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The 1A1 π→π* state of formaldehyde
CI-IEhlICAL PHYSICS LETTERS
Volume 29, number 2
I.5 November 1974
.THE ‘A I IS-+& STATE OF FORMALDEHYDE Stephen R. LAhGHOFF’,
Stephen T. ELBERTf, Charles F. JACKELS and Ernest R. DAVIDSON** Deparhleni of Chembtry, Universip of Washington, Secttle, Weshingon 98195, USA
Retied
The ‘Al POT’ state of formaldehyde
Received 20 May 1974 manuscript received 8 July 1974
is predicted
to be 11.2 eV above the ground state and not diffuse.
There have been several papers published recently which differ significantly on the energy and nature of the 1A, T-W.* state of formaldehyde. Buenker and Peyerimhoff [l] using a basis set without diffuse orbitals obtained an.excitation ener,g of 11.72 eV. In this calculation the T-W_*state appeared as the second root of the secular equation and fhe 3A, SCF orbitals were used io construct the configurations. In a later paper Peyerimhoff et al. [2] obtained an excitation energy of 11.41 eV. In this calculation diffuse basis functions were included to represent some Rydberg states. The W* state appeared as the third root of the secular equation based on 3AI SCF orbit&. Both of these calculations miss the ground state SCF energy by about 2 eV and account for less than 10 % of the correlation energy. Whitten and Hackmeyer [3] obtained 11.3 1 eV for the excitation energy based on the third root of a secular equation. They used diffuse basis functions and ground state SCF orbitals. This calculation also missed the ground state SCF energy by i eV and obtained about. 15 %.of &e correlation energy, In a later cakulation Whitten [4] obtained 9.9 eV as the second root of a secular equation. This calcul&ion used a “better” basis set that missed the ground state SCF energy by only 1 eV and enough configurations to obtain.about * Present address: Battelle Memorial Institute, Columbus, Ohio 43201, USA. $ Present address: Lzhrstuhl fiir ceoretische Chemie der UniversitZt Bonn, 53 Bonn, West Germany. ** To whom correspondence should be addressed.
2.5% of the correlation energy. Effuse basis functions, however, were used only in the bl(%) subspace. He feit this would a&w the proper size adjustment of the 7r and n* orbit&. Because diffuse b, orbit& were not included, the lower energy b, + bj Rydberg states were not present in the secular equation. He found the 71and + orbit& were not particaiarly diffuse. Recently Yeager and McKay [51 have treated this excitation with the equations of motion method. They obtained 10.1 eV from the four’& ‘A, energy level. But they also claim the state is very diffuse with an average value of Z(X,? +yf) about 15a$ larger than the ground state and with a small f due. No absolute energies are given by which the completeness of their results can be judged. Although the energy reported in this work agrees with Whitten, the large change in Z(X~ +$) does not. Whitten indic;ttes a change of only about 1.4 for Llxf. The experimental results for this state of formaldehyde are also confused. All of the smal1 amides and aldehydes were at one time belieqed to Slave a T a IT* 1 A state near 7-8 eV [6] . It has now been established for formaldehyde and fonnamide that the lowest excited IA state is really an n -, 3p Rydberg state [3, 7, 81. For formaldehyde, there is no currently accepted experimental result for the pi -+ T* gtate. One of the uncertainties in t.he SCF CI approaches to.this state has been the inability to handle the SCF problem. b order to overcome this we have-done a multi-configuration SCF calcuIation using the three configtirations r2, iii?*, and fl?_ (Merely. optimizing 247
CHEMICAL PHYSICS LX’l-ERS
Volume 29, nGmber 2 ,_ :
.; ‘.
.: ‘.
,’
..
1
,.T&le
Ericrgies
State. :
Calctilation
‘gr&nd SkF W/O .’ $~&-id~’ SCF W
;
,15 November 1974’
’
:
:h.
ground
MC .kF W/o
R--X+ li+x*
MC!SCF W/Q.. MC SCF W
.’ ground
.CI W/G
EnergL
12.5%
-113.4688
(Il$)
-114.1349
-113.7066 CI w/o (root 2, ;m*j -113.7213 CI w/o (root 2, UZ, v2) cr’w: -113.5119 (root 3, GS) ~ CIW -113.7056 (root 2,7+) extrapohted extrapohted
ground
i-r--r*
a) Compahg b, Comptiing
1.5jb)
-114.1347
;I+?r*
X+7?*
:
~y113.4676
Cl W
??+n*.
.,
,-rl 13.8505 -113.8911 -113.9276.’
ground
n--a*
.Excitation energy
:
‘i1:6’) 11.2=) 10.3d) Xzi”)
^.
-114.2Gl -113.788
11.2
MC SCF remits. MC SCF results for n * n* with SdF for ground
.’ !) z&ring CI results. . . d, Compared to ground state SCF.
0.94(&
-* does.not work since .one obtains a’ ,. the.energy of nrr
bad ~~pro~atjon to the ground state. ‘The energy ., goes well below the true excited state energy.). The ” MC SCF procedure consists oftwo steps; an adjustment of the orbit& and a small con~~rat~~n,i~terac~ tion; In order to get the excited’state we used the set.ond root of this’CI. The basis sets used were one without (labeled W/O in, table 1) diffuse functions and one with both diffuse br and b, functions (labeled Win table 1). The ground state SCF energy with these basis sets are different from the best SCF’results of Neumann and Moskowitz by 0.3 and 0.2 eV, respectively.. I..- As table 1 shows, the excitation energy cotiparing the best three configuration iesults for the ground and ’ excited states give an excitation energy of 12.5 ,eV.. Inclusion of diffuse basis~functions has very lit& ef..fe& on the energy or wavefunctions at this level df ap‘pr&m&n. Tab,le 2 gives values for x2,y2, and z2 -:24g., .; :, ..,
,, _;
‘,
for c:om~arison.’ .‘. The CI results for the ground ‘&tte.in table I are all based on single plus selected double excitations using ground state SCF orbitals and about 850 configurations. This gives about 40 lo of the correlation energy. The :xtrapoIated limiting energy for the diffuse basis set &Auding al! double and higher excitations is -1 lr1.201. The CI results for x + n* are not as straightforward. The caIculation 1abeIcd GS was based on all single esicitations from the ground state using ground state orbitah. This excitation energy, 10.1 eV, is’probably most comparable to the McKay results. The wavefunction zt this level contained an appreciable mixture of diffuse configurations. The calculation labeled z12, y2 was based on the’natural orbital form of the MC SCF wavefunction. It is weIl known that any function of the form an2 f b(n+ + n*$ t CT~*~can be factored into the form c,rrz .-kC,,Y~.All single and double excitations from both &and y2 were considered for inclusion in the wavefunction. The u, Y form of the wavefunction can be returned to a “best?’ n, n* form (the original rr;7~* form is arbitrary within a unitary transformation). This gives 7~= u + v, TP = u - Y, and a wavefunction of the form’ -I-n*?r}/d
Since t3e coefficient of rrrr* in this form’was large, a Cl based on ,single and selected double excitation from. this nti” configuration wasalso done. As can be seen in table 1, the differences betieerrthese CI results is small. The extrapolated excitation energ? based on the calculation with diffuse orbitals is 1‘1.2 eV. These resuits very clearly agree with the work of Buenker and Peyerimhoff. No excitation energy as low asthat reported by Whitten was obtainable. Thi expectation values’given in table 2 demonstrate tha& the wavefunctions arc not very sensitive to.inclusiorr of diffuse orbitafs: The sum 6rl~21n>t (n*l.r~~In*> obtaine< with’ our diffuse basis agrees w&l with Whitten’s.. value 7.01. No hint that the’ rr.+,rr* state is diffuse appears in ,iable 2. AU of the results discussed here arc subject ,to one overwhelming criticism. The ionization:energy of formaldehyde -from the.highert b2 orbital is only 10.9 eV. Consequently if the n 7 rf” state is really, at 1‘1.2eV it ,, 1
..
:
.: -,.
- 0.24(7? + rP2).
.
-.
.
.;,
‘,.
CHEMICAL PHYSICS LETTERS
Volume 29, number 2
Values forx’,
1.5 November 1974
Table 2 y*, and z2 in 0’0units a)
State
Calculation
Lx*)
(y*,
(z2)
My* 177)
ground
SCF W/O
14.96
8.65
37.15
I.637
ground ground
SCF W CI w/o
15.00 15.08
37.18 37.24
1.646
ground
CI \v
15.12
8.68 8.64 8.72
rr’i7’
Cl w/o
15.12
8.48
37.97
1.432
ir’??
CI w
15.60
9.27
38.17
1.469
a) z is along CO bond,y
k*ly*lr;t;
37.28 2.317 3.233
is out of plane, the origin is at the center of mass.
is auto-ionizing. This means that the variational prindple, strictly speaking, does not apply. If 2 complete basis set were used in the b, subspace, there would be an infmity of 3, + bb 1A1 states at lower energies. Since the variational principle only guarantees that the kth root of the secular equation is in upper bound to the true Mh energy, it would appear to be nonsensical to do straight variational calculations for this state. Two answers can be given to this. First of all, by the perturbation selection procedure used here, one can regard this calculation as one in which one set of configurations has been treated to infinite order whiie other third- and higher-order effects are omitted. Secondly, the method used here is similar to those already well-established for handling auto-ionizing states. But it is true that if more diffuse basis functions were included this calculation would break down. Particularly if it happened that a Rydberg configuration which was weakly coupled to the 5r+ n* state accidently was nearly degenerate with it, the wavefunctions vvould mix in zeroth order and no state identifiable as primarily n--f rr* would appear in the results. Nevertheless,
sensible to say that near 11.2 eV there will be a band of states which contain most of the rr --Lrr8 contribution. it is
The authors wish to thank the Petroleum Research Foundation for supporting part of this work.
References
[l]
R.J. Buenker and S.D. Peyerimhoff, I. Chem. E’hys. 53 (1970) 1368. [21 S.D. Peyerimhoff, R.J. Buenker, W.K. Rammer and H. Hsu, (Shem. whys Letters 8 (1971) 129. 131 J.L. Whitten and M. Hackmeyer, 5. Chem. Phys. 5L (19691 5584. [41 J.L Wtlitten, J. Chem. Phys. 56 (1972) 5458.. [51 P.L. Yeager and V. McKay, I. Cbem. Phys. 60 (1974) 2714. [61 A.J. Maria, D. Larson, ME. McCkviiIe and S.P. hfcC-l.ynn, Accoilnts Chem. Res. 3 (1970) 368. 171 H. Basch, hI.B. Robin and N.A. KuebIer, I. Ckem. Phys. 49 (1968) 5007. [Sl G. Herzberg, Electronic spectra of polyatomic molecules wan Nostrand, Princeton, lS66) p. 612.
Volume 29, number 2
I.5 November 1974
.THE ‘A I IS-+& STATE OF FORMALDEHYDE Stephen R. LAhGHOFF’,
Stephen T. ELBERTf, Charles F. JACKELS and Ernest R. DAVIDSON** Deparhleni of Chembtry, Universip of Washington, Secttle, Weshingon 98195, USA
Retied
The ‘Al POT’ state of formaldehyde
Received 20 May 1974 manuscript received 8 July 1974
is predicted
to be 11.2 eV above the ground state and not diffuse.
There have been several papers published recently which differ significantly on the energy and nature of the 1A, T-W.* state of formaldehyde. Buenker and Peyerimhoff [l] using a basis set without diffuse orbitals obtained an.excitation ener,g of 11.72 eV. In this calculation the T-W_*state appeared as the second root of the secular equation and fhe 3A, SCF orbitals were used io construct the configurations. In a later paper Peyerimhoff et al. [2] obtained an excitation energy of 11.41 eV. In this calculation diffuse basis functions were included to represent some Rydberg states. The W* state appeared as the third root of the secular equation based on 3AI SCF orbit&. Both of these calculations miss the ground state SCF energy by about 2 eV and account for less than 10 % of the correlation energy. Whitten and Hackmeyer [3] obtained 11.3 1 eV for the excitation energy based on the third root of a secular equation. They used diffuse basis functions and ground state SCF orbitals. This calculation also missed the ground state SCF energy by i eV and obtained about. 15 %.of &e correlation energy, In a later cakulation Whitten [4] obtained 9.9 eV as the second root of a secular equation. This calcul&ion used a “better” basis set that missed the ground state SCF energy by only 1 eV and enough configurations to obtain.about * Present address: Battelle Memorial Institute, Columbus, Ohio 43201, USA. $ Present address: Lzhrstuhl fiir ceoretische Chemie der UniversitZt Bonn, 53 Bonn, West Germany. ** To whom correspondence should be addressed.
2.5% of the correlation energy. Effuse basis functions, however, were used only in the bl(%) subspace. He feit this would a&w the proper size adjustment of the 7r and n* orbit&. Because diffuse b, orbit& were not included, the lower energy b, + bj Rydberg states were not present in the secular equation. He found the 71and + orbit& were not particaiarly diffuse. Recently Yeager and McKay [51 have treated this excitation with the equations of motion method. They obtained 10.1 eV from the four’& ‘A, energy level. But they also claim the state is very diffuse with an average value of Z(X,? +yf) about 15a$ larger than the ground state and with a small f due. No absolute energies are given by which the completeness of their results can be judged. Although the energy reported in this work agrees with Whitten, the large change in Z(X~ +$) does not. Whitten indic;ttes a change of only about 1.4 for Llxf. The experimental results for this state of formaldehyde are also confused. All of the smal1 amides and aldehydes were at one time belieqed to Slave a T a IT* 1 A state near 7-8 eV [6] . It has now been established for formaldehyde and fonnamide that the lowest excited IA state is really an n -, 3p Rydberg state [3, 7, 81. For formaldehyde, there is no currently accepted experimental result for the pi -+ T* gtate. One of the uncertainties in t.he SCF CI approaches to.this state has been the inability to handle the SCF problem. b order to overcome this we have-done a multi-configuration SCF calcuIation using the three configtirations r2, iii?*, and fl?_ (Merely. optimizing 247
CHEMICAL PHYSICS LX’l-ERS
Volume 29, nGmber 2 ,_ :
.; ‘.
.: ‘.
,’
..
1
,.T&le
Ericrgies
State. :
Calctilation
‘gr&nd SkF W/O .’ $~&-id~’ SCF W
;
,15 November 1974’
’
:
:h.
ground
MC .kF W/o
R--X+ li+x*
MC!SCF W/Q.. MC SCF W
.’ ground
.CI W/G
EnergL
12.5%
-113.4688
(Il$)
-114.1349
-113.7066 CI w/o (root 2, ;m*j -113.7213 CI w/o (root 2, UZ, v2) cr’w: -113.5119 (root 3, GS) ~ CIW -113.7056 (root 2,7+) extrapohted extrapohted
ground
i-r--r*
a) Compahg b, Comptiing
1.5jb)
-114.1347
;I+?r*
X+7?*
:
~y113.4676
Cl W
??+n*.
.,
,-rl 13.8505 -113.8911 -113.9276.’
ground
n--a*
.Excitation energy
:
‘i1:6’) 11.2=) 10.3d) Xzi”)
^.
-114.2Gl -113.788
11.2
MC SCF remits. MC SCF results for n * n* with SdF for ground
.’ !) z&ring CI results. . . d, Compared to ground state SCF.
0.94(&
-* does.not work since .one obtains a’ ,. the.energy of nrr
bad ~~pro~atjon to the ground state. ‘The energy ., goes well below the true excited state energy.). The ” MC SCF procedure consists oftwo steps; an adjustment of the orbit& and a small con~~rat~~n,i~terac~ tion; In order to get the excited’state we used the set.ond root of this’CI. The basis sets used were one without (labeled W/O in, table 1) diffuse functions and one with both diffuse br and b, functions (labeled Win table 1). The ground state SCF energy with these basis sets are different from the best SCF’results of Neumann and Moskowitz by 0.3 and 0.2 eV, respectively.. I..- As table 1 shows, the excitation energy cotiparing the best three configuration iesults for the ground and ’ excited states give an excitation energy of 12.5 ,eV.. Inclusion of diffuse basis~functions has very lit& ef..fe& on the energy or wavefunctions at this level df ap‘pr&m&n. Tab,le 2 gives values for x2,y2, and z2 -:24g., .; :, ..,
,, _;
‘,
for c:om~arison.’ .‘. The CI results for the ground ‘&tte.in table I are all based on single plus selected double excitations using ground state SCF orbitals and about 850 configurations. This gives about 40 lo of the correlation energy. The :xtrapoIated limiting energy for the diffuse basis set &Auding al! double and higher excitations is -1 lr1.201. The CI results for x + n* are not as straightforward. The caIculation 1abeIcd GS was based on all single esicitations from the ground state using ground state orbitah. This excitation energy, 10.1 eV, is’probably most comparable to the McKay results. The wavefunction zt this level contained an appreciable mixture of diffuse configurations. The calculation labeled z12, y2 was based on the’natural orbital form of the MC SCF wavefunction. It is weIl known that any function of the form an2 f b(n+ + n*$ t CT~*~can be factored into the form c,rrz .-kC,,Y~.All single and double excitations from both &and y2 were considered for inclusion in the wavefunction. The u, Y form of the wavefunction can be returned to a “best?’ n, n* form (the original rr;7~* form is arbitrary within a unitary transformation). This gives 7~= u + v, TP = u - Y, and a wavefunction of the form’ -I-n*?r}/d
Since t3e coefficient of rrrr* in this form’was large, a Cl based on ,single and selected double excitation from. this nti” configuration wasalso done. As can be seen in table 1, the differences betieerrthese CI results is small. The extrapolated excitation energ? based on the calculation with diffuse orbitals is 1‘1.2 eV. These resuits very clearly agree with the work of Buenker and Peyerimhoff. No excitation energy as low asthat reported by Whitten was obtainable. Thi expectation values’given in table 2 demonstrate tha& the wavefunctions arc not very sensitive to.inclusiorr of diffuse orbitafs: The sum 6rl~21n>t (n*l.r~~In*> obtaine< with’ our diffuse basis agrees w&l with Whitten’s.. value 7.01. No hint that the’ rr.+,rr* state is diffuse appears in ,iable 2. AU of the results discussed here arc subject ,to one overwhelming criticism. The ionization:energy of formaldehyde -from the.highert b2 orbital is only 10.9 eV. Consequently if the n 7 rf” state is really, at 1‘1.2eV it ,, 1
..
:
.: -,.
- 0.24(7? + rP2).
.
-.
.
.;,
‘,.
CHEMICAL PHYSICS LETTERS
Volume 29, number 2
Values forx’,
1.5 November 1974
Table 2 y*, and z2 in 0’0units a)
State
Calculation
Lx*)
(y*,
(z2)
My* 177)
ground
SCF W/O
14.96
8.65
37.15
I.637
ground ground
SCF W CI w/o
15.00 15.08
37.18 37.24
1.646
ground
CI \v
15.12
8.68 8.64 8.72
rr’i7’
Cl w/o
15.12
8.48
37.97
1.432
ir’??
CI w
15.60
9.27
38.17
1.469
a) z is along CO bond,y
k*ly*lr;t;
37.28 2.317 3.233
is out of plane, the origin is at the center of mass.
is auto-ionizing. This means that the variational prindple, strictly speaking, does not apply. If 2 complete basis set were used in the b, subspace, there would be an infmity of 3, + bb 1A1 states at lower energies. Since the variational principle only guarantees that the kth root of the secular equation is in upper bound to the true Mh energy, it would appear to be nonsensical to do straight variational calculations for this state. Two answers can be given to this. First of all, by the perturbation selection procedure used here, one can regard this calculation as one in which one set of configurations has been treated to infinite order whiie other third- and higher-order effects are omitted. Secondly, the method used here is similar to those already well-established for handling auto-ionizing states. But it is true that if more diffuse basis functions were included this calculation would break down. Particularly if it happened that a Rydberg configuration which was weakly coupled to the 5r+ n* state accidently was nearly degenerate with it, the wavefunctions vvould mix in zeroth order and no state identifiable as primarily n--f rr* would appear in the results. Nevertheless,
sensible to say that near 11.2 eV there will be a band of states which contain most of the rr --Lrr8 contribution. it is
The authors wish to thank the Petroleum Research Foundation for supporting part of this work.
References
[l]
R.J. Buenker and S.D. Peyerimhoff, I. Chem. E’hys. 53 (1970) 1368. [21 S.D. Peyerimhoff, R.J. Buenker, W.K. Rammer and H. Hsu, (Shem. whys Letters 8 (1971) 129. 131 J.L. Whitten and M. Hackmeyer, 5. Chem. Phys. 5L (19691 5584. [41 J.L Wtlitten, J. Chem. Phys. 56 (1972) 5458.. [51 P.L. Yeager and V. McKay, I. Cbem. Phys. 60 (1974) 2714. [61 A.J. Maria, D. Larson, ME. McCkviiIe and S.P. hfcC-l.ynn, Accoilnts Chem. Res. 3 (1970) 368. 171 H. Basch, hI.B. Robin and N.A. KuebIer, I. Ckem. Phys. 49 (1968) 5007. [Sl G. Herzberg, Electronic spectra of polyatomic molecules wan Nostrand, Princeton, lS66) p. 612.