . Weak fields were alsoused in the calculation ofa2 and B, fields produced bycharges of 100e at distances of 200a0 @14,15#.

The nine innermost orbitals were kept frozen in all elec-tron correlation calculations. Excitations to the 42 highestorbitals were not allowed in calculations with B1. For B2excitations were not allowed to the highest 33 orbitals.

All calculations were carried out with theGAUSSIAN 92setof programs@19#.

EXPERIMENT

The interferometric high-temperature measurements arecarried out in the same manner as has been done before inthe case of Cd@4,20# and Hg@5#, using an evacuated Mich-elson twin interferometer of geometrical path length of ap-proximately 2.20 m@21#. However, on account of the com-parable low vapor pressure of zinc, we have to employ hightemperatures~1260 K; see below! to ensure complete evapo-ration of the zinc samples. This makes it necessary to blockthe enormous amount of radiation in the near infrared emerg-ing from the pipe furnace in order to reduce the thermallyinduced movement of the end mirrors of the interferometer.This was achieved by using two aluminum plates in front ofthe optical elements of the interferometer. The white-lightand laser radiation used for the measurements transversesthese plates through little bores. To the best of our knowl-edge 1260 K is the highest temperature ever employed inequilibrium interferometric measurements of the refractiveindex against vacuum. Higher temperatures have only beenused in the determination of the polarizability of lead at 1480K against helium@22# and in interferometric nonequilibriumshock-tube measurements used for the determination of therefractive index of fluids@23#.

As usual, the measurements are performed in three parts:~i! preparation of the zinc sample of amount-of-substancedensity %5m/(MV), by filling zinc ~atomic massM565.393 1023 kg mol21) of massm into a quartz-cell oflength l /2 and volumeV, ~ii ! complete evaporation of thesample by heating the sample cell with a pipe furnace toapproximately 1260 K and counting the resulting interfer-ence fringe shift at three different wavelengthsl15632.99nm ~0.071 981 15EH\21), l25543.516 nm~0.083 830 75EH\21), and l35325.13 nm~0.140 138 9EH\21), whichyields directly absolute values of@n(lk ,%)21# for threedifferent wavelengths, and~iii ! use of dispersive Fouriertransform spectroscopy~DFTS!, which means recording of a

TABLE II. Experimental conditions and measured refractivities@n(v,%)21# of atomic zinc at differentfrequenciesv. In all cases, the volume of the sample cell is 26.5~2! cm3.

105@n(v,%)21#

T ~K! m~1026 kg! %(mol m23! l (T) ~m! v/EH\215 0.071 98 0.083 83 0.140 14

1085~2! 4.94~1! 2.8513 0.999 193~22! 6.6391 6.93541082~2! 0.999 192~22! 6.8038 7.09861083~2! 0.999 192~22! 6.9052 7.20741083~2! 0.999 192~22! 6.8735 7.18021084~2! 0.999 192~22! 10.13601153~2! 11.63~1! 6.7126 1.000 872~20! 16.6173 17.21441158~2! 1.000 873~20! 16.7438 17.37741163~2! 1.000 875~20! 16.7912 17.45881190~2! 16.36~1! 9.4426 0.998 635~14! 22.3592 23.30511199~2! 0.998 637~14! 22.4575 23.37591197~2! 0.998 636~14! 22.5082 23.43041192~2! 0.998 635~14! 33.40381209~2! 0.998 639~14! 33.5830

1974 54DIRK GOEBEL, UWE HOHM, AND GEORGE MAROULIS

white-light interferogram with a 100 W halogen lamp as ra-diation source, from which@n(l,%)21# can be obtained viaFourier transformation in the wavelength range between500 nm~0.091EH\21),l,910 nm~0.05EH\21) with aresolution of 7 cm21~3.23 1025EH\21). The experimentalconditions and measured refractivities are given in Table II.The refractive index curves measured with DFTS are fixed ton21 obtained atl1 and l2. More details can be found inRefs.@4,20,24#.

RESULTS

Discussion of theoretical results

All theoretical results presented refer to static~zero-frequency! values of the polarizabilities. MP4 and CCSD~T!results calculated with basis B1 are presented in Table I.MP2 results calculated with basis B2, and more SCF valuesobtained with B3 are given in Table III, where the presentexperimental and theoretical findings are compared to previ-ous efforts. The three sets B1, B2, and B3 lead to SCF dipolepolarizabilities of 53.88, 54.05, and 54.06e2a0

2EH21 , respec-

tively, quite close to the numerical Hartree-Fock~NHF! val-ues of 54.06@25# and 54.061e2a0

2EH21 @3#. The quality of B1,

B2, and B3 may also be judged by the close agreement of therespective SCFa2 and g with the NHF values of474.4e2a0

4EH21 and 6093102e4a0

4EH23 , an agreement better

than 1% in all cases. Table I shows that electron correlationchanges drastically the SCF picture of the atomic polarizabil-ities of Zn. In absolute terms, all MP4, CCSD, and CCSD~T!values are lower than the respective SCF ones. The B1CCSD~T! values ofa, g, a2, andB are 22.6, 41.2, 25.7, and42.3% lower than the respective SCF results. It is worth no-

ticing that the B1 MP2 value ofa is quite close to theCCSD~T! one. The addition of tightf -GTF to B1 results inrather important changes for the MP2 values. We rely on theobserved pattern in the B1 correlated values in order to pro-duce estimates for all properties by combining the B1 and B2results. We use the~D2 correction calculated with B2!/~D2correction calculated with B1! ratio to scale the B1 CCSD~T!electron correlation correction and propose as recommendedtheoretical values the following:a539.2(8)e2a0

2EH21 ~an

uncertainty of 2%!, g531(6)3103e4a04EH

23 ~an uncertaintyof 20%!, a25324.8(16.2)e2a0

4EH21 ~an uncertainty of 5%!,

andB52237(24)3101e3a04EH

22 ~an uncertainty of 10%!.Next, we compare our dipole polarizability results to pre-

vious efforts. Rosenkrantzet al. @26# used an effective corepotential~ECP! to obtain a relativistic SCF value of the di-pole polarizability of 43.2e2a0

2EH21 and a respective multi-

configuration SCF~MCSCF! one of 35.1e2a02EH

21 . Both re-sults are lower than our nonrelativistic values. Our values arecloser to the variation-perturbational~VP! value of40.9e2a0

2EH21 reported by Chandler and Glass@27#. We also

judge worth mentioning a CEPA~coupled electron pair ap-proximation! value of 47.83e2a0

2EH21 by Reinsch and Meyer,

referenced by Stiehler and Hinze@3#. The most recent theo-retical quasirelativistic CCSD~T! value of the static dipolepolarizability a537.86e2a0

2EH21 of Kello and Sadlej@2# is

only slightly lower than our best value. Last, we mention thevalue of 38.5e2a0

2EH21 recommended by Miller and Bederson

@11#, remarkably close to our results.Our results do not include possible relativistic effects on

the reported values. In a recent study Kello¨ and Sadlej@28#calculated the longitudinal component of the dipole polariz-ability of FCN, ClCN, BrCN, and ICN. They obtained the

TABLE III. Comparison of theory and experiment for the static polarizabilities and hyperpolarizabilitiesof Zn.

Method a 1022g a2 1021B

SCFa 54.06 603 472.54 2485SCFb 54.05 604 472.54 2485MP2b 39.02 268 299.63 2215SCFc 53.88 607 472.54 2482MP2c 41.54 319 328.84 2256CCSD~T!c 41.69 357 351.57 2296B-Rd 39.2~8! 310~60! 324.8~16.2! 2237~24!NHF 54.06e, 54.066f 609f 474.4e

Other theory 43.2g, 35.1h, 40.9i, 37.86j

Experiment 39.03k ; 38.58k

aPresent investigation. Basis set B3,@18s14p10d8 f #, 166 CGTF.bPresent investigation. Basis set B2,@18s14p10d7 f #, 159 CGTF.cPresent investigation. Basis set B1,@18s14p10d5 f #, 145 CGTF.dBest or recommended theoretical values, present study.eNumerical Hartree-Fock values of Markiewiczet al. @25#.fNumerical Hartree-Fock values of Stiehler and Hinze@3#.gRelativistic SCF with effective core potential~ECP! by Rosenkrantzet al. @26#.hRelativistic multiconfigurational SCF with effective core potential~ECP! by Rosenkrantzet al. @26#.iVariation-perturbational result of Chandler and Glass@27#.jQuasirelativistic CCSD~T! result of Kelloand Sadlej@2#.kPresent investigation, static polarizability obtained from an extrapolation of dynamic values to zero fre-quency.

54 1975THEORETICAL AND EXPERIMENTAL DETERMINATION . . .

nonrelativistic values~relativistic results in parentheses! of25.9~25.9!, 45.5~45.5!, 53.9~53.8!, and 69.0~68.2!e2a0

2EH21 ,

supporting the idea that relativistic effects on the dipole po-larizability of Zn are likely to be very small if not negligible.

Discussion of experimental results

Our absolute measurements are evaluated according to theLorentz-Lorenz relation

n2~v,% !21

n2~v,% !125

NA

3«0%a~v!, ~3!

where NA56.022 136 73 1023mol21 is Avogadro’s con-stant and«058.854 187310212 F m21 the permittivity ofvacuum. We have used a constraint fit which ensuresn[1for %50. From this fit the dynamic polarizabilitiesa(v1)543.03(32)e2a0

2EH21 , a(v2)544.77(31)e2a0

2EH21 ,

anda(v3)563.26(12)e2a02EH

21 are obtained. The rms errorof the fit is given in parentheses. Our DFTS measurementshave been performed at the lowest (%52.8513 mol m23)and highest (%59.4426 mol m23) densities used. To ensurecomplete evaporation and to avoid recondensation of the Znsample, in our measurements we have to use equilibriumtemperatures of approximately 1110 K and 1260 K, respec-tively. For each density five independent DFTS measure-

ments have been performed, from which the arithmetic meanis calculated. The two polarizability curves obtained in thisway are in very good agreement and show a maximum de-viation of 0.5% nearv50.05EH\21. A possible temperaturedependent dispersion of the polarizability, which has beenobserved in the case of xenon@29#, is not detectable for zinc.For our further studies, therefore, we assume thata(v) isindependent of temperature and pressure and the arithmeticmean of all DFTS curves obtained in the frequency rangebetween 0.05EH\21 and 0.091EH\21 is used. Thisa(v)curve, which includes the absolute valuesa(v1) anda(v2) exactly, is shown together with our recorded value ofa(v3) and with the best theoretical value of this work in Fig.1. The five diamonds represent the polarizabilities of Cuth-bertson and Metcalfe@30#, which have been obtained fromthe original data of their measurements given in Ref.@30#. Ingeneral, they are lower by approximately 3e2a0

2EH21 and

show a much larger scatter, so that their frequency depen-dence is much more uncertain.

In the case of Zn, we have explored several possibilitiesto fit the frequency dependence ofa(v). In contrast to ourmeasurements on cadmium@4#, where the frequencyv3 isvery close to the3P1←1S0 transition, and, therefore, hasbeen neglected in the fit ofa(v) in that case, here we haveto includea(v3). We have done this in four different ways.First, with a statistical weight of 1, which is equal to theweight of the other 1286 data points obtained from the DFTSmeasurements, and second, with a twofold and with a tenfoldstatistical weight. In the fourth case, we have used a con-straint fit which ensures thata(v) matches exactly the mea-sured valuea(v3). All data sets are analyzed according to athree-term Cauchy formula

a~v!5S~22!1S~24!v21S~26!v4 ~4!

and alternatively according to a one-term Kramers-Heisenberg dispersion formula

a~v!5a~0!v02

v022v2 , ~5!

wherev0 is an effective transition frequency. The results ofthese fits are presented in Table IV.

In the case of a perfect measurement,S(22) should beexactly equal toa(0). However, in the present studyS(22) is larger by approximately 0.5e2a0

2EH21 than a(0).

Additionally, the fit parameter given in Table IV depend onthe statistical weight of the uv valuea(v3). As the regionbetween 0.05EH\21 and 0.091EH\21 is represented by

TABLE IV. Cauchy momentsS(22),S(24),S(26), static polarizabilitiesa(0), andeffective eigen-frequenciesv0 for Zn obtained from the experimental results according to Eqs.~4! and ~5!. Da is the rmserror of the respective fit.

Statistical S(22) S(24) 1021S(26) a(0) v0 DaEq. ~4! DaEq. ~5!

weight ofa(v3)

1 38.961~20! 617.9~6.2! 3003~47! 38.650 7~69! 0.226 41~16! 0.09 0.072 38.998~18! 602.2~5.4! 3154~37! 38.626 6~65! 0.225 79~14! 0.10 0.0810 39.030~17! 588.6~4.5! 3284~23! 38.580 9~50! 0.224 612~86! 0.10 0.08→` 39.038~17! 585.1~4.2! 3318~17! 38.559 9~29! 0.224 082~16! 0.10 0.06

FIG. 1. Frequency dependence of the linear dipole polarizabilityof ground-state zinc. Full circles: absolute measurements, this work;full square: best calculated value, this work; open diamonds: resultsof Cuthbertson and Metcalfe@30#; full line: quasicontinuous DFTSmeasurements; dashed line; fit of the experimental values.

1976 54DIRK GOEBEL, UWE HOHM, AND GEORGE MAROULIS

1286 data points, we feel it is reasonable to take a statisticalweight larger than one fora(v3). Therefore, the results inthe second and third row of Table IV are judged to be thefinal results of the measurements ofa(v) of zinc. From thecombination of S(22) and a(0) a final experimentalstatic value of the dipole polarizability ofa(0)538.8(8)e2a0

2EH21 results, where we have included an error

of 2% resulting from the uncertainties in the refractive indexand amount-of-substance measurements. According to theresults ofS(24) andS(26) in Table IV we recommend theCauchy moments S(24)5590(10) and S(26)53.25(20)104 for the zinc 1S0 state. The experimental re-sults are used to get a first estimate of the dispersion inter-action energy constantC6 according toC6'(3/4)a(0)2v0.Using our experimental results presented in the second rowof Table IV we getC65257EHa0

6 . As experience shows,this result is likely to be too small by 5–15 %.

CONCLUDING REMARKS

We have presented a combined experimental and theoreti-cal study of some polarizabilities of the ground-state zincatom. The experiment extends to measure the frequency de-pendence of the linear dipole polarizability, while ourabinitio study is used to calculate static values of the lineardipole polarizability as well as higher and nonlinear polariz-abilities. Both studies are connected through the static valueof the linear dipole polarizabilitya(0), which is obtainablewith high precision~see the errors in Table IV! from experi-ment by extrapolation of the frequency-dependent values tozero frequency. Considering the present system, the experi-mental value ofa(0)538.8(8)e2a0

2EH21 agrees quite well

with our best theoretical one of 39.2~8!e2a02EH

21 . On accountof this agreement we conclude that the other calculated po-larizabilities a2, g, and B are superior to those presentedhitherto in the literature.

In this work, we have calculated a static second dipolehyperpolarizability g of 31(6)103e4a0

4EH23 . For a large

class of atoms and molecules we have shown that a roughestimate of g is given in terms of the Cauchymoments S(22), S(23), and S(24) viag'27.36S(23)S(24)/S(22) @31,32#. Using the estimateS(23)5@S(22)S(24)# (1/2), we get the approximationg'633103e4a0

4EH23 . This is twice the calculated result of

this work, but is roughly the numerical Hartree-Fock valueof Stiehler and Hinze@3#; see Table III. It seems that in thecase of zinc, the approximation used forg only gives theright order of magnitude. To the best of our knowledge thedipole-dipole-quadrupole polarizabilityB has been calcu-lated for the first time, and there is only one value of thequadrupole-quadrupole polarizabilitya2 to compare with.Together with the proposed value of the dispersion-interaction energy-coefficientC65257EHa0

6 all polarizabil-ities are extremely useful in evaluating collision-induced ab-sorption and collision-induced light-scattering spectra ofzinc, which, however, have not been recorded up to now.They also might provide a deeper insight in the interpretationof rovibronic spectra of the zinc dimer@33#.

As in the case of the equilibrium system NO2-N2O4 @24#we have presented again a fruitful and very successful col-laboration of experiment and theory in order to obtain high-quality electro-optical properties of matter. This collabora-tion becomes more and more important in view of gettingprecise values of higher and nonlinear polarizabilities whichare of great significance in various fields, but which arehardly measurable@34–36#.

ACKNOWLEDGMENTS

D.G. and U.H. are grateful for financial grants from theDeutsche Forschungsgemeinschaft and the Fonds der Che-mischen Industrie.

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1978 54DIRK GOEBEL, UWE HOHM, AND GEORGE MAROULIS