- Email: [email protected]
Calculation of photoionization cross sections using ab-initio wavefunctions and the plane wave approximation
:.
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OFBHO~O~ONIZATION CROSS SECT&S
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+INSrrO
W; THIEL’ M&.DEWAR k. KQMO&IKI dc&rtmen> of Ckmktry, The ‘U&ersit;.of Texq ~tAustin,
‘.
”
Austi~r, Texm 78712, USA
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16 Se~tember.1974
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,A &$hod is oiled for ~~~~cu~tio~ of photoionization ,mdimal basis set STO.NG ~ave~u~~tion &d the foal &ate by ‘. s6ra.l small mo!ccules compare well with the .btid intensities band intensities obtained from valence election wavefunctions. , : .’ dtiati&s lead to comparable results. . . : ,’ ,’
-. 1. &r&ction t
..
?hotoionizatio~ cross sectipns for the ionization of valence molecular-orb&& (MO’s) can be calculated
b$ a’recently deveioped theory [l] which describes .the $iti’al state of the ionized molecule by LCAO SCF MO’s with a vale&e basis set of Slater.atomic orMaIs (STO’s), and the fiiaI state by the plane wave approximation. This theoryreproducks the observed ‘band intensities in‘~ photoelictrdn spectroscopy fairly well, especially their variation in going from He I to ze II excitation f’kl, and can be useftil in assignkent probIems [3]. In thd soft X-ray regioil, ho+fever, application of .:lhe theqy is not feasible. The transition moment foi’ ‘_the iqnization$s proporti&al~to the qve&p between the ionized MO and the piane vjave of the ejected ph6toekction:‘[l, 41, Since in ESCA spectroscopy +&e .wavelen&h”of the photoelectron is typically in the ordec of 0.3 a, a reliable pre~cti~n of &CA band in-‘,
.. .’ . : . *it3&6seque& of 3 &timi& se& of papers on%&or$ I :. and applicati?~ ofphtifoeiectmn spectroscopy by A. Schtieig a&d co-wdrkeq $-ds~paper wiil b-e t&ga:rdedaS part-. ..?X bj;t$atlaboratory.. : ; .’ ,_’ ,. _, . .‘,
cross sections iepres.enti& the initial state by a thy plane w&e appro~~ation. Numericlll results for 1 in published ESCA spectra, and are superior to ESCA For the UV region, ab-initio and valence electron cd:
tenSities requires a good representation’ of the electron densiiy ~s~~bution in the immediate vicinity of the nuclei which of course is not obtained in the valence electron approbation, To overcome this ~f~cul~ it has been suggested [5,6f to associate the LCAO co-
efficitmts obtained by valence electron calculations with :sTO’s which ar~,ort~ogonal to the core AO’s,
thereby effectively introducing some “core character” the valence eIectron wavefunctions. :, Ln our opinion, the description of the initial state by ab%nitio waveftinctions offers’s more direct appro& to.the’ca.lculation of ESCAband intensities. in the following sections we therefore derive the necessaqr formalism, report numerical results fo; several molecules and discuss the connection between the present work tid the semi-empirical model (7-101 for ESCA, band intdn&ies?. ’ into
,.
,:.
..
.‘ * .+fier:co&letin~ our work; similar results hsve been pub ;li&kd [ 151 .obtaine$ by ysing ab-initio’wavefyxtjoqs of ,.’the gaussian lobe typq for thk in&l state, and orthogonalizel plane w&es’for t&e fqal state. The reported theore% -&I Yalues and k+eriment+l est+&es agree well with auf ‘_‘.’ m-&s listedin tab16 1.‘: : ._ ., . . _,.‘_ : .‘:, .: .. “.. .’ ,‘~ ..
Volume 3 1, nukbcr 2
CHEMICAL PHYSICS LETTERS
2. Theory
As shown previously [I] the photoionization section Ui of the jth MO.is given by
cross
where e and m, aie the charge and mass 3f an electron, fi is Planck’s rationalized constant, c the velocity of light, ki the wavenumber of the photoelectron,,.and Eph the photon energy. QD and Qnabare one- and twocenter terms (atoms a and b) originally defined for MO’s with a valence basis set of STO’s. In our present study, we use a minimal basis set expanded in terms of the ab-initio ST0 NG wnvefunctions [l 11, the basis orbitals xnlnl being represented by linear combinations of NG gaussians (GTO’s)
where N ,],,i, dn/,ijanl,i and 5,) are the normalization constants, contraction coefficients, orbital exponents and scaling factors, respectively, for the basis orbital. Y, (0 ,@)is a normalized spherical harmonic, and a0 Bohr’s radius. The introduction of ST0 NC wavefunctions leads to the following modifications in our previous formalism [l] : The expressions for the one- and two-center terms must include not only valence contributions,. but also core contributions and valence-,core cross terms. Summing over all orbit& in the minimal basis. set we obtain:
canIn is the LCAO coefficient of the basis A0 x&i. at atom a; aft&otating the internuclear axiS between
atoms Q andb (diStanCeR~b)intOtheZ-axiSofthe ccordinate system, the transformed coefficients are denoted by ?anl,,,: The interference terms GrillI,m (k,&ab) have been’given explicitly [I]. The t&sition moments Manlm for the gaussian basis AO’s are proportional to the Fourier transforms of the radia1 part of (2) and can be evaluated analytically:
X exp (-kf/4
cu’,,,,J.
(6)
3. Results and discussion Photoionization cross sections have been calculated for a variety of molecules at different levels of the STONG approximation (N = 3-6). The input data consisted of the experimental geometries, vertical ionization potentials, and ST0 NG eigenvectors; the contraction coefficients, orbital exponents, and scaIing factors were taken from refs. [11,12], the orbital exponents being identical for s- and p-orbit& of the same shell. We summarize the numerkal results for UV excitation (He I, He II) and for X-ray excitation (Mg Kci): In the UV region the results for the difFerent ST0 NG levels are almost the same since the transition moments (6) converge very rapidly to the limiting ST0 value, the deviations usually being less than 1% even for the 3G basis. The calculated values for the relative band intensities are usually quite similar to those obtained in the valence electron approximation [2] and therefore need not be reproduced here. There is one notab!e exception. In the valence electron calculations a semi-empir@al correction for the upp two. center contribution was introduced in order to restore the expected relation between the sign of this contributiqn and the app overlap integral [ I]. This correc.lion hlrns out to be unnecessary in the present abinitio calculations which is not surprising since overlap effects are now included in the normalization of the eigenvectors. In the valence electron cal&Iat$ons, the correction can likewise be avoided by using the normalize! coefficient matrix C’=EwV2 C (overlap matrix S) titead of the original coefficient matrix C. ,In the ESCA region the‘calculated photoionization
Volume
31, number 2
CHEMICAL
PHYSICS LETTERS
,.’
1 March 1975 ‘:
.
‘,
discrepancies between, e.g., 3G This is due to the fact that the
are some quantitative
Tabre ‘1
/+bso!ute and relative ,photoidnkation ESCAsp&a (Mg Keexcitation)
cross sections for
and 6 G calculations. kansition
moments
(6)
for ESCA.ionizations
are
nlahly detumincd by the shape of the,basti orbital tin the immediate vicinity of the nucleus which should be represented accurately enough only, by the higher order.expansions. Therefore, STO. 6G eigenvectors 'N2
15.60 16.98 .1&78 31.3. .409.9 409.9
CO
14.01 16.91 19.72
38.3 295.9 542.1 Hz0
1262 14.74 18.51. 32.2 539.7
50 1 ir 4 a
30 2 0 :io 1 br 3 21 1 b2 2a, 1 al
170.9 234.5
2a1.
,.
.22.2
m4
14.35 2291
230.7 c2H4
1 nTu 20, 2 Lrg 1lJ” l.og’
2 bl 5 a1 2 b2 4 ai Ibl,lb2,3al
10.47 13.33 15.47
H2S
lO.51 -1285 14.66 15.87. 19.1
23.6 290.6
T90.6
0.581
3gg
1 tp 2al 1 al 1 bzu 1b2g 3 ag 1 bl, 2bau
2ag
1 blu 1 ag
0.95
0.126 3.004 3.150 61.187 57.055
0.21 4.31 5.15 100 93
0.974 .’ 0.249 2186 4.137 36.801 77.532
1.26 0.32 282 5.34 47 100
0.83 0.42 2.08 4.16 46 100
0.166 0.935 0.085 3.903 77.435
0.24 1.21’ 0.11 5.04 100
0.50 1.37 .0.43 5.26 100
8.20 4.68 3.24 1.27 300
.6.43 8.26 4.60 5.51 300
2.984 1.704 1.178 0.461 108.816 4.470
0.071 1.499 36.459
should be used for the calculation of ESCA band intensities, whereas ST0 3 G functions are sufficient in
1.31 0.67 6.65 4.66
the UV region.
Table 1 contains ST0 6G results for the relative photoionization cross sections of severall small molecules being ionized by Mg KU radiation (1253.6 ev). Fig. 1 shows a plot of the energy dependence of the
; 193
ST0 6 G cross sections
and MO’s with high coefficients at hydrogen (see the hydrides) show low ESCA intensities; for fist-row
._
12
0.19 4.11 100
for the valence MO’s of meth-
are. The results of table 1 compare well with the published ESCA spectra of these molecules 17, 131. The following trends are reasonably well predicted: Valence ionizations are at least one order of magnitude weaker than core ionizations; oxygen Is ionizations are about twice as strong as carbon 1s ionizations (see CO); in the valence she!, rr MO’s (se,e Nz, CO, C2H4)
atoms, the 2s type MO’s yield considerably stronger ESCA bands than the 2p type MO’s (see N2, CO, H20, CH4, C2H4); on the. other hand, for second-row atoms, 3s type bands are predicted to be somewhat weaker &tn 3p type bands (see H,S), although our calcula-
tions might overestimatethis effect, comparedto expeLtient
1131. calculation.of ESCA band intensities, abinitio wavefunctions prove to be superior to valence electron wavefunctions, even in the’case of valence MO’s: The major contributions to the cross sections in -he ESCA region usually arise from the core orbitaIs and the valence-core cross terms which both are ne-’
1.35 5.00 100
In the
0.027 0.07 O.G27 0.07 0.043 0.11 0.019 o.c.5 1.213. 3.22 l.a56 4.93 .35.292. 94 37.646 100
glejted
in a formalism
based upon the valence electron
approximation. Associating the LCAO coefficients from valence electron methods with STO’s orthogonal$zed to the core AO’s, as recently suggested [S, 6’1,
a) Taken from refs [ 13,143.
-does improve the calculated ESCA band intensities in the frametiork of the valence electron approximation. To obtain differential cross sectiohs for err&ion at righi This method, however, has certain drawbacks. First, a.r$es to the photon &, divide by 8 n/3. it cannot predict the cross sections for the ionization .C)T&enfromref. [15]. of core MO’s. Secondly, for valence MO’s, the orthogonalization procedure inconsistently introduces some class sections-show similar~qualitative trends for the vhious ST0 NG wavefunctions (N = 3-6), but there ’ .’ effective core population into LCAO SCF eigenvectors b"l$e calculated total cross sections are given in 10m21cm2.
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Volume 31, numbx
2
CiiEhfICAL
PHySKS
‘i Fig. 1. Logarithmic p16t of calculated total photoionizatjon cross sections (cm2) Yersus photon energy (em, for the MO’sof methane.
va-
lence
obtained from a valence basis set. Thirdly, for a given atom, the one-center core and valence contributions always ap&ar linthe same ratio, due to the orthogonality requirement, whereas they are determined in dependentIy in our calculations. Because of these deficiencies,,themodified ESCA
ioniz-ations
valence
electron
[S, 61 s-tilt seems
treatment
to be inferior
for
to
the ab-initio treatment_ The established semi-empirical model for ESCA band intensities [7-101 can be considered as a sim-, plification of our approach. First, this model completely negIects the two.center terms in (1) which is cfearly justified by tbe,fact that &ratio Z &,/L: Q=‘for the thirty-six ionizations listed in table 1 averages to 0.022, the ~rnurn value being 0.106. Secondly, the rern~~g one-center terms are obta+ed in this model as the sum over products between theoretical gross atomic populations and empirically determined cross sections for the basisAO:s, In view of our present work, the use of net atomic populations would seem more appropriate [7]. The main difference between. the two aFproaches~ however, concerns the cross secy tions of the basis AO’swhich in our ~otation,are proportional to MrmrmPm&. Using a minimal basis set, .our for-&a&m tieats the core contributions and the ..
.~ ..’
‘;
.. .
1 :&itch 1975
LETTERS
v~ence~5re cross terms explicitly. On t&e other hand, the semi-empirical model for ESCA band intensities includes only contributions from the vaknce shel.I,,and accoun& for the other usutiy q.tite large terms by adjusting the cross sections of the valence basis AO’sas empirical parameters. Therefore, the quantizes M&m itPafm actually refer to different basis AO’s and differ completely in the se~~rnpi~c~ model and in our treatment, although both approaches seem to predict similar qus&ative trends for the band intensities of valence MO’sin ESCA spectra. Ln the Mg Kn spectrum of neon, e.g., the measured [,I31 .2s/2p intensity ratio is 8.7 which compares well with the calculated ST0 6G value of 9.15. In the semi-empirical model, the relative values for the crr\ss sections of the basis AO’sare then 8.7 (2s): 1.0 (Zp), by deftition, whereas, for the ST0 6G basis orbitaIs, they are calculated to be. 89.0 (Is): 0.4 (&): I .O (2~1. Finally it should be noted that the direct cakulation of the cross sections by our formalism can be used to predict their energy dependence (cf. fig.. 1) while the semi~mpi~c~ model is restricted to the particular photon energy for which it was par~ete~ed.
Acknowledgement This work was supported by the Air Farce Office of ScientifiC Research (Contract No. F44620-73.X0119) and the Deukche Forsch~~sg~me~~ch~t. The c~c~ations were carried out using the CDC 6400,’ 6600 computer at the University of Texas Computation Center. One of us (W.T.) thanks the “Studienstiftung des deutschen Volkes” for a past-doctoral feBowsbip.
References [l]
[2]
A. Schweigand iv. ThielJ. A. Schwek and W. Thiel, I.
131ZSchwig
Chem phys, 60 (19?$} 9.31. EtecQon Spectry. 3 (19741
and Pr., Thiel, C&n.
Phys.
Letters 22 (t973)
541.
[4] WK. P-rice, A.W. ?otts and DE. Streets, in: Elkon
spectroscopy, ed D.A. Shkley Q4or&HoUand, Amsteriiam, 1972) pp. l&7-33; [S] J.T.J. Huan,o and F.0. ELliron, Chem. phys. Letters 25 (1974143. ‘.
..
,289 ‘.
.‘
~.
_ “&I&
:.-
.
31;&-riber
:,.-
7
’
;
‘.
: ., &tlEhllCAL
[6] J.T.J. Hung,
.,
: ..
1,March 1975
[12] ?;.I. Hchre, R DitdMeld; RF. Stewart and J.A. Fo&, J. Chem. Phys. 52 (1970) 2769. [13]-bi Siegbahn; C. N&&g, G. Johansson, J. Hedman, P.F. Hede’n, K. Hamrin.U. Gelius, T. Bergmark, L.O. Wehe. R hlanne tid Y. Baef, ESCA, applied to free molecules (North-Hopand, Amsterdam, 1971). [14] D.W. TtirnFr, C. BakFr, A.D. Bakeraand C.R. Brundle, Alolecular photoelectron spectrdscopy (Wiley-Inters:ience, New York, 1970).
F.O:EUisorI a;ld ;.W. Rabalais, 3. .Elee tiorl,Spectry. 3. (1974) 339. .[VI U. Geljus, in: Elwtron spe+os&py, ed D.A, Shirlei’ (North-Holland;Amsterdam, 1972) pp:311-334. :- : [ 81 U~Gelius, C.J. Allen, G: Johansson; H. Siegbahn, D.A; Allison tid K. Siegb&;Physica Scripta 3 (1971) 237. .- 1 [9j U. Gelius, C.J. A.llen,‘H. Siegbahri tid k Siegbahn, Chern Phys Letters 11 (1.971) 224. : [lOj.R F’rins,Chcm. Pljys, Letters 19 (1973) 355. ’ [ 111 W,J, Hchre; R.F. Srewart end J.A. Pople, J. Chcin. ‘. Phys 51 (1969) 2657, an” sub;sequcnt papers.
.:
”:
PF;YSICS LETTERS
[15]
J.W.
Rabdais, T.P.
and F.O:Ellison,
Dcbics,
J.L.
Bcrkivsky,
J.T.J.Huvlg
J. Chem Phys. 6i (1974) 516.
:
.-. .
., .:
,.. ..
:
.’
‘i.
.
5
: ,Vol&ie 3l,number 2 ;. .,: .. ,.: ,. .. -,. ,, ,.-’ :. ,‘. .~’ . ;...; ‘. .’ .‘. ,=’ ” : .- -. ... . ‘_ ‘;
&JLATiON ‘.
.. _““s;
.;-
_I.&IERS ,.: .: ,,
,CHEXfICilL PI&KS ..;. ,‘, ‘. ‘-
‘.
..
‘.
..
‘,...
1
,,
.’
.
..
;
‘.
OFBHO~O~ONIZATION CROSS SECT&S
.
‘. and’ .‘.
&xiveti
.
.;
; _‘. ,’
‘.:_ ,.‘,,.‘..i .’ )..
,’ .;.,.
‘.‘.
,. ..’ .
.,
:-
;
:-I Marc11,l97.5 ‘. ‘, ,-‘. ‘. ., ‘.
‘. :
‘.
WAVEn;UNClIONSAND TFB P&NI: ivavlE MPROXiMAT16N* :
+INSrrO
W; THIEL’ M&.DEWAR k. KQMO&IKI dc&rtmen> of Ckmktry, The ‘U&ersit;.of Texq ~tAustin,
‘.
”
Austi~r, Texm 78712, USA
(,
: .,
‘.
..
.’ .,
_
16 Se~tember.1974
._
.’ :
_:
.
.
)’
..,
_:
I._,
...
,A &$hod is oiled for ~~~~cu~tio~ of photoionization ,mdimal basis set STO.NG ~ave~u~~tion &d the foal &ate by ‘. s6ra.l small mo!ccules compare well with the .btid intensities band intensities obtained from valence election wavefunctions. , : .’ dtiati&s lead to comparable results. . . : ,’ ,’
-. 1. &r&ction t
..
?hotoionizatio~ cross sectipns for the ionization of valence molecular-orb&& (MO’s) can be calculated
b$ a’recently deveioped theory [l] which describes .the $iti’al state of the ionized molecule by LCAO SCF MO’s with a vale&e basis set of Slater.atomic orMaIs (STO’s), and the fiiaI state by the plane wave approximation. This theoryreproducks the observed ‘band intensities in‘~ photoelictrdn spectroscopy fairly well, especially their variation in going from He I to ze II excitation f’kl, and can be useftil in assignkent probIems [3]. In thd soft X-ray regioil, ho+fever, application of .:lhe theqy is not feasible. The transition moment foi’ ‘_the iqnization$s proporti&al~to the qve&p between the ionized MO and the piane vjave of the ejected ph6toekction:‘[l, 41, Since in ESCA spectroscopy +&e .wavelen&h”of the photoelectron is typically in the ordec of 0.3 a, a reliable pre~cti~n of &CA band in-‘,
.. .’ . : . *it3&6seque& of 3 &timi& se& of papers on%&or$ I :. and applicati?~ ofphtifoeiectmn spectroscopy by A. Schtieig a&d co-wdrkeq $-ds~paper wiil b-e t&ga:rdedaS part-. ..?X bj;t$atlaboratory.. : ; .’ ,_’ ,. _, . .‘,
cross sections iepres.enti& the initial state by a thy plane w&e appro~~ation. Numericlll results for 1 in published ESCA spectra, and are superior to ESCA For the UV region, ab-initio and valence electron cd:
tenSities requires a good representation’ of the electron densiiy ~s~~bution in the immediate vicinity of the nuclei which of course is not obtained in the valence electron approbation, To overcome this ~f~cul~ it has been suggested [5,6f to associate the LCAO co-
efficitmts obtained by valence electron calculations with :sTO’s which ar~,ort~ogonal to the core AO’s,
thereby effectively introducing some “core character” the valence eIectron wavefunctions. :, Ln our opinion, the description of the initial state by ab%nitio waveftinctions offers’s more direct appro& to.the’ca.lculation of ESCAband intensities. in the following sections we therefore derive the necessaqr formalism, report numerical results fo; several molecules and discuss the connection between the present work tid the semi-empirical model (7-101 for ESCA, band intdn&ies?. ’ into
,.
,:.
..
.‘ * .+fier:co&letin~ our work; similar results hsve been pub ;li&kd [ 151 .obtaine$ by ysing ab-initio’wavefyxtjoqs of ,.’the gaussian lobe typq for thk in&l state, and orthogonalizel plane w&es’for t&e fqal state. The reported theore% -&I Yalues and k+eriment+l est+&es agree well with auf ‘_‘.’ m-&s listedin tab16 1.‘: : ._ ., . . _,.‘_ : .‘:, .: .. “.. .’ ,‘~ ..
Volume 3 1, nukbcr 2
CHEMICAL PHYSICS LETTERS
2. Theory
As shown previously [I] the photoionization section Ui of the jth MO.is given by
cross
where e and m, aie the charge and mass 3f an electron, fi is Planck’s rationalized constant, c the velocity of light, ki the wavenumber of the photoelectron,,.and Eph the photon energy. QD and Qnabare one- and twocenter terms (atoms a and b) originally defined for MO’s with a valence basis set of STO’s. In our present study, we use a minimal basis set expanded in terms of the ab-initio ST0 NG wnvefunctions [l 11, the basis orbitals xnlnl being represented by linear combinations of NG gaussians (GTO’s)
where N ,],,i, dn/,ijanl,i and 5,) are the normalization constants, contraction coefficients, orbital exponents and scaling factors, respectively, for the basis orbital. Y, (0 ,@)is a normalized spherical harmonic, and a0 Bohr’s radius. The introduction of ST0 NC wavefunctions leads to the following modifications in our previous formalism [l] : The expressions for the one- and two-center terms must include not only valence contributions,. but also core contributions and valence-,core cross terms. Summing over all orbit& in the minimal basis. set we obtain:
canIn is the LCAO coefficient of the basis A0 x&i. at atom a; aft&otating the internuclear axiS between
atoms Q andb (diStanCeR~b)intOtheZ-axiSofthe ccordinate system, the transformed coefficients are denoted by ?anl,,,: The interference terms GrillI,m (k,&ab) have been’given explicitly [I]. The t&sition moments Manlm for the gaussian basis AO’s are proportional to the Fourier transforms of the radia1 part of (2) and can be evaluated analytically:
X exp (-kf/4
cu’,,,,J.
(6)
3. Results and discussion Photoionization cross sections have been calculated for a variety of molecules at different levels of the STONG approximation (N = 3-6). The input data consisted of the experimental geometries, vertical ionization potentials, and ST0 NG eigenvectors; the contraction coefficients, orbital exponents, and scaIing factors were taken from refs. [11,12], the orbital exponents being identical for s- and p-orbit& of the same shell. We summarize the numerkal results for UV excitation (He I, He II) and for X-ray excitation (Mg Kci): In the UV region the results for the difFerent ST0 NG levels are almost the same since the transition moments (6) converge very rapidly to the limiting ST0 value, the deviations usually being less than 1% even for the 3G basis. The calculated values for the relative band intensities are usually quite similar to those obtained in the valence electron approximation [2] and therefore need not be reproduced here. There is one notab!e exception. In the valence electron calculations a semi-empir@al correction for the upp two. center contribution was introduced in order to restore the expected relation between the sign of this contributiqn and the app overlap integral [ I]. This correc.lion hlrns out to be unnecessary in the present abinitio calculations which is not surprising since overlap effects are now included in the normalization of the eigenvectors. In the valence electron cal&Iat$ons, the correction can likewise be avoided by using the normalize! coefficient matrix C’=EwV2 C (overlap matrix S) titead of the original coefficient matrix C. ,In the ESCA region the‘calculated photoionization
Volume
31, number 2
CHEMICAL
PHYSICS LETTERS
,.’
1 March 1975 ‘:
.
‘,
discrepancies between, e.g., 3G This is due to the fact that the
are some quantitative
Tabre ‘1
/+bso!ute and relative ,photoidnkation ESCAsp&a (Mg Keexcitation)
cross sections for
and 6 G calculations. kansition
moments
(6)
for ESCA.ionizations
are
nlahly detumincd by the shape of the,basti orbital tin the immediate vicinity of the nucleus which should be represented accurately enough only, by the higher order.expansions. Therefore, STO. 6G eigenvectors 'N2
15.60 16.98 .1&78 31.3. .409.9 409.9
CO
14.01 16.91 19.72
38.3 295.9 542.1 Hz0
1262 14.74 18.51. 32.2 539.7
50 1 ir 4 a
30 2 0 :io 1 br 3 21 1 b2 2a, 1 al
170.9 234.5
2a1.
,.
.22.2
m4
14.35 2291
230.7 c2H4
1 nTu 20, 2 Lrg 1lJ” l.og’
2 bl 5 a1 2 b2 4 ai Ibl,lb2,3al
10.47 13.33 15.47
H2S
lO.51 -1285 14.66 15.87. 19.1
23.6 290.6
T90.6
0.581
3gg
1 tp 2al 1 al 1 bzu 1b2g 3 ag 1 bl, 2bau
2ag
1 blu 1 ag
0.95
0.126 3.004 3.150 61.187 57.055
0.21 4.31 5.15 100 93
0.974 .’ 0.249 2186 4.137 36.801 77.532
1.26 0.32 282 5.34 47 100
0.83 0.42 2.08 4.16 46 100
0.166 0.935 0.085 3.903 77.435
0.24 1.21’ 0.11 5.04 100
0.50 1.37 .0.43 5.26 100
8.20 4.68 3.24 1.27 300
.6.43 8.26 4.60 5.51 300
2.984 1.704 1.178 0.461 108.816 4.470
0.071 1.499 36.459
should be used for the calculation of ESCA band intensities, whereas ST0 3 G functions are sufficient in
1.31 0.67 6.65 4.66
the UV region.
Table 1 contains ST0 6G results for the relative photoionization cross sections of severall small molecules being ionized by Mg KU radiation (1253.6 ev). Fig. 1 shows a plot of the energy dependence of the
; 193
ST0 6 G cross sections
and MO’s with high coefficients at hydrogen (see the hydrides) show low ESCA intensities; for fist-row
._
12
0.19 4.11 100
for the valence MO’s of meth-
are. The results of table 1 compare well with the published ESCA spectra of these molecules 17, 131. The following trends are reasonably well predicted: Valence ionizations are at least one order of magnitude weaker than core ionizations; oxygen Is ionizations are about twice as strong as carbon 1s ionizations (see CO); in the valence she!, rr MO’s (se,e Nz, CO, C2H4)
atoms, the 2s type MO’s yield considerably stronger ESCA bands than the 2p type MO’s (see N2, CO, H20, CH4, C2H4); on the. other hand, for second-row atoms, 3s type bands are predicted to be somewhat weaker &tn 3p type bands (see H,S), although our calcula-
tions might overestimatethis effect, comparedto expeLtient
1131. calculation.of ESCA band intensities, abinitio wavefunctions prove to be superior to valence electron wavefunctions, even in the’case of valence MO’s: The major contributions to the cross sections in -he ESCA region usually arise from the core orbitaIs and the valence-core cross terms which both are ne-’
1.35 5.00 100
In the
0.027 0.07 O.G27 0.07 0.043 0.11 0.019 o.c.5 1.213. 3.22 l.a56 4.93 .35.292. 94 37.646 100
glejted
in a formalism
based upon the valence electron
approximation. Associating the LCAO coefficients from valence electron methods with STO’s orthogonal$zed to the core AO’s, as recently suggested [S, 6’1,
a) Taken from refs [ 13,143.
-does improve the calculated ESCA band intensities in the frametiork of the valence electron approximation. To obtain differential cross sectiohs for err&ion at righi This method, however, has certain drawbacks. First, a.r$es to the photon &, divide by 8 n/3. it cannot predict the cross sections for the ionization .C)T&enfromref. [15]. of core MO’s. Secondly, for valence MO’s, the orthogonalization procedure inconsistently introduces some class sections-show similar~qualitative trends for the vhious ST0 NG wavefunctions (N = 3-6), but there ’ .’ effective core population into LCAO SCF eigenvectors b"l$e calculated total cross sections are given in 10m21cm2.
28.8;..-
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:, ;, j -!
Volume 31, numbx
2
CiiEhfICAL
PHySKS
‘i Fig. 1. Logarithmic p16t of calculated total photoionizatjon cross sections (cm2) Yersus photon energy (em, for the MO’sof methane.
va-
lence
obtained from a valence basis set. Thirdly, for a given atom, the one-center core and valence contributions always ap&ar linthe same ratio, due to the orthogonality requirement, whereas they are determined in dependentIy in our calculations. Because of these deficiencies,,themodified ESCA
ioniz-ations
valence
electron
[S, 61 s-tilt seems
treatment
to be inferior
for
to
the ab-initio treatment_ The established semi-empirical model for ESCA band intensities [7-101 can be considered as a sim-, plification of our approach. First, this model completely negIects the two.center terms in (1) which is cfearly justified by tbe,fact that &ratio Z &,/L: Q=‘for the thirty-six ionizations listed in table 1 averages to 0.022, the ~rnurn value being 0.106. Secondly, the rern~~g one-center terms are obta+ed in this model as the sum over products between theoretical gross atomic populations and empirically determined cross sections for the basisAO:s, In view of our present work, the use of net atomic populations would seem more appropriate [7]. The main difference between. the two aFproaches~ however, concerns the cross secy tions of the basis AO’swhich in our ~otation,are proportional to MrmrmPm&. Using a minimal basis set, .our for-&a&m tieats the core contributions and the ..
.~ ..’
‘;
.. .
1 :&itch 1975
LETTERS
v~ence~5re cross terms explicitly. On t&e other hand, the semi-empirical model for ESCA band intensities includes only contributions from the vaknce shel.I,,and accoun& for the other usutiy q.tite large terms by adjusting the cross sections of the valence basis AO’sas empirical parameters. Therefore, the quantizes M&m itPafm actually refer to different basis AO’s and differ completely in the se~~rnpi~c~ model and in our treatment, although both approaches seem to predict similar qus&ative trends for the band intensities of valence MO’sin ESCA spectra. Ln the Mg Kn spectrum of neon, e.g., the measured [,I31 .2s/2p intensity ratio is 8.7 which compares well with the calculated ST0 6G value of 9.15. In the semi-empirical model, the relative values for the crr\ss sections of the basis AO’sare then 8.7 (2s): 1.0 (Zp), by deftition, whereas, for the ST0 6G basis orbitaIs, they are calculated to be. 89.0 (Is): 0.4 (&): I .O (2~1. Finally it should be noted that the direct cakulation of the cross sections by our formalism can be used to predict their energy dependence (cf. fig.. 1) while the semi~mpi~c~ model is restricted to the particular photon energy for which it was par~ete~ed.
Acknowledgement This work was supported by the Air Farce Office of ScientifiC Research (Contract No. F44620-73.X0119) and the Deukche Forsch~~sg~me~~ch~t. The c~c~ations were carried out using the CDC 6400,’ 6600 computer at the University of Texas Computation Center. One of us (W.T.) thanks the “Studienstiftung des deutschen Volkes” for a past-doctoral feBowsbip.
References [l]
[2]
A. Schweigand iv. ThielJ. A. Schwek and W. Thiel, I.
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Chem phys, 60 (19?$} 9.31. EtecQon Spectry. 3 (19741
and Pr., Thiel, C&n.
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541.
[4] WK. P-rice, A.W. ?otts and DE. Streets, in: Elkon
spectroscopy, ed D.A. Shkley Q4or&HoUand, Amsteriiam, 1972) pp. l&7-33; [S] J.T.J. Huan,o and F.0. ELliron, Chem. phys. Letters 25 (1974143. ‘.
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31;&-riber
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7
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[6] J.T.J. Hung,
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1,March 1975
[12] ?;.I. Hchre, R DitdMeld; RF. Stewart and J.A. Fo&, J. Chem. Phys. 52 (1970) 2769. [13]-bi Siegbahn; C. N&&g, G. Johansson, J. Hedman, P.F. Hede’n, K. Hamrin.U. Gelius, T. Bergmark, L.O. Wehe. R hlanne tid Y. Baef, ESCA, applied to free molecules (North-Hopand, Amsterdam, 1971). [14] D.W. TtirnFr, C. BakFr, A.D. Bakeraand C.R. Brundle, Alolecular photoelectron spectrdscopy (Wiley-Inters:ience, New York, 1970).
F.O:EUisorI a;ld ;.W. Rabalais, 3. .Elee tiorl,Spectry. 3. (1974) 339. .[VI U. Geljus, in: Elwtron spe+os&py, ed D.A, Shirlei’ (North-Holland;Amsterdam, 1972) pp:311-334. :- : [ 81 U~Gelius, C.J. Allen, G: Johansson; H. Siegbahn, D.A; Allison tid K. Siegb&;Physica Scripta 3 (1971) 237. .- 1 [9j U. Gelius, C.J. A.llen,‘H. Siegbahri tid k Siegbahn, Chern Phys Letters 11 (1.971) 224. : [lOj.R F’rins,Chcm. Pljys, Letters 19 (1973) 355. ’ [ 111 W,J, Hchre; R.F. Srewart end J.A. Pople, J. Chcin. ‘. Phys 51 (1969) 2657, an” sub;sequcnt papers.
.:
”:
PF;YSICS LETTERS
[15]
J.W.
Rabdais, T.P.
and F.O:Ellison,
Dcbics,
J.L.
Bcrkivsky,
J.T.J.Huvlg
J. Chem Phys. 6i (1974) 516.
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., .:
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‘i.
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