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The C3 radical: A theoretical study of the electronic spectrum of C3 via the equations of motion method
.Volume
33, number3
CHEhllCAL
RADKAL: A THEORETICAL STUDY OF THE ELECTRONIC VJA THE EQUATIONS OF lMOTPON MEWSOD
THE
15 June 1975
PHYSICS LETTERS
C,
SPECI-RUM
OF C,
Geoffrey R.J. ‘WILL.iAMS Res-asck School of Chemistry. Australtin
Nationd
Liniveuity.
Canberra. A.C T.. 2600. Ausrralfiz
Received 17 March 197.5
The excited states of the Cj radical have been studied using the quations of motion method. This study confums the experimental assiggunents of the two lowest ener,~ excitations observed for this radical and suggests an assignment for hvo other weak transttians that have been detected for this system. The expected ‘_‘: - 1 Zi transition is predicted to Be at an energy of 7.68 eV and to have an oscillator strength of 0.92.
2. Method One of the first spectra observed of a triatomic nonhydride radicai, consisting of atoms of the first period of the perisdic table, was that of the C3 radical. It was first observed in emission in the spectra of comets [l], and much later in the laboratory in discharges through methane and other hydrocarbons [2]. The spectrum was first identified as that of C3 by Douglas [3]. Up to that time, the free C3 radical had noi even been postulated in chemical reactions or equilibria, but since then it has been shown to be one of the most important constituents of carbon vapour and to be present in a number of photochemical reactions. The C, radical is also of considerable astrophysical importance because of its presence in the spectra of comets. tie spectrum of C, has also been investigated at low temperatures in inert gas matrices by Barger and ,&oida [4] and Weltner and his collaborators [5,6]. Apart from Some esr!y work by Pitzer and Clementi ‘[7] ud.Nibler and Linnett [8] there has not been any rigorous ab initio study of the exciied states of C3, BecaWof the impor&nce tif the C, radical ir! phptochemistry and astrophysics, we felt it worthwhile to investigate theoretically the electronic sp_actrum of @3 in some detail.
The equations of motion (EOM) memod [9-I 6] has been developed to obtain directly the excitation energies, transition moments and other response properties of closed shell electronic systems. A comparative study of the EOM method and the more traditional configuration interaction (CI) approach, for the calculation of excitaticn properties of molecules, is presented in the paper by Williams and Poppinger [9]. in the EOLMmethod, we define an excitation operator Oi, such that OIlO> = IX) ) where IX>is some excited state and IO>is the ground state. It can then be shown that 0: satisfies an equation of motion [‘17] given by
where the double commutation
is defined by
2[/I,B,C]
f&Cl1 3
= [[.4,Bl,CI
f [A,
wx is the excitation frequency and SO: represents a variation on the amplitudes specifyjng 0:. If 0: is assumed .to be composed of single particle-hole pairs (Ip-lhj, the equatibns of motion. become
CHEMICAL
Volume 33, number 3
PHYSICS
LETTERS
15 June 197.5
Table 1
where the matrices A, B and D are ground state expectatioil values of second-quantized operators, and Y(A) and Z(X) are the amplitudes defining the operation 0:. In this work we have taken the ground state 10, to be a correlated ground state, i.e., we have included in the ground state most of the important pair correlations [12]. The necessary correlation coefficients were evaluated using first-order Ray!ei&--Schrijdinger perturbation theory. It also can be shown [Z 11, to a very good approximation, that the matrix D can be taken to be effectively diagonal. The equations of motion then reduce to the form
I_;:*
_J(
:;;;‘j
=-&;:)
y
State
actions on the excitation energies have been included using 3 perturbational approach [9,11]. Further details of the EOM method are available
in
refs. [9-1771. 2.1. Basis set The basis set employed in this work was the extended 4-3 IG basis developed by Pople and co-workers [IS 1. This basis gives rise to a total of 27 molecular orbitals for CQ. In this work we have considered all particle-hole pairs arising from the excitation of electrons from the four highest-energy hole states to the thirteen lowest energy particle states. This procedure produces a total of 52 particle-hole pairs. All calcu!ations were performed at the experimental geometry [19] of the ground state (IZ:,‘) of C,.
3. Results and discussion in in
hE (CV)
f”)
1.73
sf sdf sf
2.66 2.95
t2)
where A’, B’, Y’(h) and Z’(X) are re-normalized versions of those matrices appearing in eq. (1). The effects of double particle-hole (2p-2h) inter-
The results of these calculations are presented table 1. The !owest predicted singlet-singlet transition
Vertical excitation energies (ti) and osciktor strengths (fl for the excited states ot’the C3 rzdicaf CIScornput& from the qualions of motion
a)
3.61
sdf
3.63
0.129
4.07
Sdf
4.08
df
4.18
df
5.02
df
7.68
0.918
a.58
sdf
9.26 9.52 9.93 9.98
df
.
sf sdf df
df = dipole forbidden; sf= spin forbidden; sdf = spin and dipole forbidden.
this work is ‘II, ++‘3: at an energy of 3.63 eV. This may be compared with the experimental transition energy of3.06 eV for the lIIU + 1Z.Z; transition, obtained by Herzberg et al. [20]. So far, this is the only band that has been observed for C3 in the gas phase. However, in the spectrum of Cj obtained from low temperature matrix isolation studies [5,6] there is, in addition to the 4050 A (3.06 eV) band system, a strong phosphorescence observed a; about 5900 A (2.12 eV). This band has been tentatively assigned by Weltner and McLeod as due to the spin forbidden %I,, tj. 1C.p’transition. The EOM calculations predict the 311u ft ‘Zi transition to occur at 1.73 eV, which supports the assignment of the observed phosphorescence at 5900 t9 to this transition. The 31T, 6 ‘2: transition is probably made allowed by borrowing its intensity from the ‘IT, ++ l,$ transition if spin-orbit coupling is the mechanism of bceakdorvn of the multiplicity selection rule. In addition to the I&, ++ ‘2: a search has been car&d and 311U ++ 1.Z’ transitions, out by Weltner et al; [6] for the expected ‘E;f tf ‘Ci transition. One would expect this to be 2 strong adsorption system since it is a V-N transition. However, 533
Voluine
33,
numlxr 3
CHEMICAL
up to an energy of 6.4 eV only two _small bands at 2812 A (4.41 eV) and 2320 A (5.34 eV) have so far been observed. The EOM results predict the very strong (f= 0 ’92‘1 to occur at , IX+ + lC+ transition 7.68 eV, which is coisistenfwith the experimental obseivation of the absence of a strong transition at energies below 6.4 eV. However, the EOLMstudy does predict
a ID
x+ Lx;
transition
at 4.18
eV and a 1110
U D transition at 5.02 eV. Both of these transitions are dfpole forbidden but could acquire intensity via vibronic interactlcn. On the basis of these results one might then tentztively assign the weak band observed _ I at 4.41 eV to the (AU + ‘Zp transition, and the weak tand at 5.34 eV io the Ills ++12: transition. These assignments are consistent wiviththe weak nature and positions of the observed transitions. In order to observe the expe’cted 1X; * lIZi transition the experimental spectrum of C, wili have to be extended up to energies of about 8 eV. One other property of C3 that is of some importance is the ionization potential. On the basis of Koopmans’ theorem, one predicts from these calculations the ionization potential of C3 to be 12.7 eV. Thus the EOM study provides a set of results which form a firm basis for the interpretation of the electronic spectrum of the C3 radical. The results confirm the experimental assignments of the I&, ++ ‘2: and 311u ++‘IZz transitions, and suggest possible assignments of rhe weal: bands observed at 2812 8, and 2320 A. On extension of the experimental spectrum io energies of about 8 eV, one should be able to observe the strong 1X; * 1x; transition at 7.68 eV. e ‘Xi
PHYSICS LEI-I’ERS
15 June 1975
References [I] P. Swings, Publ. Astron. Sot. Pacific 54 (1942) 123; Monthly NoticesRoy. Astron. Sot. 103 (1943) 86. [2] C. Herzberg, Asttophys. J. 96 (1942) 314. [3] A.E. Dou&s, Astrophys. 5. 114 (1951) 466; K. Clusius and A.E. Douglas, Can. J. Phys. 32 (1954) 319. [4] R.L. Barger and H.P. Broida, J. Chem. Phys. 37 (1962) 1152. 1.51 W. Weltner and D. hicieod, J. Chem. Phys. 40 (1964) 1305. [6] W. Weltner and D. McLeod, J. Chem. Phys. 45 (1966) 3096. . [i] K.S. Pitzer and E. Clementi, J. Am. Chem. Sot. 61 (1953) 4477. [8] J.\V. Nibler and J.W. Linnett, Trans. Faraday Sot. 64 (1966) 1153. [9] G.R. Williams and D. Poppinger, Mol. Phys. (1975), to be published. [lo] G.R. Wilhms, Chem. Phys. Letters 30 (1975) 495. [11] T. Shiiuya, J. Rose and V. McKay.. , J. Chem. Phvs. _ 58 (1973) sbo. [121 T. Shibuya and V. McKay, Phys. Rev. A2 (1970) i208. 1131 J. Rose, T. Shibuya and V. McKay, J. Chem. Whys.58
(1973) 74. and V. hidCoy, Chem. Whys. 1141 P.H.S. Martin, D.L. Yqer Letters 25 (1974) 182. [I51 D.L. Yeager and V. McKay, J. Chem. Fhys. 60 (1974) 2714. f161 J.B. Rose, T. Shibuya and V. McKay, J. Chem. Phys. 60 (1974) 2700. 1171 D.J. Rowe, Rev. Mod. Phys. 40 (1968) 153. 11g1 R. Ditchfield, W.J. Here and J.A. Pople, J. Chem. Phys. 54 (1971) 724. f 191 G. Herzberg, Electronic spectra and electronic structure of potyatomic molecules (Van Nostrand, Princeton, 1966). [20! L. Gaussel, G. Henberg, A. Lagerquist and B. Rosen, Discussions Faraday,Soc. 35 (1963) 113.
33, number3
CHEhllCAL
RADKAL: A THEORETICAL STUDY OF THE ELECTRONIC VJA THE EQUATIONS OF lMOTPON MEWSOD
THE
15 June 1975
PHYSICS LETTERS
C,
SPECI-RUM
OF C,
Geoffrey R.J. ‘WILL.iAMS Res-asck School of Chemistry. Australtin
Nationd
Liniveuity.
Canberra. A.C T.. 2600. Ausrralfiz
Received 17 March 197.5
The excited states of the Cj radical have been studied using the quations of motion method. This study confums the experimental assiggunents of the two lowest ener,~ excitations observed for this radical and suggests an assignment for hvo other weak transttians that have been detected for this system. The expected ‘_‘: - 1 Zi transition is predicted to Be at an energy of 7.68 eV and to have an oscillator strength of 0.92.
2. Method One of the first spectra observed of a triatomic nonhydride radicai, consisting of atoms of the first period of the perisdic table, was that of the C3 radical. It was first observed in emission in the spectra of comets [l], and much later in the laboratory in discharges through methane and other hydrocarbons [2]. The spectrum was first identified as that of C3 by Douglas [3]. Up to that time, the free C3 radical had noi even been postulated in chemical reactions or equilibria, but since then it has been shown to be one of the most important constituents of carbon vapour and to be present in a number of photochemical reactions. The C, radical is also of considerable astrophysical importance because of its presence in the spectra of comets. tie spectrum of C, has also been investigated at low temperatures in inert gas matrices by Barger and ,&oida [4] and Weltner and his collaborators [5,6]. Apart from Some esr!y work by Pitzer and Clementi ‘[7] ud.Nibler and Linnett [8] there has not been any rigorous ab initio study of the exciied states of C3, BecaWof the impor&nce tif the C, radical ir! phptochemistry and astrophysics, we felt it worthwhile to investigate theoretically the electronic sp_actrum of @3 in some detail.
The equations of motion (EOM) memod [9-I 6] has been developed to obtain directly the excitation energies, transition moments and other response properties of closed shell electronic systems. A comparative study of the EOM method and the more traditional configuration interaction (CI) approach, for the calculation of excitaticn properties of molecules, is presented in the paper by Williams and Poppinger [9]. in the EOLMmethod, we define an excitation operator Oi, such that OIlO> = IX) ) where IX>is some excited state and IO>is the ground state. It can then be shown that 0: satisfies an equation of motion [‘17] given by
where the double commutation
is defined by
2[/I,B,C]
f&Cl1 3
= [[.4,Bl,CI
f [A,
wx is the excitation frequency and SO: represents a variation on the amplitudes specifyjng 0:. If 0: is assumed .to be composed of single particle-hole pairs (Ip-lhj, the equatibns of motion. become
CHEMICAL
Volume 33, number 3
PHYSICS
LETTERS
15 June 197.5
Table 1
where the matrices A, B and D are ground state expectatioil values of second-quantized operators, and Y(A) and Z(X) are the amplitudes defining the operation 0:. In this work we have taken the ground state 10, to be a correlated ground state, i.e., we have included in the ground state most of the important pair correlations [12]. The necessary correlation coefficients were evaluated using first-order Ray!ei&--Schrijdinger perturbation theory. It also can be shown [Z 11, to a very good approximation, that the matrix D can be taken to be effectively diagonal. The equations of motion then reduce to the form
I_;:*
_J(
:;;;‘j
=-&;:)
y
State
actions on the excitation energies have been included using 3 perturbational approach [9,11]. Further details of the EOM method are available
in
refs. [9-1771. 2.1. Basis set The basis set employed in this work was the extended 4-3 IG basis developed by Pople and co-workers [IS 1. This basis gives rise to a total of 27 molecular orbitals for CQ. In this work we have considered all particle-hole pairs arising from the excitation of electrons from the four highest-energy hole states to the thirteen lowest energy particle states. This procedure produces a total of 52 particle-hole pairs. All calcu!ations were performed at the experimental geometry [19] of the ground state (IZ:,‘) of C,.
3. Results and discussion in in
hE (CV)
f”)
1.73
sf sdf sf
2.66 2.95
t2)
where A’, B’, Y’(h) and Z’(X) are re-normalized versions of those matrices appearing in eq. (1). The effects of double particle-hole (2p-2h) inter-
The results of these calculations are presented table 1. The !owest predicted singlet-singlet transition
Vertical excitation energies (ti) and osciktor strengths (fl for the excited states ot’the C3 rzdicaf CIScornput& from the qualions of motion
a)
3.61
sdf
3.63
0.129
4.07
Sdf
4.08
df
4.18
df
5.02
df
7.68
0.918
a.58
sdf
9.26 9.52 9.93 9.98
df
.
sf sdf df
df = dipole forbidden; sf= spin forbidden; sdf = spin and dipole forbidden.
this work is ‘II, ++‘3: at an energy of 3.63 eV. This may be compared with the experimental transition energy of3.06 eV for the lIIU + 1Z.Z; transition, obtained by Herzberg et al. [20]. So far, this is the only band that has been observed for C3 in the gas phase. However, in the spectrum of Cj obtained from low temperature matrix isolation studies [5,6] there is, in addition to the 4050 A (3.06 eV) band system, a strong phosphorescence observed a; about 5900 A (2.12 eV). This band has been tentatively assigned by Weltner and McLeod as due to the spin forbidden %I,, tj. 1C.p’transition. The EOM calculations predict the 311u ft ‘Zi transition to occur at 1.73 eV, which supports the assignment of the observed phosphorescence at 5900 t9 to this transition. The 31T, 6 ‘2: transition is probably made allowed by borrowing its intensity from the ‘IT, ++ l,$ transition if spin-orbit coupling is the mechanism of bceakdorvn of the multiplicity selection rule. In addition to the I&, ++ ‘2: a search has been car&d and 311U ++ 1.Z’ transitions, out by Weltner et al; [6] for the expected ‘E;f tf ‘Ci transition. One would expect this to be 2 strong adsorption system since it is a V-N transition. However, 533
Voluine
33,
numlxr 3
CHEMICAL
up to an energy of 6.4 eV only two _small bands at 2812 A (4.41 eV) and 2320 A (5.34 eV) have so far been observed. The EOM results predict the very strong (f= 0 ’92‘1 to occur at , IX+ + lC+ transition 7.68 eV, which is coisistenfwith the experimental obseivation of the absence of a strong transition at energies below 6.4 eV. However, the EOLMstudy does predict
a ID
x+ Lx;
transition
at 4.18
eV and a 1110
U D transition at 5.02 eV. Both of these transitions are dfpole forbidden but could acquire intensity via vibronic interactlcn. On the basis of these results one might then tentztively assign the weak band observed _ I at 4.41 eV to the (AU + ‘Zp transition, and the weak tand at 5.34 eV io the Ills ++12: transition. These assignments are consistent wiviththe weak nature and positions of the observed transitions. In order to observe the expe’cted 1X; * lIZi transition the experimental spectrum of C, wili have to be extended up to energies of about 8 eV. One other property of C3 that is of some importance is the ionization potential. On the basis of Koopmans’ theorem, one predicts from these calculations the ionization potential of C3 to be 12.7 eV. Thus the EOM study provides a set of results which form a firm basis for the interpretation of the electronic spectrum of the C3 radical. The results confirm the experimental assignments of the I&, ++ ‘2: and 311u ++‘IZz transitions, and suggest possible assignments of rhe weal: bands observed at 2812 8, and 2320 A. On extension of the experimental spectrum io energies of about 8 eV, one should be able to observe the strong 1X; * 1x; transition at 7.68 eV. e ‘Xi
PHYSICS LEI-I’ERS
15 June 1975
References [I] P. Swings, Publ. Astron. Sot. Pacific 54 (1942) 123; Monthly NoticesRoy. Astron. Sot. 103 (1943) 86. [2] C. Herzberg, Asttophys. J. 96 (1942) 314. [3] A.E. Dou&s, Astrophys. 5. 114 (1951) 466; K. Clusius and A.E. Douglas, Can. J. Phys. 32 (1954) 319. [4] R.L. Barger and H.P. Broida, J. Chem. Phys. 37 (1962) 1152. 1.51 W. Weltner and D. hicieod, J. Chem. Phys. 40 (1964) 1305. [6] W. Weltner and D. McLeod, J. Chem. Phys. 45 (1966) 3096. . [i] K.S. Pitzer and E. Clementi, J. Am. Chem. Sot. 61 (1953) 4477. [8] J.\V. Nibler and J.W. Linnett, Trans. Faraday Sot. 64 (1966) 1153. [9] G.R. Williams and D. Poppinger, Mol. Phys. (1975), to be published. [lo] G.R. Wilhms, Chem. Phys. Letters 30 (1975) 495. [11] T. Shiiuya, J. Rose and V. McKay.. , J. Chem. Phvs. _ 58 (1973) sbo. [121 T. Shibuya and V. McKay, Phys. Rev. A2 (1970) i208. 1131 J. Rose, T. Shibuya and V. McKay, J. Chem. Whys.58
(1973) 74. and V. hidCoy, Chem. Whys. 1141 P.H.S. Martin, D.L. Yqer Letters 25 (1974) 182. [I51 D.L. Yeager and V. McKay, J. Chem. Fhys. 60 (1974) 2714. f161 J.B. Rose, T. Shibuya and V. McKay, J. Chem. Phys. 60 (1974) 2700. 1171 D.J. Rowe, Rev. Mod. Phys. 40 (1968) 153. 11g1 R. Ditchfield, W.J. Here and J.A. Pople, J. Chem. Phys. 54 (1971) 724. f 191 G. Herzberg, Electronic spectra and electronic structure of potyatomic molecules (Van Nostrand, Princeton, 1966). [20! L. Gaussel, G. Henberg, A. Lagerquist and B. Rosen, Discussions Faraday,Soc. 35 (1963) 113.