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A note on the orbital energy-splitting of closed-shell systems in the DODS scheme
VoGmc 31,.numbcr
I.5 Fcbruxiy
CI~BMICAL PHYSICS LETTERS
1
1975
.’
‘A NOTEoN ..
THE ORBITAL ENERGY-SPLITTING OF CLOSED-SHELL SYSTEMS
1N THE DODS SCHEME
Received 12 June 1974 .Revised manuscript received 18 October 1974
investigating respectively, in results obtairicd decrease of the
the lowering of the totd energy and the orbital energy-splittings of electrons with spine LTand p, the DODS scheme, ab initio calculations have been performed for the formaldehyde molecule. The by changing the C=O distance and the orbital exponents of the C and 0 atoms indiute that the electron density in the GO bond region causes larger orbital energy-splittings and iowcring of the
energy.
of a closed-shell system with a single unprojected Sister determinant in some cases a splitting of Q:and 0, the orbital energies belonging to the electrons with spins, respectively (and with it the lowering of the total energy) has occurred and in some other cases the solutions converged back in the SCF procedure to those of the conventional Hartree-Fock scheme. Thus, 1~electron DODS calculations of the nucIeotide bases and of their infinite stacked chains with the aid of the PPP MO and CO metho& 161, respectively, have given different Q and fi levels and bands, rcspectively [5] . An all-valence electron (CNDO/2) DODS ca.lcu!ation of the same molecules has not given any orbitaI,energy-splittings whatsoever [6] . The same difficulty arose by the ab initio DODS treatment of the l& molecule at the equilibrium bond distance (1.40 au) [7] and by an ,ab initio AM0 (unprojected) treatment.of an Li crystal by.Calais and Sperber [8] using the experimental lattice constants. On the other hand, by increasing the distances between the Li atoms they have obtained orbital energy-splitti&s. in connection with the latter calculations it should be mentioned that if the number of electrcns (n) ,of the system goes to ~mfiiity, ?I + m an+ II S s {the multipIicity), the solutions of ‘-he EHF equations, a5 bne ca,n show [9], go over to those of. ... the .unprojected UHF equations. This means that in
1. Introduction The method of different orbitals for different spins (DODS) is considered as one possible way to take into account the correlation. However, using one single Slater determinant built up from different spatial orbitals for the electrons with spins cr and p, respectively, the many-electron wavefunction, as is well-known, will not be an eigenfunction of the operator s2 of the total spin. To overcome this difficulty, one can either project out from the resulting SIater determinant the component with the desired multiplicity after the variation (unrestricted. Hartree-Fcck, UHF method) or more consequently, as Ldwdin [l] has pointed out, one-could perform the variation with an &ready’ spin-projected Slater determinant (spin-projected extended Hartree-Fock, El-IF method). Treating a closed-shell system, with the EHF method [l-3] one would always expect a splitting of the one-electron levels of the electrons with spins (Yand p, respectively. Namely, the solutions of rather complicated spin-projected EHF equations (which one gets from the variation of a spin-projected Slater determinant [3]) a&the solutions of a specific multi-. ..: ‘configurational SCF problem. The resuIts obtained 'ir! this way fulfil this expectafick [4]:: On the other hand? if one starts the.calculation .’ ._: ” :. .’ :.‘
s3 ,’
:-
..
‘,
the cases of solids, if, no orbital energy-splittings OCCIX in the simple unprojected DOQS (UHF) scheme, one cannot cspcct that a’spin-projcctcd calculation would help to obtain a part.of the corielation energy. Consideritig the different cases when orbital criergy.splitt&s occur and do not cccur, one’!,first in-rpression would be that this depends on,the average electron density pioduced by the electrons taken into account explicitly in the.system (in the n-electron calculations with qne &lectron.per atom one gets splitt.ings, the difficulties start already by the all-, valence-electron calculations with more electrons
at the larger C=O,distance
about 90 iteiations were needed to achieve self-consistency. For i..le &CO ko!ecule-the geometry determined : “by Taka:g ana Oka has beeil used with a.C:O distance of 1.209 t$ (2,282 au),‘C-H bond distances of 1.116 9, and with’s HCO val&cc aiigle of 121.74” [l?] .‘Bcsides the abqve given equilibrium C=O dis-’ tance the calculations have been repeated for the i C=.O boi:d lengtll? Cl.944 a( 1.782 au) and 1.739 A (3.282 alj, respectively, keeping the C-H distances and the lcalence angles unchanged. For the two small& C=.O distances,-tihcre the total.encr& obtained in the UHF scheme is within z 2 X low5 au the same as in the conventional HF scheme [see below), further ca!culations have been performed multiplying Lhe exponents of all the carbon and oxygen .orbitals by.a factor of 2 and l/10, respectively (while the exponents of the hydrogen orbitals have been kept constant). In the case of the larger GO distance of 1.739 A [when a splitting has occurred; see-table 1) the exponents of the C and 0 orbitals ha& been multiplied only by l/10. Atternpis to perforin the cdl-
per atom) and perhaps a criterion formulated in terms 0: t&quantity, similar to hlolt’s.criterion [lo] for-the metal-non-metal &nsition, co&l be e&blished *. To’investigatc this possibility ‘as first step we have investigated the formaldehyde molecule with the aid of the ab initio DODS SCF LCAO ,MO method (unprojectcd) at different GO bond distances and changing the orbital exponznts of the applied gaus-1 sian lobes basis functions.
culgtions by the two smaller GO distances with 2. Method
larger mu.ltiplicative factors of the C and 0 orbitals than 2 (3 and 5) have failed because the HF calculation has not given convergent results. In the cases when thkorbitul energy-splittings were very small they couid be obtained by adding small perturbation
For the calculations a gaussian ,lobes’7/3 + 4 (7 s furictions and 3 &, p,, and pz functions, respectively, fir the ?tor& C and O.and 4 s functions for the
hydrogen atom) basis set has beer-i applied contracted
terms to ,:ach element of theXF
to 4 s and 2 p functions at the hearj ilTorns and. to 2 s functions at tl,ie, H atoms. Th’c orbital exponents 2nd .conk&rl’coefficients were those given by Huzinaga [l, l] . The restricted SCF LCAO MO and, PODS SCF LCqO. MO calculations have been performed with the aid of the program MOLPRO written by IMeyer and.Pulay [ 123 . The convergence criteria applied in these programs are
matrix P/2 (obtafned in the restricted. HI; calcu13tion) and using. this matrix as starting matrix PC(O) for the electrons with spin d, whereas for Pp@) the unchanged matrix P/2 has been applied. ..
!p(rl+l) r,s
-p;‘:$ao-10, .‘
,p”“‘7 r,s
_~(“)~~<‘l()-~o(r=ol r,s
-3. Discussion
(1) 30)
(2)
-for e&h correspo;iding element of the charge-bond order niatri&,.P and.PT (7 =.a,@, respectively, obtained in two s!lbscquent ite’rdion steps, These criteria were usually fulfilIed by’ tl& equilibrium arid the $naller:C=O diqtance in 30 iteratiori steps,whereas .‘ * S&estion~ofF. : .x4: : ., -..
Martino. .’
__
”
. . ., .‘
,:
.’
charge-bond order
of the results
In table 1 we present the total cl!ergies z&the HOMO and LEMO qne-electron energies obtained both in the conventional HF and in tl!e UHF (DODS) schemes, respectively, at differellt C=O distances and ” .‘. . different.orbital exponents. Looking at the table ‘we find that at the equilibriurn C=O distance (I? = 2.282 au) @it11%= 1. the. tote1 ener.5 is only by.2 X 10~~ au = 5 X. 10W4 eV lower,in riie PODS case thanin the ‘conventional HF case. Increasing the orbital exp&ents by j factor of2 (that is localizl’ng more. the.AO’s on the C and’0. ._ ..,.
Volume
I
31, number
Table 1 The total cncrgY orbitaj exponents
CHE?JICAL.I’HYSICS
and the HO>10 and LChlO one-clcctron in the ab initio HF and UHF (DODS)
energies of tlw I’ormaldchydc schemes (in atomjc units)
Etota1
R = 1.782b)
R =2.282”)
R = 3.282
;;i = 1 a)
I5 [:chru~Y
LISTTERS
UHF
HE:
UHF d)
-1 Z3.47385
-113.47385
-0.37708
-0.37695
i;=2
-11 i.88099
-112.88099
-0.30246
z = 0.1
-
-
-0.58602
;i = 1
-113.68889
-113.68891
-0.43167
ii,= 2
-113.02500
-113.03292
-0.32947
z = 0:1
-
-
84.13269
-0.56371
Z=l
-113.51333
-I 13.60387
-0.39117
z = 0.1
-
-
-0.46530
84.13269
E4.55860
82.95106
84.55860
at diffcrcnt
C=O distancci
2nd
‘LEMO
EHOh10
1IF
82.95106
molccr~lc
1955
HI: 0.205
-0.31124 -0.30223 -0.30268 -0.53599 -d.58604
17
0.205
LI
3.20523
0.20 LO2
0.29114
0.19168 -0.04676 -C1.C4675
-0.04676
-0.42951 -0.43435 -0.31214 -3.34077 -0.56374 -0.56371
0.14606
0.14988 0.14447 0.28687 0.30253 -0.06603 -0.06602
0.29384 -0.06602
-0.43756 -0.5239! -0.46523 -0.46536
a) Z is the factor with which the original exponents [ 1l] of IIIC orbit& belonging tiplied. b) The GO distance. c) Equilibrium GO distance. d) The fust value refers always to the a-level, the second one to the p-level.
UHF’)
0.03140
0.17440 0.14988 -0.09138 -0.09 125
-0.09131
to the C and 0 atoms
-
respectively,
wete mul-
the lowering of the total energy in the UHF cn~e becomes larger (A.&,,, = S X 1 O-3 au = 0.2 ev>:
wouid bc ri&t,
other hand, when WC have decreased the exponents of the carbon and oxygen orbit& multiply-
which increnses the augpdge electron density in the molecule, would cause - contrary to the obtained
density
atoms)
On
the
ing them by a factor of 0.1, the HF and UHF total energies became equal within tke accuracy of the caJcukition (1 0P6 au):Looking tit the HOMO and LEMO one-electron energies (which usually show somewhat iarger splittings than the lowering of the total energy values) we find a similar behaviour: the stronger localization of the AO’s (;;; = 2) increases the original orbital energy-splittings (both in the HOMO and LEMO cases = 5 X 10e3 au) to 28.6 X 1O-3 au (HOMCJ and 15.7 X iOd4 au (LEMO), whereas in the case of the more deloca@d AO’s with smaller exponents the level-splittings become extretiely small (1-3 X 10m5 au). AU these results indichte that tile orbital energy-splittings (and the lowering of the tota,1 energy) in the DODS case depend much more on the electronic density in the.bond regiqn than Y as it was previously assumed - on the average:electron-
.’
of the molecule.
(If the latter assumption
the increase
of tile
orbital
exponents,
results - a strong decrease of the orbital encrgysplittings.) Turning now to the smaller GO distance the hehaviour of the total energy values and of the HOMO and LEMO !evels is consistent with thz above discussed results: the total energies are for all the three Z&‘-valuesthe same in the HF and UHF (DODS) cases, whereas the,remaining very small orbital energysplittings (l-3 X lo-$ au for Z = I).become again larger in the more localized iii = 2 case and practically disappear (orbital energy-splittings of I-S X 10-S au) in the more delocalized z = 0.1 case). Finally, at the rather large C=O distance of 3.282 au there is a lowering of the, total energy in the-UHF case (= 9 X 1@-2 au = 2.50 eV) as compared to the liF energy ‘fo; Z’= 1, whereas forZ = 0.1 this energy lowering disappears., Correspondingly ti;e ratI;er large orbital
.. .. . ,‘,’ : ClIE~ZICALPIIYSIC:S LITTERS; .L5 f-‘cbru~ry 1975, V&unic 31. riumbcr i .: ..’ ., .,’ .. ‘I- : .gratefu& acknowledged~~Fi&lly, we’should like to _’;ner,v-splittings of 0.02-0.08 au decrease.to’10-4.. des Max-PlanckrInstituts ‘,: :, thank the-Rcchenzentrum 3u when we changeZ from 1 to 0.1: ., fir PlasmAphysik for giving &co,tiputer time on Iheir : To broke that these fiist results’show a ge&nl T IBlA 360/91 computer. .,. ; tendency also,in other molecuIes,(w~zat.i~e.consideE : ~. : .. -3s probable),si&&r &lcLlat!,ons have to be-performed in ‘somewhat liiger molec+s dorit&ing more heavy ;. _’ Rel‘erences. (non-hydrogen) atoni.%. To be able to formulate a
genera! c$erion for the drbital energy-?mdittings8, ,‘the bODS scl~cme it will be necessary also to perform h detailed~analysis~of tile eleclronic density in diFf&nt r@ions bf-t@ investigated moiecules. We plan to perform thkse rather extensive investigations.-
: Acknowledgqment
.We should like to e;ipress our gratitude to Professor. G .L. Hofackcr.
:‘or &continuous
interest
and
.’
support which made it possible td perform this investigation.
We are further indebted
to Profcssdr
Meyer for his help by putting his programs to our disposaj and for useful dis2ussions. Sincere thanks are due to-Professor F. Martin0 for the stimulating
.W.
disc&ions on tile problem. The financial support : of the Deutsche Forscllungsgemeinscllaft should be :
‘.
.’
..
[i 1 P.a. LEwdin, Phys. Rev. 97 (1955) 1474, 1490. [2] W.A. Goddard III, PJhys. Rev. 157 (1967) 73, 81;.J. a^ ,_^. Chem. PhyS .48 (iYti7) 450,5537. (3 ] F. Martin0 ~UU -A J.’ ~4 r -dik, J. Chem. Phys. 52 (1970)
2262;, . . 1.__, __, _. _ I I. L1Id G. Bicz6, Intern. J, Quantum Chem. 7 (1973) 583. .[4] F. Mnrtino; to be published. [5] M. KertCsz, S. Suhai and J. Lad&, Rcta Piiys. Hung. Aud. Sci., to be published. IAl A _.Al.ILY Fbrtb ~nrl I IWV qdikunpubfisI,ed. L.2, r.. VA‘”I_ [T] W..Meyer, private co mmunichiion. [S] J--L. Calais and G. Sperbcr, In&n. J. Quantum Chem. 7 (1973) 501. [9] F. Martin0 and J. L;ldik, Phys. Rev. A3 (1971) 862. [IO] N.1:. Mott and E.A. Davis, Electronic processes in noncrystalline materials (ClarencIon Press. Oxford, 1971) p. 121. [ 111 S. Huzinaga, J. Chem. Phys. 42 (1965) 1293. [ 121 W. ,M.eyer and P. Pulay, unpubEshed. 1131 I<. ‘Tagki and.T.Okn. J. Phys. Sot. Japan ;8 (1963) .. 1174.
I.5 Fcbruxiy
CI~BMICAL PHYSICS LETTERS
1
1975
.’
‘A NOTEoN ..
THE ORBITAL ENERGY-SPLITTING OF CLOSED-SHELL SYSTEMS
1N THE DODS SCHEME
Received 12 June 1974 .Revised manuscript received 18 October 1974
investigating respectively, in results obtairicd decrease of the
the lowering of the totd energy and the orbital energy-splittings of electrons with spine LTand p, the DODS scheme, ab initio calculations have been performed for the formaldehyde molecule. The by changing the C=O distance and the orbital exponents of the C and 0 atoms indiute that the electron density in the GO bond region causes larger orbital energy-splittings and iowcring of the
energy.
of a closed-shell system with a single unprojected Sister determinant in some cases a splitting of Q:and 0, the orbital energies belonging to the electrons with spins, respectively (and with it the lowering of the total energy) has occurred and in some other cases the solutions converged back in the SCF procedure to those of the conventional Hartree-Fock scheme. Thus, 1~electron DODS calculations of the nucIeotide bases and of their infinite stacked chains with the aid of the PPP MO and CO metho& 161, respectively, have given different Q and fi levels and bands, rcspectively [5] . An all-valence electron (CNDO/2) DODS ca.lcu!ation of the same molecules has not given any orbitaI,energy-splittings whatsoever [6] . The same difficulty arose by the ab initio DODS treatment of the l& molecule at the equilibrium bond distance (1.40 au) [7] and by an ,ab initio AM0 (unprojected) treatment.of an Li crystal by.Calais and Sperber [8] using the experimental lattice constants. On the other hand, by increasing the distances between the Li atoms they have obtained orbital energy-splitti&s. in connection with the latter calculations it should be mentioned that if the number of electrcns (n) ,of the system goes to ~mfiiity, ?I + m an+ II S s {the multipIicity), the solutions of ‘-he EHF equations, a5 bne ca,n show [9], go over to those of. ... the .unprojected UHF equations. This means that in
1. Introduction The method of different orbitals for different spins (DODS) is considered as one possible way to take into account the correlation. However, using one single Slater determinant built up from different spatial orbitals for the electrons with spins cr and p, respectively, the many-electron wavefunction, as is well-known, will not be an eigenfunction of the operator s2 of the total spin. To overcome this difficulty, one can either project out from the resulting SIater determinant the component with the desired multiplicity after the variation (unrestricted. Hartree-Fcck, UHF method) or more consequently, as Ldwdin [l] has pointed out, one-could perform the variation with an &ready’ spin-projected Slater determinant (spin-projected extended Hartree-Fock, El-IF method). Treating a closed-shell system, with the EHF method [l-3] one would always expect a splitting of the one-electron levels of the electrons with spins (Yand p, respectively. Namely, the solutions of rather complicated spin-projected EHF equations (which one gets from the variation of a spin-projected Slater determinant [3]) a&the solutions of a specific multi-. ..: ‘configurational SCF problem. The resuIts obtained 'ir! this way fulfil this expectafick [4]:: On the other hand? if one starts the.calculation .’ ._: ” :. .’ :.‘
s3 ,’
:-
..
‘,
the cases of solids, if, no orbital energy-splittings OCCIX in the simple unprojected DOQS (UHF) scheme, one cannot cspcct that a’spin-projcctcd calculation would help to obtain a part.of the corielation energy. Consideritig the different cases when orbital criergy.splitt&s occur and do not cccur, one’!,first in-rpression would be that this depends on,the average electron density pioduced by the electrons taken into account explicitly in the.system (in the n-electron calculations with qne &lectron.per atom one gets splitt.ings, the difficulties start already by the all-, valence-electron calculations with more electrons
at the larger C=O,distance
about 90 iteiations were needed to achieve self-consistency. For i..le &CO ko!ecule-the geometry determined : “by Taka:g ana Oka has beeil used with a.C:O distance of 1.209 t$ (2,282 au),‘C-H bond distances of 1.116 9, and with’s HCO val&cc aiigle of 121.74” [l?] .‘Bcsides the abqve given equilibrium C=O dis-’ tance the calculations have been repeated for the i C=.O boi:d lengtll? Cl.944 a( 1.782 au) and 1.739 A (3.282 alj, respectively, keeping the C-H distances and the lcalence angles unchanged. For the two small& C=.O distances,-tihcre the total.encr& obtained in the UHF scheme is within z 2 X low5 au the same as in the conventional HF scheme [see below), further ca!culations have been performed multiplying Lhe exponents of all the carbon and oxygen .orbitals by.a factor of 2 and l/10, respectively (while the exponents of the hydrogen orbitals have been kept constant). In the case of the larger GO distance of 1.739 A [when a splitting has occurred; see-table 1) the exponents of the C and 0 orbitals ha& been multiplied only by l/10. Atternpis to perforin the cdl-
per atom) and perhaps a criterion formulated in terms 0: t&quantity, similar to hlolt’s.criterion [lo] for-the metal-non-metal &nsition, co&l be e&blished *. To’investigatc this possibility ‘as first step we have investigated the formaldehyde molecule with the aid of the ab initio DODS SCF LCAO ,MO method (unprojectcd) at different GO bond distances and changing the orbital exponznts of the applied gaus-1 sian lobes basis functions.
culgtions by the two smaller GO distances with 2. Method
larger mu.ltiplicative factors of the C and 0 orbitals than 2 (3 and 5) have failed because the HF calculation has not given convergent results. In the cases when thkorbitul energy-splittings were very small they couid be obtained by adding small perturbation
For the calculations a gaussian ,lobes’7/3 + 4 (7 s furictions and 3 &, p,, and pz functions, respectively, fir the ?tor& C and O.and 4 s functions for the
hydrogen atom) basis set has beer-i applied contracted
terms to ,:ach element of theXF
to 4 s and 2 p functions at the hearj ilTorns and. to 2 s functions at tl,ie, H atoms. Th’c orbital exponents 2nd .conk&rl’coefficients were those given by Huzinaga [l, l] . The restricted SCF LCAO MO and, PODS SCF LCqO. MO calculations have been performed with the aid of the program MOLPRO written by IMeyer and.Pulay [ 123 . The convergence criteria applied in these programs are
matrix P/2 (obtafned in the restricted. HI; calcu13tion) and using. this matrix as starting matrix PC(O) for the electrons with spin d, whereas for Pp@) the unchanged matrix P/2 has been applied. ..
!p(rl+l) r,s
-p;‘:$ao-10, .‘
,p”“‘7 r,s
_~(“)~~<‘l()-~o(r=ol r,s
-3. Discussion
(1) 30)
(2)
-for e&h correspo;iding element of the charge-bond order niatri&,.P and.PT (7 =.a,@, respectively, obtained in two s!lbscquent ite’rdion steps, These criteria were usually fulfilIed by’ tl& equilibrium arid the $naller:C=O diqtance in 30 iteratiori steps,whereas .‘ * S&estion~ofF. : .x4: : ., -..
Martino. .’
__
”
. . ., .‘
,:
.’
charge-bond order
of the results
In table 1 we present the total cl!ergies z&the HOMO and LEMO qne-electron energies obtained both in the conventional HF and in tl!e UHF (DODS) schemes, respectively, at differellt C=O distances and ” .‘. . different.orbital exponents. Looking at the table ‘we find that at the equilibriurn C=O distance (I? = 2.282 au) @it11%= 1. the. tote1 ener.5 is only by.2 X 10~~ au = 5 X. 10W4 eV lower,in riie PODS case thanin the ‘conventional HF case. Increasing the orbital exp&ents by j factor of2 (that is localizl’ng more. the.AO’s on the C and’0. ._ ..,.
Volume
I
31, number
Table 1 The total cncrgY orbitaj exponents
CHE?JICAL.I’HYSICS
and the HO>10 and LChlO one-clcctron in the ab initio HF and UHF (DODS)
energies of tlw I’ormaldchydc schemes (in atomjc units)
Etota1
R = 1.782b)
R =2.282”)
R = 3.282
;;i = 1 a)
I5 [:chru~Y
LISTTERS
UHF
HE:
UHF d)
-1 Z3.47385
-113.47385
-0.37708
-0.37695
i;=2
-11 i.88099
-112.88099
-0.30246
z = 0.1
-
-
-0.58602
;i = 1
-113.68889
-113.68891
-0.43167
ii,= 2
-113.02500
-113.03292
-0.32947
z = 0:1
-
-
84.13269
-0.56371
Z=l
-113.51333
-I 13.60387
-0.39117
z = 0.1
-
-
-0.46530
84.13269
E4.55860
82.95106
84.55860
at diffcrcnt
C=O distancci
2nd
‘LEMO
EHOh10
1IF
82.95106
molccr~lc
1955
HI: 0.205
-0.31124 -0.30223 -0.30268 -0.53599 -d.58604
17
0.205
LI
3.20523
0.20 LO2
0.29114
0.19168 -0.04676 -C1.C4675
-0.04676
-0.42951 -0.43435 -0.31214 -3.34077 -0.56374 -0.56371
0.14606
0.14988 0.14447 0.28687 0.30253 -0.06603 -0.06602
0.29384 -0.06602
-0.43756 -0.5239! -0.46523 -0.46536
a) Z is the factor with which the original exponents [ 1l] of IIIC orbit& belonging tiplied. b) The GO distance. c) Equilibrium GO distance. d) The fust value refers always to the a-level, the second one to the p-level.
UHF’)
0.03140
0.17440 0.14988 -0.09138 -0.09 125
-0.09131
to the C and 0 atoms
-
respectively,
wete mul-
the lowering of the total energy in the UHF cn~e becomes larger (A.&,,, = S X 1 O-3 au = 0.2 ev>:
wouid bc ri&t,
other hand, when WC have decreased the exponents of the carbon and oxygen orbit& multiply-
which increnses the augpdge electron density in the molecule, would cause - contrary to the obtained
density
atoms)
On
the
ing them by a factor of 0.1, the HF and UHF total energies became equal within tke accuracy of the caJcukition (1 0P6 au):Looking tit the HOMO and LEMO one-electron energies (which usually show somewhat iarger splittings than the lowering of the total energy values) we find a similar behaviour: the stronger localization of the AO’s (;;; = 2) increases the original orbital energy-splittings (both in the HOMO and LEMO cases = 5 X 10e3 au) to 28.6 X 1O-3 au (HOMCJ and 15.7 X iOd4 au (LEMO), whereas in the case of the more deloca@d AO’s with smaller exponents the level-splittings become extretiely small (1-3 X 10m5 au). AU these results indichte that tile orbital energy-splittings (and the lowering of the tota,1 energy) in the DODS case depend much more on the electronic density in the.bond regiqn than Y as it was previously assumed - on the average:electron-
.’
of the molecule.
(If the latter assumption
the increase
of tile
orbital
exponents,
results - a strong decrease of the orbital encrgysplittings.) Turning now to the smaller GO distance the hehaviour of the total energy values and of the HOMO and LEMO !evels is consistent with thz above discussed results: the total energies are for all the three Z&‘-valuesthe same in the HF and UHF (DODS) cases, whereas the,remaining very small orbital energysplittings (l-3 X lo-$ au for Z = I).become again larger in the more localized iii = 2 case and practically disappear (orbital energy-splittings of I-S X 10-S au) in the more delocalized z = 0.1 case). Finally, at the rather large C=O distance of 3.282 au there is a lowering of the, total energy in the-UHF case (= 9 X 1@-2 au = 2.50 eV) as compared to the liF energy ‘fo; Z’= 1, whereas forZ = 0.1 this energy lowering disappears., Correspondingly ti;e ratI;er large orbital
.. .. . ,‘,’ : ClIE~ZICALPIIYSIC:S LITTERS; .L5 f-‘cbru~ry 1975, V&unic 31. riumbcr i .: ..’ ., .,’ .. ‘I- : .gratefu& acknowledged~~Fi&lly, we’should like to _’;ner,v-splittings of 0.02-0.08 au decrease.to’10-4.. des Max-PlanckrInstituts ‘,: :, thank the-Rcchenzentrum 3u when we changeZ from 1 to 0.1: ., fir PlasmAphysik for giving &co,tiputer time on Iheir : To broke that these fiist results’show a ge&nl T IBlA 360/91 computer. .,. ; tendency also,in other molecuIes,(w~zat.i~e.consideE : ~. : .. -3s probable),si&&r &lcLlat!,ons have to be-performed in ‘somewhat liiger molec+s dorit&ing more heavy ;. _’ Rel‘erences. (non-hydrogen) atoni.%. To be able to formulate a
genera! c$erion for the drbital energy-?mdittings8, ,‘the bODS scl~cme it will be necessary also to perform h detailed~analysis~of tile eleclronic density in diFf&nt r@ions bf-t@ investigated moiecules. We plan to perform thkse rather extensive investigations.-
: Acknowledgqment
.We should like to e;ipress our gratitude to Professor. G .L. Hofackcr.
:‘or &continuous
interest
and
.’
support which made it possible td perform this investigation.
We are further indebted
to Profcssdr
Meyer for his help by putting his programs to our disposaj and for useful dis2ussions. Sincere thanks are due to-Professor F. Martin0 for the stimulating
.W.
disc&ions on tile problem. The financial support : of the Deutsche Forscllungsgemeinscllaft should be :
‘.
.’
..
[i 1 P.a. LEwdin, Phys. Rev. 97 (1955) 1474, 1490. [2] W.A. Goddard III, PJhys. Rev. 157 (1967) 73, 81;.J. a^ ,_^. Chem. PhyS .48 (iYti7) 450,5537. (3 ] F. Martin0 ~UU -A J.’ ~4 r -dik, J. Chem. Phys. 52 (1970)
2262;, . . 1.__, __, _. _ I I. L1Id G. Bicz6, Intern. J, Quantum Chem. 7 (1973) 583. .[4] F. Mnrtino; to be published. [5] M. KertCsz, S. Suhai and J. Lad&, Rcta Piiys. Hung. Aud. Sci., to be published. IAl A _.Al.ILY Fbrtb ~nrl I IWV qdikunpubfisI,ed. L.2, r.. VA‘”I_ [T] W..Meyer, private co mmunichiion. [S] J--L. Calais and G. Sperbcr, In&n. J. Quantum Chem. 7 (1973) 501. [9] F. Martin0 and J. L;ldik, Phys. Rev. A3 (1971) 862. [IO] N.1:. Mott and E.A. Davis, Electronic processes in noncrystalline materials (ClarencIon Press. Oxford, 1971) p. 121. [ 111 S. Huzinaga, J. Chem. Phys. 42 (1965) 1293. [ 121 W. ,M.eyer and P. Pulay, unpubEshed. 1131 I<. ‘Tagki and.T.Okn. J. Phys. Sot. Japan ;8 (1963) .. 1174.