Metastable optical excitations of linear C−5

Volume 180. number 5

CHEMICAL PHYSICS LETTERS

31 May 1991

Metastable optical excitations of linear Cc Ludwik Adamowicz Department ofChemistry, University ofArizona, Tucson, AZ 85721, USA

Received 8 January 1991; in final form 3 April 1991

It is theoretically predicted that the Cc electronic vertical excitations are metastable with respect to the electron ejection. The energies of the lowest %;, “EL, *I;, ‘IX;, 24 and ‘Ae states evaluated using first-order correlation orbitals and the coupled cluster method are reported.

1.

Introduction

This work is a continuation of our theoretical studies on the carbon cluster anions and their properties [ 1,2] including optical excitation spectra. The formation of stable anionic excited states of Cq lying below the electron ejection threshold and below the photofragmentation level has been a matter of our previous theoretical investigation [ 3 1. The electronic structure of metastable excited states of CT, which will decay with the formation of the neutral cluster and a free electron, is the subject of the present study. Many reports have recently appeared concerning the formation and spectroscopic properties of neutral carbon clusters [ 4-101. It has been well documented that carbon clusters of different sizes and shapes exhibit considerable electron affinities. Smalley and co-workers’ [4] results, obtained by means of ultraviolet photoelectron spectroscopy (UPS), indicated that even-numbered chains having open-shell electronic structures possess significantly higher electron affinities in comparison to the odd-numbered chains, which are closed-shell systems. This alternating trend was also observed in our recent theoretical studies [ 21. To our knowledge, there has been no experimental attempt to study photoexcitation of carbon cluster anions, with an exception of the investigations on the vertical electron affinities [ 41, which correspond to the most extreme form of the photoexcitation the excitation to the continuum. Our recently com466

pleted theoretical study on the electronic excitations of the linear CT anion [ 31 indicated that there are four electron excitations with energy below the electron ejection threshold leading to stable anionic excited states. Following this finding, a question was raised whether Cc will also have similar spectral properties. A significantly lower ionization potential for C; than for CT seems to suggest that there will be fewer (if any) bound excited states for CT than for C; _ It is also possible that the excitations of CI will lead to the formation of metastable states with energies above the electron ejection threshold. In the present study we elucidate this problem. The methodology used in the present work was recently developed in our group [ 1 ] and later used in studies on small carbon chains [ 2,3]. It combines the first-order correlation orbital (FOCO) method for the spin-unrestricted Hartree-Fock (UHF) reference wavefunction with the coupled cluster (CC) method. In the first part of this article we briefly discuss the UHF/FOCO/CC methodology and its application to CT calculations. In the second part the results are discussed and commented upon.

2. Methodology 2.1. FOCO method The FOCO method offers a way for contracting the virtual orbital space to a significantly smaller space of correlation orbitals, which are used in post-

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SCF higher-order correlation calculations. Due to the reduction of the number of orbitals the higher-order calculations can be carried out for larger systems and with better basis sets. The calculations presented in this report exemplify the application of the FOCO procedure in conjunction with the coupled cluster method in studies on larger molecular systems. For the variational methods, the orbital optimization scheme results directly from the variational principle leading to the MCSCF method. For the nonvariational approaches, such as many body perturbation theory (MBPT) or the coupled cluster method, there is no obvious way for generating an optimal orbital manifold. The most direct approach, which we have pursued for some time, is to utilize the Hylleraas functionals, which constitute upper bounds to the even-order MBPT correlation energies. Since the CC method can be considered an infinite summation of certain classes of MBPT diagrams, the correlation orbitals obtained through the minimization of the Hylleraas functionals should provide adequate basis sets for CC calculations. Our recent results for both neutral and anionic, open- and closed-shell molecular systems [ 2,3,11] provide evidence to document this point. The FOCO procedure employed in this work is the same one used in our previous study on C4 and CT [ 1,3] _The reader is referred to these papers for a more complete overview of the method and its application to treat electronic excited states. 2.2. CC calculations In the present work we applied the single-reference coupled cluster method, which offers a non-variational procedure to find size-extensive solutions of the Schriidinger equation [ 121. The CC scheme can be viewed as a method of determining the CC operator usually presented in an exponential form, exp T, which when operating on the reference function - for the single-reference CC method this is the SCF wavefunction - produces a more exact approximation to the exact wavefunction for the state under consideration. In essence, both ground and excited states can be determined this way provided that good reference wavefunctions are known. Obviously, for many excited states and also for some ground states, it is not possible to find a single de-

terminantal reference wavefunction - a typical example is an open shell singlet state - and then one needs to use a CC method which allows for a multiconfiguration form of the reference function. In spite of much work done in this area, including our recently proposed method [ 131, a computational implementation on a practical scale has not yet been accomplished. In the present study we have used a single-reference approach, which works well for highspin states, but for lower-spin states usually leads to solutions which are combinations of states with different multiplicities. There is an approximate procedure for resolving impure states into their components [ 141, which has been applied in the present study. The results of this procedure should be, however, viewed with caution for the following reason: The spin is the local property and for a spin-pure state the Sz operator should give the same eigenvalue when operating on the wavefunction at all points in space. The average value of the spin operator is, therefore, only a very approximate measure of the spin impurity of the wavefunction. The version of the single-reference CC method used in this work was the CCSD+T( CCSD ) scheme 1151#I. The CCSD+ T( CCSD ) model allows the triple excitation to affect the single excitation amplitudes, p,, and double excitation amplitudes, ri;, only in the last iteration of the CCSD calculation. This allows us to account for the triple excitation contribution for larger molecular systems. The CCSD+T(CCSD) calculation was performed independently for each state considered in this work using a set of FOCOs generated for that state. The final value of the total energy was determined using the following formula: -%T =&,+~(Z)+&.o.

>

(1)

where the E(*) energy was calculated with all virtual orbitals, and the higher-order contribution, Eh.o.rwas obtained from the FOCO CCSD t T (CCSD) result using the following scaling procedure: li’ The CC calculations presented in this paper were accomplished with the PROPAGATOR program system consisting of the MOLECULE Gaussian integral program of Almkif and program GRNFNC, which does SCF/MPBT/CC calculations written by R.J. Bartlett, G.D. Purvis, Y.S. Lee, S.J. Cole and R. Harrison.

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CHEMICAL PHYSICS LETTERS

=r/(E$$::+T(CCsD)- E#c,)

~=E’*‘fE&,

,

31 May 1991

3. Numericaldetails and results

.

(2)

3.1. Molecular geometries and basis set

The scaling factor, g, was taken as the ratio of the second-order energy calculated with all virtual orbitals, EC’), and the second-order energy determined with FOCOs, &?)oco. Our experience indicates that the above procedure produces an improved estimation of the correlation contribution from higher orders [ 161, although for C, the scaled and unscaled results lead to virtually identical excitation energies.

The bond distances of the linear C,’ anion were optimized at the MBPT ( 2) level using the finite difference approach with the 6s/4p/ Id basis employed in our previous studies on electron affinities of small carbon chains [ 21. Our computer program was used for this purpose. The same procedure and the basis set was also used to optimize the linear structure of C3 and CT used in the calculations of the photo-

Table 1 Molecular geometries used in the present calculations

G 0-l”)

R, .- Y? . -.-.-.

c, ( ‘2: ) c, (‘l-I,)

R .-.-.

R

R .-.-,

R

RI

R, =l .3225 8, R2

R,=l .2958 A R =1.3162A R =1.3169A

Table 2 UHF results for Cc ground and excited states State

cy

E UHF

GIOMO

08

Lll

=P

6,

(J”

%g

xys

6,

6 6

2 1

1 1

0’5 0

1 1

1 1

0 0

- 189.147929

x, (ar), -0.16525

5

” 4X_

6 6

2 1

2 1

0 0

5 4

1 1

1 1

0 0

-189.065715

o1 (j?), -0.03209

% 4X_

6 5

2 1

2 1

0 0

5 5

1 1

1

0

-189.062153

B, (8). -0.02772

I

0

6 6

2 1

1

0

1 1

1 1

0 0

or (/3), -0.07887

0

5 4

- 188.989408

2

6 5

2 1

1 2

0 0

5 5

1 1

I

0

- 188.983930

IS”(/I), -0.07238

1

0

6 6

2 2

1 1

0 0

5 4

1 1

1

0 b,

- 188.948164

o* (8), -0.07970

1

0”

6 5

2 2

1

0

1 1

1 1

- 188.942869

6, (B), -0.07337

0

5 5

;

1

6 6

1 1

1

0

0 0

xxB,nB (a,b), -0.40723

1

1 1

- 189.035002

0

5 5

1

I

xm,

2+4z" a)

2+4% p) 2A,+22- Y 2$+2X;

CS

Configuration

% x ‘z+

aJ Mixture of the doublet and quartet states. b, In the calculation, in which real x orbitals are used, this configuration leads to a state which is a combination ofthe 2A, and 2Z; states: j(2A,+2C;). c, In the calculation, in which real n orbitals are used, this configuration leads to a state, which is a combination of the ‘4 and *C; states: f(‘$+‘%).

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of C;, which were found spatially bound, are two quartet states, 4C; and 4C;, two doublet states, 2A,, and *$, and two impure spin states, which in the first approximation can be considered as linear combinations of the doublet and quartet IE; and IX; states, respectively.

fragmentation energies. The optimal geometries of Cr, C3 and C,- are presented in table 1. The calculation on the ground state of the C5 neutral cluster was done using the C, geometry to enable the evaluation of the vertical ionization potential of CT. The UHF/FOCO/CCSD t T( CCSD) calculations, which led to determination of the excitation energies of C; , were accomplished using the 5s/3p/ 2d basis set. The same basis was previously used in the study of photoexcitations of CT [ 3 1, and was shown to reproduce almost exactly the experimental value of the electron affinity for this system.

3.3. FOCO generation In table 3 we present the results of the FOCO generation for the ground and excited states of Cr and the ground state of C5. The number of FOCOs was selected to be less than one-half of the size of the UHF virtual orbital space. The FOCO symmetry distribution, shown in table 3 for each state, was selected based on our previous analysis of the performance of the FOCO procedure applied to several carbon chain molecules and their anions [ 21, and based on some trial FOCO optimization for the ground state involving different symmetry distributions. Upon examining the symmetry distributions for different states in table 3, one notices that the sum of occupied orbitals and correlation orbitals in a particular sym-

3.2. UHF calculations In table 2 we present the configurations of the C, states for which a converged UHF solution was accomplished with negative energies for all occupied orbitals (the last entry in the table shows the energy, symmetry and spin of the highest occupied spin orbital). Several other configurations were considered but were rejected due to the highest occupied orbitals having positive orbital energies. The excited states Table 3 Generation

c,

of the first-order

correlation

E’2’

&go

%

2 2

-0.653432

-0.612435

93.7

5

2

-0.608856

-0.566205

93.0

5

2 2 2

-0.606105

-0.563608

93.0

5 5

2 2

-0.701753

-0.657973

93.8

5

5

2

-0.700706

-0.656787

93.7

5

5

2

IO II

5

5

2

-0.751270

-0.705993

94.0

5

5

2

3 3

10 10

5 5

5 5

2 2

-0.749927

-0.704509

94.0

3 3

10 10

5 5

5 5

2 2

-0.694415

-0.654960

94.3

FOCO distribution

XTI”

12 12

4

5

3

5

5

3

10 IO

5 5

12

4

4

5

5

3 3

10 11

5

12

5

12 13

4 5

4 5

3 3

10 10

5 5

5 5

12 12

4 5

5 4

3 3

10 11

5 5

12

4

10

5

5 4

3

13

3

10

12

4 4

5

3

12

5

3

12 13

4 4

5 5

12 12

5 5

5 5

42;

2+4z

2+4G

a)

a)

*A,+‘C- ” b’

2Ag+2X,

C5

orbitals for the ground and excited states of the Cc anion

State

42m ”

c’

x ‘c+8

31 May 1991

5 5

a-c’ See footnotes to table 2.

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Volume 180, number 5

Table 4 Vertical electron excitation energies for the C? anion

c,

CS

State

EToT(au )

u

XTI, 4X; % %I; KX; ‘AU=’ 2A9*’

- 189.880152 - 189.788278 - 189.784245 -189.771858 -189.767114 - 189.745026 - 189.740666

0 2.50 2.61 2.95 3.08 3.68 3.80

x ‘z+8

- 189.785770

2.57 a’

(eV)

31 May 1991

3.4. Vertical electron excitations of Cy Exp.

2.80 b’

a) Vertical ionization potential. b, Experimental vertical electron affinity, see ref. [4]. ‘) The energy of this state was determined as: 2E(*A, + ‘Z;) E(Q;) [ 141. d, The energy of this state was determined as: 2E(lA, + 2C;) J?(Q) [14].

metry representation is the same for all considered states of both C5 and CF. The optimal values of the second-order Hylleraas functional, denoted as E,L&, in table 3, can be compared with the second-order energies calculated with all virtual orbitals - entry _E@’in the table. One notices that the contraction of the virtual space by nearly 60% leads to a rather small decrease of the secondorder correlation energy for all states considered.

The total energies of the ground and excited states of CT and the ground state of C5 calculated according to eq. (2) are presented in table 4. In the case of *C; and ‘Xi states of CT, the energies of the pure spin states were determined using the approximate procedure of Magers et al. [ 141 mentioned above. The total energies in the first column produced the values of the electron excitations of the second column. The energy of the vertical excitation of the CT excessive electron to the continuum of 2.57 eV, which is equivalent to the vertical ionization potential, is compared to the experimental CSvertical electron affinity of 2.80 eV determined by Smalley and co-workers [ 41. The most striking observation based on the results in table 4 is that all electron excitations of C, are very near or above the electron ejection threshold. This fact is discussed in section 3.5. In table 5 some details of the CCSD+T( CCSD) calculations, are shown. Of particular interest to the reader might be the variation of the contribution due to triple excitations for different states, as well as the size of the largest T, and T2 CC amplitudes, which do not indicate a significant multireference character for neither of the states.

Table 5 Details of the coupled cluster calculations of C, and CT. Energies in au State

E@!

ECORR T(CcsD)

Maximum CC amplitudes T,

CY

x 2l-l” 91; 2X, 1+4G ai) WE- PI 2A,;2E, ” 2A8+2Z, d’

-0.651876 -0.632443 -0.631661 -0.688522 -0.688728 -0.705514 -0.705312

-0.034407 -0.039503 -0.03980 1 -0.05 1675 -0.052268 -0.055930 -0.056590

G

x ‘CC 8

-0.661717

-0.046395

0.119 0.165 0.182 0.146 0.148 0.133 0.136 -0.046

CCSD spin b1 multiplicity

T, - 0.080 0.138 0.142 -0.151 -0.156 -0. I75 -0.181

2.078 4.177 4.178 3.019 3.026 2.060 2.065

- 0.089

1.000

a} Mixture of the doublet and quartet states. h, Spin multiplicity, 2S+ 1, of the coupled cluster wavefunction is calculated using the positive root of the following equation: S(S+I)=(~“Y,,,IS21(ltt,+~tf~~)~“V,,,). ‘) j(*A”+ ‘Z;). d’ j(*$+%;g).

CHEMICAL PHYSICS LETTERS

Volume 180, number 5

Table 6 FOCO/CC calculation of the total energies of the products of photofragmentation reactions of CF.

E (au) CT (2z:)tc3 (‘rg) C* (‘El) tc,

(‘I$)

-189.683184 - 189.640996

3.5. Photofragmentation One question which arises in studies of photoexcitations of anionic systems is whether the energy of the state under consideration exceeds the energy of the photofragmentation. To address this question we performed UHF/FOCO/CC calculations on the ground states of C3 (‘C: state) and CT (*I’&state), using the energies of C2 (X ‘C.$) and C,- (X ‘Z: ), calculated before [ 3 ] and we determined the dissociation energies C; . The results are summarized in table 6. The lowest dissociation energy corresponds to separation of Cc into the CT (X ‘C,) and C3 (X ‘C,) fragments. This energy is significantly higher than the energies of all the C; excited states calculated here.

4. Conclusions The electronic excitations of C; can become the subject of experimental measurements in the near future. Our theoretical calculations indicate that all the lowest Cc excited states are metastable with respect to an electron ejection, with a possible exception of the 42; state, which seems to be marginally bound. The situation is very different for the CT anion, where several stable bound excited states were theoretically predicted [ 3,171. The difference in the spectral properties of C; and C; is surely due to considerably different ionization potentials of these anions (higher for CL than for C; ). The prediction of metastable states for C; near the electron ejection threshold opens an interesting possibility of an ex-

31 May 1991

perimental investigation, which we hope to initiate with the present study. The present work represents an extensive computational effort. Calculations for each state required about 30 CPU hours on the CONVEX 2 10 mini-supercomputer. Work is currently in progress on larger carbon and silicon cluster anions.

References [ I ] L. Adamowicz, J. Chem. Phys. 93 (1990) 6685. [2] L. Adamowicz, J. Chem. Phys. 94 (1990) 1241. [ 31 L. Adamowicz, Chem. Phys., submitted for publication. [4] S. Yang, K.J. Taylor, M.J. Craycroft, J. Conceicao, C.L. Pettiette, 0. Cheshnovsky and R.E. Smalley. Chem. Phys. Letters 144 ( 1988) 43 I. [ 51E.A. Rohling, D.M. Cox andA. Kaldor, J. Chem. Phys. 81 (1984) 3322. [6] H.W. Kroto, J.R. Heath, SC. O’Brien, R.F. Curl and R.E. Smalley, Nature 318 (1985) 162. [7]S.C. O’Brien, J.R. Heath, R.F. Curl and R.E. Smalley, J. Chem. Phys. 88 (1988) 220. [ 81 D.M. Cox, KC. Reichmann and A. Kaldor, J. Chem. Phys. 88 (1988) 1588. [9] P.P. Radi,T.L. Bunn, P.R. Kemper,M.E. Mo1chanandM.T. Bowers, J. Chem. Phys. 88 ( 1988) 2909. [lo] M. Algranati, H. Feldman, D. Kella, E. Malkin, E. Miklazky, R. Naaman, Z. Vager and J. Zjafman, J. Chem. Phys. 90 (1989) 4617; H. Feldman, D. Kella, E. Malkin, E. Miklazky, Z. Vager, J. Zjafman and R. Naaman, I. Chem. Sot. Faraday Trans. 86 ( 1990) 2469. [ 1I] L. Adamowicz, J. Comput. Chem. IO (1989) 928; Chem. Phys. Letters 161 (1989) 73; A. LeSand L. Adamowicz, Chem. Phys. Letters 175 ( 1990) 187. [ 12I R.J. Bartlett, Ann. Rev. Phys. Chem. 32 ( 1981) 359. [ 131N. Oliphant and L. Adamowicz, J. Chem. Phys., in press. [ 141D.H. Magers, R.J. Harrison and R.J. Bartlett, J. Chem. Phys. 84 (1986) 3284; S. Binkley and C. Melius, unpublished. [ I5 ] Y.S. Lee, S.A. Kucharski and R.J. Bartlett, J. Chem. Phys. 81 (1984) 5906. [ 161L. Adamowicz, J. Phys. Chem. 93 (1989) 1780. [ 171J.D. Watts, I. Cemusak and R.J. Bartlett, Chem. Phys. Letters 178 (1991) 259.

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