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Localized and delocalized molecular orbitals within the model of single-orbital relaxation energies
Volume 83. number 3
LOCALIZED
CHEhliCAL
AND DELOCALlZED
WLTHKN THE MODEL
PHYSICS
MOLECULAR
OF SINGLE-ORBLTAL
1 November 1981
LEITERS
ORBlYALS
RELXXATION
ENERGIES
T FICKER
Rece+zd
16 May 1981,
III f-iicil Form 1 July 1981
Ab IN:IO LCAO UO SCI’ calcularions habe been pertormed on CO znd Nz Smgle-orbttal relaxtlon energy conu~buAlthou_eh satisfactory tlons obtxncd by means of molecular orbttals wth Mierent locahzat~on character are compared agreement x%tth duectiy calculated relaxation enzrgx has been found m all casts. ddferenr part~rlonmg of the total relrtuatlon eneru_y ~1x0 orbttal conrnbutzons appears
1. Introduction ReIa\anon phenomena accompanymg core lonlzatlon tn tnolecules haw been mresrlgated by Clark et ai [I-S) m terms of smgle-orbltal contnbutrons to the total tela\atlon energy The computational model for the smgle-orbltal contrlbutlons. described III more detad ptevlousiy [ 1 .I!]. uses the ground- and corehole-state MOs as a starrmg pomt for addItIona Hartree-Fock (HF) con~putat~ons performed wlthm a smgle SCF rterntlon From tfus pomt of vxew, results obramed from rlus model depend &rectly on the character of the startmg MOs In addltlon, wlthm the model there IS no physical reason for preferrmg some kmd of MOs Fmally it IS of interest whether the check cntenon of the single relaxation model, 1 e a_rreement of the directly calculated total 1SCF rela_xatlon energy and the sum of the smglearbltai relavation enerles, IS sat&led if a change of the hi0 locahzauon character occurs The prune mottvatlon of the present paper IS to show that the use of MOs w~rh different character of locahzation wlthm the model mentxoned above leads to bfferent partltlonmg of the total relaxation energy mto smgle*rbital contrlbutlons As lllustratlve examples we have picked two simple molecules, CO and N,, for wfuch the smgle-orbltal relaxation energtes are avtiable [I ,2] _ Furthermore, these examples 578
serve to pomr out that the check crltenon of the smgli: relaxation model can be satisfied IO a satlsfactory degree of accuracy even for different smgieorbital reiatatlon energy drsrributlons. obtauted as ;1 consequence of different MO localtzatlon, 1-e co point out that the smgle rel&\atron model does not contam any cnterlon for the detemunatlon of the unarnblguous partrtlorung of smgle-arbltal relakatlon energies. To our knowledge. such a discussion concerning computational features of the singte-telaxatron model has not been given before. although thrs model IS often used [l--51 for studying the reiasatton process accompanymg core loruzatlon m simple molecules
2 Computational
details
We have performed ab mitlo LCAO MO SCF calculations on the gtoucd and core-hole states of the CO and N2 molecules usmg the MOLECULE-ALCHEMY program package [6] The triple-zeta basis set [7j of contracted gausstan functions (wlthout poianzatIon functions) has been employed. Calcularxons have been carned out uang experimental equlhbnum bond length [8], 1 e. for C-O and N-N 2 132 and 2 074 au, respectively, for both the ground and core-hole states 0 009-26 14/Sl/OOOO-0000/S
02.75 0 1981 North-Holland
Volume 83. number 3
The singleorbital rela?tatron model [ 1,221 actually is a two-step computational procedure. The first step represents HF computatrons of the ground and corehole states for each molecule. The hlOs of the ground state are often called unrelaxed orbrtals, and those of the core-hole state are relaxed orbltals. They both have symmetry-adapted (delocahzed) characterwithrn the symmetry-restricted
I November 1981
CHEhUCAL PHYSICS LETTERS
HF calculatrons.
Neverthe-
less, If the molecule contams several symmetryequlvalent atoms, e g NZ, It is necessary to perform symmetry-unrestricted HF calculanons for the core-hole state. Such a calculation provrdes relaxed orortals which are delocahzed m the valence regron but strongly localized m the core molecular region. Thrs fact IS well known [9 ] and classrfied as a computar~onal artefact accompanying the symmetry-unrestncted HF treatment of molecules wrth several
putatronal agreement (calculated
al reI&atron energes (calculated wrthin the singlerelaxation model) - as rt is prescribed by the check
criterion The question IS whether such a choice of the starting orbltals can ensure also an unamblguity m thz partrtiomng of the srnglearbital relaxauon energies, I e. whether thus choice can remove influence of the MO localization on the values of the singleorbitalrelaxation energres. The answer to thrs question 15gven m sectron 3 _
3. Results
symmetryequrvalent atoms. Other molecules, e g. CO. do not require symmetry-unrestncted HF calculatrons for a good descnptron of the core-hole state, I e. their hlOs are symmetry adapted. Properly chosen relaxed and unrelaxed MOs (cf.
refs. [I ,223) are used m the second computational step of the smgle-relaxation model as startmg orbrtals for further HF calculatrons performed w~hr.n a smgle SCF Iteration. Naturally, the single SCF iteration yrelds the total energy which 1s duectly dependent on the character of the startmg orbitals, so that the character of locahzation of the Mos used 111such calcula-
trons should influence the final results. However, the proper chorce of the relaxed and unrelaxed orbrtals, wluch IE necessary for transrtron from the first com-
step into the second one [ 1,121. ensures between the total ASCF relaxation energy separately) and the sum of the singlearbit-
and discussion
Tables 1,2 and 3 contam our results and data taken from the papers of Clark et al [ i,3,4], who mvestrgated extensively CO as well as N, w&in the model of smgle-orbital relaxatron con&butions. Clark et al. [I ,3] emphasrzed the importance of a balanced basrs set for calculattons
of such a kind.
Their basis set (ST06,33G plus polarizatron functions) is comparable m quahty to a double-zeta one. On the other hand, our basis set* is essentially of mple-zeta quahty but without polanzation functions_ As we can see from table 1, our basis set provrdes bet-
ter values for electron bmding energres (by = I eV as *We trave used (C/11,7),(0/11,7),(N/11,7) ot‘gaussran-rype iunchons 171 contracted (O/6,3) and (N/6,3) schemes
primitive sets the K/6.3),
mto
Table 1 Electron bmdrng and to+A relaxauon energies (ev) of the CO and Nz molecules Molecule
Orbuai
NZ
co
a) Theoretical.
Relaxauon energy
Bmdmg energy thrs work
other work
41100 1%
411.00
lo
54250
20
298.63
thiS-AOrk
other
412 88 [3] a) 409.92 [3] b)
16.60
15.35 131 a)
543.71 542.30 299.64 296.20
20.40
19.91 [4] a)
11.54
10.79 [4] a)
[4] a) [2] b) 141 a) [2] b,
work
16.50
b, E.xpenmental. 579
1 November 1981
CHEMICALPHYSICSLETTERS
Volume 83. number 3
Table 2 Single-orbital relaxation energies (eV) of the CO molecules .; ____-..--_._ _.-.--. MOs used in previous calculations [ 11 Symmetry adapted MOs (this work) Cirbital --_-_-.--_-Cr s core hole 0%s core hole 2~ core hole 10 core hole ___...~____.P_____II_.________________.___. __ _______“______^___.~_._______ 4.60 13.00 4.68 12.70 1Yl 3.49 -0.54 3.70 1.35 5a 0.95 4.54 1.34 3.12 40 1.66 2.66 0.77 2.18 :a 0.09 0.00 1.12 0.00 20 0.00 0.25 0.01 1.11 10 19.91
10.79
19.91 11.54 20.40 -___.. ---.l_l_~....___,"_ .__.~._. __.___.____~ ...."_.~_.__
10.79
1X.62
20.46
sum Total
ASCF
relaxation energy
compared with experiment) and thus better total relaxation energies (tables 2 and 3), our agreement with the directly calculated total relaxation energy is worse than in the case of Clark et al. It is probable that inclusion of polarization functions in our basis set would improve this agreement. Nevertheless, the
~s~re~ancies do not exceed =O.l eV, i.e. = 1% of the total relaxation energies, and thus we consider our results as satisfactory. Let us focus our attention on the singleorbital relaxation energies of the core-ionized CO moIecuJe (tabfe 2). it is not a straightforward matter to explain differences between results obtained with our symmetry-adapted MOs and those of Clark et al. [1,33. T;iolc 3 Stngle~rbitd relaxation energies (eV) of the N, molecule -_1_-
..-...-
%a1
.-..--.
--_._
-
log core hole
1 t ?*I I iv; 2ug hJ
10, SlJJ
~__^_.___._____
-
_
Symmetry adapted MOs (this work)
.
.
-
.__.
The MOs of the CO molecule have inherently localized character mainly in the core and inner valence re. gions in both treatments. Nevertheless, the diverse basis sets used in either calculations, undoubtedly do not form the same localized character of MOs,especially for the core region, where our basis set is essentially larger, So we have associated differences jn thr single-orbital energies for carbon monoxide with the different character of the MOs employed in the calctllations. We do not discuss the characteristic features of the single-orbital relaxation phenomenon since such a discussion for carbon monoxide has been done elsewhere [I] in substantial detail. Our results confirm this discussion in its main directions.
_” - _._...___. ._-- .._._ --.-___--_l_-_-___l__ Orbital
1ou core hole
3.30 1.66 1.26 0.55 9.74 0.05 -..-
3.30 1.66 1.26 0.55 0.08 9.60
16.65
16.45
_-II_-
MOs us-d in previous calculations [ 3) N t s core hole
In 50 4P 30 20 lo f
8.74 0.30 3.62 1.90 0.80 15.36
to1111 AS3 -7.
580
relaxation energy 16.60 .- I---..---...----._.________
16.50
_-___
-V--.1-_--
15.35
Volume 83, number 3
CHEMICALPHYSICS LETTERS
An interesting situation appears in the core region of the N, molecule (table 3). ‘Ihe highest values of the singlearbital relaxation energy correapondt+he corelu~andlb,MOI.Inthisarewe~vle~~wrrelaxed orbit.& with’symmetry&tpted (d&calized) character. Our relaxed orbitalshave also been delocalized in the valence region and, however, localized in the core region as a consequence of the symmetry-unrestricted HF treatment of the core-h& molecular state. Clark et al. [3] used the localized MOs for Nz. At this point we mention once more that there is no physical reason for preferring some kind of MOawithin the single-relaxation model. The different MO localization in both cases is mirrored in the different distribution of relaxation energy into the orbital contributions. The two most deeply lying MOs consists of N,, atomic orbitals and thus, in our treatment, the unrelaxed lug and lo, core orbit& are symmetrically delocalized over both the N centers while the corresponding relaxed orbitals are localized on these centers, As a consequence of the delocalization and localization of the unrelaxed and relaxed core orbitals, respectively, we can expect - in comparison with the calculations of Clark et al. (table 3) who have employed localized orbitals - the greatest difference in the relaxation energies just in the core molecular region. Our different character of localization used for relaxed and unrelaxed core orbitals within the model causes substantial energy change8 which require redistribution of the singleorbital relaxation energies. Although more detailed discussion of the relaxation energies for N2 has been given [3], we consider it useful to illustrate an interesting behaviour of these energies for the highest MOs In,, and In of both the relaxation energy distributions under study (table 3). The In, orbital is delocalized over both the nitrogen centers, so that only about one-half of the whole orbital charge can relax towards the core hole#. On the other hand, all the orbital charge of the localized In
* These energy changes refer to a method B (see mf. [ 11) in the second computational step of the single&axationmodel. #The core hole has no physical interpretation and is only a computational artefact appearingin the symmetry-broken HF calculations 191. Nevertheless, we use this expression to simplify the treatment.
1 November 1981
orbital can relax quite effectively to the center bearing the core hole. These facts explain why the relaxation energy of t+e Jrru orbitalIs 8pproxiniately oneUf of the fr relaxatiori eneqy_in the l&liz.+d,case: .fie’~~.exam$e# C!p and Ni sh& that results obtained.from the computati&uI model of singleorbital reluution eneqies are‘dkpendent on the characterof the star&g MOs. Whili the different MO character for the CO,mtilecule is caused only.by two distinct A0 basis seta used in the calculations, the different MO character for Nz ls also given through the additional lo&lW.ion procedure (cf. ref. (31). Nevertheless, the different characters of the MOs yield an ambiious distribution of single-relaxation energies in spite of maintaining agreement with the total directly computed relaxation energy. So we conclude that the absence of a physical reason for preferring ,MOswith cert$n character of localization within the single-relaxation model leads to ambiguous partitioning of the ainglearbital relaxation energies; the proper choice of starting MOs used in the second computational step of the model does not remove the influence of the MO localization on the values of the singlearbital energies. In such a way, the singlearbital relaxation energies within the model have only a restricted meaning since their values depend on the MO localization, i.e. on the character of the MO basis.
References [l] D.T. Clark, B.J. Cromarty nnd A. Sgamellotti, Chem. Phys. Letters 51 (1977) 356. [2] D.T. Clark, B.J. Cromarty and A. Sgamellotti, J. Electron Spectry. 14 (1978) 175. (3 j D.T. Clark, B.J. Cromarty and A, Sgameliotti, J. Electron spectry. 14 (1978) 49. 14) D.T. Clark, B.J. Cromarty and A. Sgamellotti, Chem. Phys. 26 (1977) 179. (51 D.T. Chuk, B.J. .Cromartyand A. Sgamellotti, J. Electron Spectry. 13 (1978) 85; 17 (1979) 237. [6] J. Abnbf, P.S. Bagus, B. Liu, A.D. McLean, U. Wah&#rcn and M. Yoshomhre, IBM San Jon! Research Laboratory Report. [7] F.B. van Du&tenveldt, IBM J. Res. Develop. 945 (1971). [S] L.E. Sutton, ad., Tables of interatomic distances (The
Chemical Sodety, London, 1958). [ 91 A. Denis, J. LangIet and J.-P. Malrieu, Theoret. Chim. Acta 38 (1975) 49.
581
LOCALIZED
CHEhliCAL
AND DELOCALlZED
WLTHKN THE MODEL
PHYSICS
MOLECULAR
OF SINGLE-ORBLTAL
1 November 1981
LEITERS
ORBlYALS
RELXXATION
ENERGIES
T FICKER
Rece+zd
16 May 1981,
III f-iicil Form 1 July 1981
Ab IN:IO LCAO UO SCI’ calcularions habe been pertormed on CO znd Nz Smgle-orbttal relaxtlon energy conu~buAlthou_eh satisfactory tlons obtxncd by means of molecular orbttals wth Mierent locahzat~on character are compared agreement x%tth duectiy calculated relaxation enzrgx has been found m all casts. ddferenr part~rlonmg of the total relrtuatlon eneru_y ~1x0 orbttal conrnbutzons appears
1. Introduction ReIa\anon phenomena accompanymg core lonlzatlon tn tnolecules haw been mresrlgated by Clark et ai [I-S) m terms of smgle-orbltal contnbutrons to the total tela\atlon energy The computational model for the smgle-orbltal contrlbutlons. described III more detad ptevlousiy [ 1 .I!]. uses the ground- and corehole-state MOs as a starrmg pomt for addItIona Hartree-Fock (HF) con~putat~ons performed wlthm a smgle SCF rterntlon From tfus pomt of vxew, results obramed from rlus model depend &rectly on the character of the startmg MOs In addltlon, wlthm the model there IS no physical reason for preferrmg some kmd of MOs Fmally it IS of interest whether the check cntenon of the single relaxation model, 1 e a_rreement of the directly calculated total 1SCF rela_xatlon energy and the sum of the smglearbltai relavation enerles, IS sat&led if a change of the hi0 locahzauon character occurs The prune mottvatlon of the present paper IS to show that the use of MOs w~rh different character of locahzation wlthm the model mentxoned above leads to bfferent partltlonmg of the total relaxation energy mto smgle*rbital contrlbutlons As lllustratlve examples we have picked two simple molecules, CO and N,, for wfuch the smgle-orbltal relaxation energtes are avtiable [I ,2] _ Furthermore, these examples 578
serve to pomr out that the check crltenon of the smgli: relaxation model can be satisfied IO a satlsfactory degree of accuracy even for different smgieorbital reiatatlon energy drsrributlons. obtauted as ;1 consequence of different MO localtzatlon, 1-e co point out that the smgle rel&\atron model does not contam any cnterlon for the detemunatlon of the unarnblguous partrtlorung of smgle-arbltal relakatlon energies. To our knowledge. such a discussion concerning computational features of the singte-telaxatron model has not been given before. although thrs model IS often used [l--51 for studying the reiasatton process accompanymg core loruzatlon m simple molecules
2 Computational
details
We have performed ab mitlo LCAO MO SCF calculations on the gtoucd and core-hole states of the CO and N2 molecules usmg the MOLECULE-ALCHEMY program package [6] The triple-zeta basis set [7j of contracted gausstan functions (wlthout poianzatIon functions) has been employed. Calcularxons have been carned out uang experimental equlhbnum bond length [8], 1 e. for C-O and N-N 2 132 and 2 074 au, respectively, for both the ground and core-hole states 0 009-26 14/Sl/OOOO-0000/S
02.75 0 1981 North-Holland
Volume 83. number 3
The singleorbital rela?tatron model [ 1,221 actually is a two-step computational procedure. The first step represents HF computatrons of the ground and corehole states for each molecule. The hlOs of the ground state are often called unrelaxed orbrtals, and those of the core-hole state are relaxed orbltals. They both have symmetry-adapted (delocahzed) characterwithrn the symmetry-restricted
I November 1981
CHEhUCAL PHYSICS LETTERS
HF calculatrons.
Neverthe-
less, If the molecule contams several symmetryequlvalent atoms, e g NZ, It is necessary to perform symmetry-unrestricted HF calculanons for the core-hole state. Such a calculation provrdes relaxed orortals which are delocahzed m the valence regron but strongly localized m the core molecular region. Thrs fact IS well known [9 ] and classrfied as a computar~onal artefact accompanying the symmetry-unrestncted HF treatment of molecules wrth several
putatronal agreement (calculated
al reI&atron energes (calculated wrthin the singlerelaxation model) - as rt is prescribed by the check
criterion The question IS whether such a choice of the starting orbltals can ensure also an unamblguity m thz partrtiomng of the srnglearbital relaxauon energies, I e. whether thus choice can remove influence of the MO localization on the values of the singleorbitalrelaxation energres. The answer to thrs question 15gven m sectron 3 _
3. Results
symmetryequrvalent atoms. Other molecules, e g. CO. do not require symmetry-unrestncted HF calculatrons for a good descnptron of the core-hole state, I e. their hlOs are symmetry adapted. Properly chosen relaxed and unrelaxed MOs (cf.
refs. [I ,223) are used m the second computational step of the smgle-relaxation model as startmg orbrtals for further HF calculatrons performed w~hr.n a smgle SCF Iteration. Naturally, the single SCF iteration yrelds the total energy which 1s duectly dependent on the character of the startmg orbitals, so that the character of locahzation of the Mos used 111such calcula-
trons should influence the final results. However, the proper chorce of the relaxed and unrelaxed orbrtals, wluch IE necessary for transrtron from the first com-
step into the second one [ 1,121. ensures between the total ASCF relaxation energy separately) and the sum of the singlearbit-
and discussion
Tables 1,2 and 3 contam our results and data taken from the papers of Clark et al [ i,3,4], who mvestrgated extensively CO as well as N, w&in the model of smgle-orbital relaxatron con&butions. Clark et al. [I ,3] emphasrzed the importance of a balanced basrs set for calculattons
of such a kind.
Their basis set (ST06,33G plus polarizatron functions) is comparable m quahty to a double-zeta one. On the other hand, our basis set* is essentially of mple-zeta quahty but without polanzation functions_ As we can see from table 1, our basis set provrdes bet-
ter values for electron bmding energres (by = I eV as *We trave used (C/11,7),(0/11,7),(N/11,7) ot‘gaussran-rype iunchons 171 contracted (O/6,3) and (N/6,3) schemes
primitive sets the K/6.3),
mto
Table 1 Electron bmdrng and to+A relaxauon energies (ev) of the CO and Nz molecules Molecule
Orbuai
NZ
co
a) Theoretical.
Relaxauon energy
Bmdmg energy thrs work
other work
41100 1%
411.00
lo
54250
20
298.63
thiS-AOrk
other
412 88 [3] a) 409.92 [3] b)
16.60
15.35 131 a)
543.71 542.30 299.64 296.20
20.40
19.91 [4] a)
11.54
10.79 [4] a)
[4] a) [2] b) 141 a) [2] b,
work
16.50
b, E.xpenmental. 579
1 November 1981
CHEMICALPHYSICSLETTERS
Volume 83. number 3
Table 2 Single-orbital relaxation energies (eV) of the CO molecules .; ____-..--_._ _.-.--. MOs used in previous calculations [ 11 Symmetry adapted MOs (this work) Cirbital --_-_-.--_-Cr s core hole 0%s core hole 2~ core hole 10 core hole ___...~____.P_____II_.________________.___. __ _______“______^___.~_._______ 4.60 13.00 4.68 12.70 1Yl 3.49 -0.54 3.70 1.35 5a 0.95 4.54 1.34 3.12 40 1.66 2.66 0.77 2.18 :a 0.09 0.00 1.12 0.00 20 0.00 0.25 0.01 1.11 10 19.91
10.79
19.91 11.54 20.40 -___.. ---.l_l_~....___,"_ .__.~._. __.___.____~ ...."_.~_.__
10.79
1X.62
20.46
sum Total
ASCF
relaxation energy
compared with experiment) and thus better total relaxation energies (tables 2 and 3), our agreement with the directly calculated total relaxation energy is worse than in the case of Clark et al. It is probable that inclusion of polarization functions in our basis set would improve this agreement. Nevertheless, the
~s~re~ancies do not exceed =O.l eV, i.e. = 1% of the total relaxation energies, and thus we consider our results as satisfactory. Let us focus our attention on the singleorbital relaxation energies of the core-ionized CO moIecuJe (tabfe 2). it is not a straightforward matter to explain differences between results obtained with our symmetry-adapted MOs and those of Clark et al. [1,33. T;iolc 3 Stngle~rbitd relaxation energies (eV) of the N, molecule -_1_-
..-...-
%a1
.-..--.
--_._
-
log core hole
1 t ?*I I iv; 2ug hJ
10, SlJJ
~__^_.___._____
-
_
Symmetry adapted MOs (this work)
.
.
-
.__.
The MOs of the CO molecule have inherently localized character mainly in the core and inner valence re. gions in both treatments. Nevertheless, the diverse basis sets used in either calculations, undoubtedly do not form the same localized character of MOs,especially for the core region, where our basis set is essentially larger, So we have associated differences jn thr single-orbital energies for carbon monoxide with the different character of the MOs employed in the calctllations. We do not discuss the characteristic features of the single-orbital relaxation phenomenon since such a discussion for carbon monoxide has been done elsewhere [I] in substantial detail. Our results confirm this discussion in its main directions.
_” - _._...___. ._-- .._._ --.-___--_l_-_-___l__ Orbital
1ou core hole
3.30 1.66 1.26 0.55 9.74 0.05 -..-
3.30 1.66 1.26 0.55 0.08 9.60
16.65
16.45
_-II_-
MOs us-d in previous calculations [ 3) N t s core hole
In 50 4P 30 20 lo f
8.74 0.30 3.62 1.90 0.80 15.36
to1111 AS3 -7.
580
relaxation energy 16.60 .- I---..---...----._.________
16.50
_-___
-V--.1-_--
15.35
Volume 83, number 3
CHEMICALPHYSICS LETTERS
An interesting situation appears in the core region of the N, molecule (table 3). ‘Ihe highest values of the singlearbital relaxation energy correapondt+he corelu~andlb,MOI.Inthisarewe~vle~~wrrelaxed orbit.& with’symmetry&tpted (d&calized) character. Our relaxed orbitalshave also been delocalized in the valence region and, however, localized in the core region as a consequence of the symmetry-unrestricted HF treatment of the core-h& molecular state. Clark et al. [3] used the localized MOs for Nz. At this point we mention once more that there is no physical reason for preferring some kind of MOawithin the single-relaxation model. The different MO localization in both cases is mirrored in the different distribution of relaxation energy into the orbital contributions. The two most deeply lying MOs consists of N,, atomic orbitals and thus, in our treatment, the unrelaxed lug and lo, core orbit& are symmetrically delocalized over both the N centers while the corresponding relaxed orbitals are localized on these centers, As a consequence of the delocalization and localization of the unrelaxed and relaxed core orbitals, respectively, we can expect - in comparison with the calculations of Clark et al. (table 3) who have employed localized orbitals - the greatest difference in the relaxation energies just in the core molecular region. Our different character of localization used for relaxed and unrelaxed core orbitals within the model causes substantial energy change8 which require redistribution of the singleorbital relaxation energies. Although more detailed discussion of the relaxation energies for N2 has been given [3], we consider it useful to illustrate an interesting behaviour of these energies for the highest MOs In,, and In of both the relaxation energy distributions under study (table 3). The In, orbital is delocalized over both the nitrogen centers, so that only about one-half of the whole orbital charge can relax towards the core hole#. On the other hand, all the orbital charge of the localized In
* These energy changes refer to a method B (see mf. [ 11) in the second computational step of the single&axationmodel. #The core hole has no physical interpretation and is only a computational artefact appearingin the symmetry-broken HF calculations 191. Nevertheless, we use this expression to simplify the treatment.
1 November 1981
orbital can relax quite effectively to the center bearing the core hole. These facts explain why the relaxation energy of t+e Jrru orbitalIs 8pproxiniately oneUf of the fr relaxatiori eneqy_in the l&liz.+d,case: .fie’~~.exam$e# C!p and Ni sh& that results obtained.from the computati&uI model of singleorbital reluution eneqies are‘dkpendent on the characterof the star&g MOs. Whili the different MO character for the CO,mtilecule is caused only.by two distinct A0 basis seta used in the calculations, the different MO character for Nz ls also given through the additional lo&lW.ion procedure (cf. ref. (31). Nevertheless, the different characters of the MOs yield an ambiious distribution of single-relaxation energies in spite of maintaining agreement with the total directly computed relaxation energy. So we conclude that the absence of a physical reason for preferring ,MOswith cert$n character of localization within the single-relaxation model leads to ambiguous partitioning of the ainglearbital relaxation energies; the proper choice of starting MOs used in the second computational step of the model does not remove the influence of the MO localization on the values of the singlearbital energies. In such a way, the singlearbital relaxation energies within the model have only a restricted meaning since their values depend on the MO localization, i.e. on the character of the MO basis.
References [l] D.T. Clark, B.J. Cromarty nnd A. Sgamellotti, Chem. Phys. Letters 51 (1977) 356. [2] D.T. Clark, B.J. Cromarty and A. Sgamellotti, J. Electron Spectry. 14 (1978) 175. (3 j D.T. Clark, B.J. Cromarty and A, Sgameliotti, J. Electron spectry. 14 (1978) 49. 14) D.T. Clark, B.J. Cromarty and A. Sgamellotti, Chem. Phys. 26 (1977) 179. (51 D.T. Chuk, B.J. .Cromartyand A. Sgamellotti, J. Electron Spectry. 13 (1978) 85; 17 (1979) 237. [6] J. Abnbf, P.S. Bagus, B. Liu, A.D. McLean, U. Wah&#rcn and M. Yoshomhre, IBM San Jon! Research Laboratory Report. [7] F.B. van Du&tenveldt, IBM J. Res. Develop. 945 (1971). [S] L.E. Sutton, ad., Tables of interatomic distances (The
Chemical Sodety, London, 1958). [ 91 A. Denis, J. LangIet and J.-P. Malrieu, Theoret. Chim. Acta 38 (1975) 49.
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