Ab initio propagator analysis of triple-charge-transfer spectra for carbon disulphide

27June

1997

CHEMICAL PHYSICS LETTERS

ELSEVIER

Chemical Physics Letters 272 (1997) 148-154

Ab initio propagator analysis of triple-charge-transfer carbon disulphide

spectra for

D.E. Parry Department

of Chemistry, UniuersiQ of Wales Swansea, Singleton Park. Swansea SA2 8PP, UK Received 7 April 1997; in final form 25 April 1997

Abstract Ab initio propagator calculations for triple ionizations of the carbon disulphide molecule to quadruplet states of CSZ’ have been performed for transition energies up to 80 eV, using the second-order algebraic diagrammatic construction (ADC(2)) method. Energies predicted for main transitions, arbitrarily defined as those to CSz+ states having at least 20% weight in 3-hole configurations, are found to correlate well with those of a range of peaks exhibited in triple-charge-transfer spectra for the collisions of 6 keV Cl *+ ions with gas-phase CS,. The ADC(2) results also enable an appraisal of the predictions of a semi-empirical method previously used to analyse the spectra. 0 1997 Elsevier Science B.V.

1. Introduction The first triple-ionization spectrum of a molecule obtained using triple-charge-transfer (TCT) spectroscopy was reported recently [ll. Peaks in the TCT spectrum produced when gas-phase CS, was bombarded with Cl*+ ions having 6 keV kinetic energy were interpreted in terms of vertical transitions from the ground state of neutral CS,. These transitions populate, due to spin conservation in the collisions [2], various triply ionized S = 3/2 quadruplet states of its molecular ion by the concerted transfer of three valence electrons from the molecule to the projectile ion. The experimental resolution enabled the TCT peak positions, and therefore the corresponding triple-ionization energies (TIES) to the final states, to be determined to typically kO.5 eV [l]. An initial analysis of the triple-ionization energies of these transitions was carried out [l] with a straightforward semi-empirical method (SEM), previously effective 0009-2614/97/$17.00 0 1997 Elsevier Science B.V. All rights reserved. PII SOOO9-2614(97)005071

in the analysis of double-charge-transfer (DCT) spectra [3]. In the SEM each transition to a quadruplet CS:’ state is modelled by the creation of three holes of parallel spin in the electron distribution of the neutral molecule, and its energy taken to be the sum of the experimentally determined single ionization energies associated with those holes, augmented by a significant interhole interaction term calculated using the MSXCZ approximation [4]. The SEM predicted TIES of CS;’ that matched well the TCT peaks observed. However, the SEM has some intrinsic shortcomings which limit its applicability: each transition is considered to populate a particular main 3-hole(3h) configuration, any configuration interaction with other main configurations and the satellite configurations [4hlp, 5h2p, etc., that are combinations of 3 holes and additional hole-particle (hp) pairs associated with “shake up” excitations to unoccupied orbitals] being neglected; in its current form it allows only one final spin state for each spin

D.E. Parv/Chemical

Table 1 Predicted

triple-ionization

energies to quadruplet

Physics Letters 272 (1997) 148-154

states of CSz+, calculated

using the ADC(2) method (see text) Leading configuration

El

Main (a)

52.90

79

54.47 55.98 56.22

78 54

79% 27F;‘2?rs-2 75% 5a; ’ 2 n, ? 54% 2n,22n, ’ 70% 6us- ’ 2 nr; ’ 77% 27y ‘2Qy j3TU 71% 5~,‘2rr,‘2a, 73%550;‘2a;‘2?r;’ 88% 2rrU- ‘Zrrs- ‘37rU 50% 5o, ’ 2 a, ’ 2 lrs- ’ 78% 2rr,- ‘2iy j3?TU 64% 6a,‘2a,‘2~-’ 66% 609 ’ 2 rr; ’ 2 ir; ’ 42% 2nU- 227rs- ‘37rU 42% 6~s~ ’ 2 a, ’ 2 ns- ’ 38% 27y 22Xy ?37rU 78% 5oU- ‘2?r- i3rrU 80% 5a,,- ‘2rrs- -‘3rr,, 76% 6us- ’ 50; ’ 2719 ’ 64% 2nU- *2ry 23zy, 17% 50;‘27~;~ 38% 6oR- ‘2rrR- .‘37rU,,31% 27r,,- ‘2rr- -j7uK 51% 6uR- ‘27r- ‘37r,, 59% 2a,‘2rrsTg3n,, 21% 2~;~2irs-’ 47% 5u; ’ 2 a, ’ 2 ns- =3rr” 21% 5~;‘27r;‘2ns-~3a,, 20% 5u;‘2n;’ 37% 2n;- ‘2rr- j7U x 14% 6~?;- ‘2n--’ 8% 4UU,- ‘2Tr- 2 23% 5oU- ‘2rr- ‘27rR- ‘3~ 66% 2rr- ‘27y ‘7us 80% 5a,- ‘2GrU_‘2?r- 23rrU 29% 5q,- ‘2~-~,- ‘2~~- ‘37rU 65% 27rU- ‘2rr- ‘7~ 56% 517,- ‘2rr,- ‘27r- ‘31r~ 40% 6uR- ‘27rU- ‘27$ ‘37rU 17% 6~a-“na,~ 53% 6os- ‘5~; ‘2~; ’ 56% 5a,,- ‘27r- ‘2rrs- 237rU 12% 5as- ’ 2?r- 2 27% 6~~~ ‘2rrUTT, ‘27r- ‘37rU 78% 5UU,- ‘2rr,,- ‘27r- 23?TU 32% 6uX- ‘2rrU- ‘2~~~ 237rU 24% 6a,- ‘27r,- ‘2~s~ ‘3rrU 38% So,- ‘27~~~.‘7uX 34% 2ny 22ns- 23rrU, 31% 5UU,- ‘2rr- 370 R 22% 6un- ‘27rUm‘2~s~ 23nU 23% 4~;’ 2°F’ 66% 5a,,- ‘2ir- ‘2xX- 23rrU 49% 5UU,- ‘27r- ‘27r*- 23TrU 81% 2lr- 227y 23rr” 21% 6us-‘2n,’ no dominant conjiguration

Term

57.11 57.69 57.84 58.02 58.05 58.90 59.13 59.29 59.55 59.72 60. IO 60.30 60.40 60.67 61.17 61.28 61.49 61.61 61.61 61.91 61.94 62.19 62.51 62.54 62.59 62.63 62.82 62.89 62.94 62.96 63.08 63.09 63.78 63.81 63.85 63.85 63.98 64.00 64.01 64.03 64.07 64.19 64.30 64.42 64.51 64.53 64.84 64.97

76 2 72 74 0 53 0 64 66 0 52 0 2 0 77 0 26 0 2 21 1 25 0 18 15 I

0 0 I II I 0 21 55 I 18 0 0 0 0 0 0 0 38 I

6 0 32 I

weights

149

150

D.E. Parry/Chemical

Term

Physics Letters 272 (1997) 148-154

Main

Leading configuration

weights

(%‘c) 65.-70 eV range has 72 transitions, 65.91 67.53 67.58 67.59 68.91 69.74

4 X”X 4X,4 8, p+ 4z‘Q;

70-75

eV range has 110 transitions,

70.05 70.11 71.11 72.44 73.26 73.38 73.51 74.73 75-80

46 4 2,u+ 4k 4X4rI” 4 2” ”

Jrl 4

%

”

eV range has 183 transitions,

4x+

76.11 77.99

49

78.02 78.06

411” 4-X8Y

those below and 66 satellites 25 40 37 44 38 34

17% 4a;’ 2fZ 38% 4a,‘2q ‘271;’ 35% 5a,‘2Tr;* 42% 4~; ’ 2 71; ’ 2 TV- ’ 31%44a;‘2~,‘2~r,’ 27% 4~; ‘5~; ‘21r, ’

those below and 102 other satellites. 18 27 26 26 31 23 21 24

18% 5ap- ‘2q,- ‘27r- ’ 26%5o;‘21~;‘2~r,’ 19% 4a;‘6ug-‘22;’ 23%5u;’ 2~;‘2$’ 20% 4u;‘5u;‘2rr;’ 22% 4u; ’ 2 q- = 13% 5~;’ 6~; ’ 27rg’ 21% 5up’ 2?r;l

those below and 179 other satellites. 36 33 24 16

35% 33% 22% 15%

4~; 5u, 5~; 4q-

’ 6~; ’ ’ 6u, ’ ’ 6~; ’ ’ 60 -

’ ’ ’ ’ 3-n,-’

5~; 5~; 2a,

Each configuration weight given for a transition is the square of its normalized coefficient in the ADC(2) eigenvector for that transition expressed as a percentage, and each main character is the sum of the weights of all the 3h configurations for that transition. Entries for satellite transitions, arbitrarily defined as having less than 20% main character (see text), have only been included, italicized, for TIES less than 65 eV.

multiplicity, so while the S = 3/2 quadruplet state of a 3h configuration is properly described there are, unless two of the holes are in the same molecular orbital, two S = l/2 doublet 3h states which are not; energy expectation values of configurations, not terms, are calculated; the range of final states that can be analysed is restricted to those for which experimental values of the appropriate single ionization energies are known - in the case of CS, this was particularly relevant as only those for the 2 nTTg, 2n,, 5a, and 6crs ionizations were available [5], so that only the first eight peaks in the TCT spectrum were assigned [l]. None of these restrictions should apply in an ab initio analysis of TCT spectra. Here the results of ab initio Green’s function calculations of the triple-ionization energies of CS, are reported, and the predicted values matched to peaks in the TCT spectrum corresponding to TIES higher than those given previously. In addition, the

ab initio results allow the reliability of, and the importance of the factors not accounted for in, the SEM to be assessed.

2. Method The second-order algebraic diagrammatic construction (ADC(2)) method of constructing the three-particle propagator, or equal-times Green’s function, of the valence electronic structure of a molecule [6] has been adopted for the calculation of the vertical triple-ionization energies of CS, and other molecules now being studied, as it offers a straightforward method of locating the positions of the poles of the Fourier transform of a propagator. In practice, that method amounts to the construction and diagonalization of an Hermitian interaction matrix, the eigenvalues of which provide the vertical

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Physics Letters 272 (1997) 148-154

transition energies from the initial state of the neutral molecule to various final states of the triply-charged ion having the same geometric structure. At second order the representation of that matrix consists of all the 3h main and 4hlp first-order satellite configurations that may be constructed from the reference configuration, the closed-shell restricted HartreeFock ground configuration of the neutral molecule [6]. While the number of main 3h configurations is modest for quite large molecules, the number of 4hlp configurations increases both with the number of occupied molecular orbitals and the size of the basis set employed. In the strict ADC(2) approximation the interaction between different 4hlp configurations is zero, but here the non-zero ADC(3) 4hlp4hlp matrix elements were included to take better account of those interactions [6]. Expressions for all the ADC(2) matrix elements have been published [6] and the method has been successfully applied quite recently to the prediction of satellite peak positions in the Auger spectra of small molecules [7-lo]. The basis set used in the calculation of the quadruplet states of CS2 was the “double-zeta plus polarization” basis incorporated in the GAUSSIAN94 ab initio molecular orbital program [ 111, which was used to perform the Hartree-Fock calculation for CS, at its linear equilibrium geometry with C-S bond length 0.1553 nm [ E(RHF) = - 832.8804996 Hartree] and to prepare 2-electron integrals in the SCF molecular-orbital basis. The ADC(2) calculations were then carried out using a program recently developed by the author, in which ADC(2) matrices with the following dimensions were set up and diagonalized separately for six different quadruplet irreducible representations of the largest Abelian symmetry subgroup D,, for CS, (the relevant irreducible representations of Dmh are noted): 1854 (4x; and ‘A,), 1968 (“C; and 4A,>, 1911 c411,>, 1854 (“2: and “A,), 1968 (4x; and “A,), 19 11(4 II ,) (the remaining two irreducible representations of D,, also correspond to the 4 IIp and 4110 transitions). The eigenvalues up to 80 eV were located by an in-core Sturm sequence algorithm which is appropriate for diagonalizations where only a small fraction of the eigenvaiues is required. The calculations were carried out on the DEC Alpha 8400 “Columbus” computer at the CLRC Rutherford Appleton Laboratory.

151

3. Results The calculated TIES, with their term symbols and configuration characters, are presented in Table 1. As the neutral molecule reference state is of singleconfiguration form [6], the eigenvector associated with a given TIE provides a configuration-interaction description of the final state of CSi+ for that transition, the configuration weights given being the squares of normalized eigenvector coefficients. The ion-molecule TCT collision process involves a change in occupancy of at least three spin-orbitals of the target molecule. In the absence of a detailed theory of its scattering amplitude it is reasonable to assume that the coupling between the initial and final channels of a collision will be much greater when a main 3h configuration dominates the CSG’ state, which suggests that the observed TCT peaks will be matched by the TIES of main transitions, while final states dominated by satellite 4hlp and higher excitations which involve changes in occupation of at least five CS, spin-orbitals in the transition are much less likely to be populated. A recent comparison, of the DCT spectrum of the ethyne molecule with the results of ab initio ADC(2) calculations analogous to those described here, supported that hypothesis [12]. The ADC(2) method predicts main transition energies to second order in the correlation interaction, but the ADC(3) interactions between satellite configurations, included as noted above, are only first order [6] which suggests that this method is well suited to the quantitative analysis of TCT spectra. Naturally, few CS:+ states lack any 3h character and so the threshold for distinguishing between main and satellite transitions must be arbitrary. In this and other ADC(2) calculations for multiple ionizations [ 12- 141 relatively few transitions are found with total 3h weights ca. 20%, so those transitions predicted to have weights greater than this are deemed main and, experience indicates, are candidates for matching the observed TCT peaks. The lowest energy transitions are predominantly 3h in character. However, the lowest satellite transition is only ca. 4 eV above the lowest TIE and there are three more satellites with TIES less than 60 eV, all associated with shake-up from the highest occupied 21r~ to the lowest unoccupied 37r, molecular orbital. In comparison with typical double-ionization

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manifolds predicted for molecules of this size [ 13,141, the density of transitions is much higher because an additional hole is created in the final ion. However, configuration interaction appears significant only for

Table 2 Comparison of the positions of peaks A-N observed in the TCT spectra, for the collision of 6 keV Cl*+ ions with gas-phase CS,, with (i) ADC(2) and (ii) SEM predictions of the TIES of transitions to main quadruplet states of CSz+ (see text)

TCTleV

A

54.3 kO.5

ADC(2) I ev + 1.40

TCt-Rl

SEM/eV+

54.30

‘n.

54.1

4 z

56.1

B

56.7.?0.5

55.87

c

57.7 +a.4

57.38 51.62

1.2

57.8 56.9

59.09 D

58.8+0.4

59.24

$+mg

59.45 E

60.2kO.3

60.53 60.69

F

61.5rO.3

61.12 62.01

+$L

%.7 ‘n.

606

62.6

62.68 G

62.9k0.5

63.01 63.34

H 65.2f0.7

$161.4

64.49

63.1

65.18

65.2

transitions above 60 eV. Calculations of the TIES up to 80 eV were performed because data on TCT peaks for CS, additional to those reported previously [1] are now available. As only main TIES are matched to TCT peak positions, details of the large number of satellite transitions predicted in the range 65-80 eV, apart from those of a couple complementing related main transitions, have been omitted for brevity. The additional TCT data are included in Table 2, where TCT peak positions have been matched to the ADC(2) predictions of the TIES of main transitions. A typical TCT spectrum was presented previously [I] and is not reproduced here because the peak positions quoted are not obtained from that alone but rather are deduced from the analysis of a large number of such spectra. Following previous practice [ 12-141 a uniform shift, here + 1.40 eV, has been applied to the ADC(2) TIES to align the lowest predicted TIE with that of peak A so that the distributions of theoretical and experimental TIES may be compared readily. Also included in Table 2 for comparison are the previously reported predictions of the SEM for the eight peaks A-H [Il. Each SEM TIE was calculated for a particular main configuration and here has been matched to the ADC(2) transitions for which that configuration is dominant, rather than to a TCT peak.

65.70 I

66.4i0.5

66.24

I

67.8 IfO.4

67.31 68.93

K 69.5 t 0.6

68.98 68.99 70.31

L

71.4+0.7

71.34 71.51

M

73.2+0.5

72.51 73.84 74.66

N

75.3*?

74.78 74.91 76.13

TCT and SEM data for peaks A-H were reported previously [l]. The statistics for peak N were very poor. Both ADC(2) and SEM TIES have been shifted uniformly as shown to facilitate comparison of the theoretical and experimental transition energies. Term I symbols, to which the SEM values have been matched here, are as predicted by the ADC(2) calculations.

4. Discussion The agreement between the ADC(2) TIES and the observed TCT peak positions given in Table 2 is very good. Each of the peaks A-L is well matched by one or more TIES, most of which are within 0.5 eV of the peak position. The spread of TIES for peaks M and N is a little wider, probably reflecting both poorer experimental statistics and ADC(2) predictions, but even for the weak peak N, at the upper end of the reaction window [1] more than 20 eV above peak A, deviations are less than 1 eV. The overall pattern of TCT peak separations is well reproduced. Clearly, most observed peaks are unresolved combinations of peaks representing individual TCT transitions and the ADC(2) predictions indicate that a significant improvement in the experimental resolution would be required to resolve them satisfactorilv. These results are consistent with the hv-

D.E. Partyv/Chemical

Phwics Letters 272 (19971 148-154

pothesis that only main transitions are significant in TCT spectra; if satellite transitions were to have comparable intensities then a TCT spectrum would, given the ADC(2) predictions of their TIES, be quasi-continuous rather than exhibiting several welldefined peaks. There is also no evidence here that spin conservation [2] is violated in these TCT collisions, as the predicted main transitions to quadruplet final states match all the observed peaks. Confidence in the assignments made here rests on two arguments: (i) analogous ADC(2) predictions of the positions of well-separated peaks in the DCT spectra of small molecules such as ethyne can be in good quantitative agreement with experiment over the 15 eV range offered by a typical DCT reaction window [ 121; and (ii) for CS,, variations in the energy separations of adjacent TCT peaks reflect the predictions to a fair extent, e.g. the 2.3 eV separation of Cl and H matches the larger than average gap between the corresponding groups of ADC(2) TIES. Both arguments indicate that ADC(2) TIES and TCT peak positions in the energy range examined should coincide acceptably on the same relative energy scale once those for the lowest energy transition are aligned. Studies of the TCT spectra of molecules with very few valence electrons are planned, for such spectra are expected to exhibit relatively few, well separated peaks that will provide unambiguous tests of ADC(2) predictions of TIES. Peak D is predicted to be a combination of all the transitions for the 5~; ’ 2 nUm’ 2 nB ’ configuration, but those for the 6~~:’ 2rrL’ 27~;’ appear to be shared between peaks E and F. The SEM should, when configuration interaction is not significant, provide a reasonable estimate of the energy expectation value of a single configuration, but being dependent on the one-electron MSX (Y approximation it cannot, if there are different terms for that configuration, predict the separation of their term energies. Analogous cases have already been noted in the double ionization spectra of acetonitrile [ 131 and propyne [14] and uncertainty in that respect will arise if the SEM alone is used to predict multiple ionization energies to configurations that are not irreducible representations of the symmetry group of the molecule studied. Above 61 eV the configuration interaction evident in the ADC(2) results is reflected in the SEM being unable to predict both the TCT

153

peak positions and the ADC(2) TIES consistently. Main-satellite configuration interaction accounted for in the ADC(2) calculations results in two “2; transitions being assigned to peak G, whereas the SEM predicted peak F to correspond to one 4C; at 61.4 eV [I] and so, in sequence, its ‘II” prediction at 62.6 eV was assigned [I] to peak G. The 411g transition for peak G results from main-satellite configuration interaction and was not predicted by the SEM which, however, assigned a strong main 4flI, at peak H in agreement with ADC(2). In many respects the SEM can play a useful role in the initial analysis of multiple ionization spectra, subject to the caveats noted above. In this case it has performed well in the energy region extending about 6 eV above the lowest transition, within which each final state is dominated by one 3h configuration and the effects of mainsatellite configuration interaction are not significant. To summarize, TCT spectra for the collisions of Cl’+ with CSZ have been analysed satisfactorily, over the entire energy range observed, with ADC(2) calculations for the transitions to quadruplet states of CS;‘. Predicted transitions to main states, defined arbitrarily has having at least 20% 3h character, have been correlated consistently with peaks observed in the TCT spectra. It is anticipated that ADC(2) predictions of the TCT spectra of other molecules now being investigated will prove equally effective.

Acknowledgements The author thanks (i) Iwan Griffiths, Frank Harris and Nadine Jeffreys for many useful discussions during the course of this project, (ii) the Engineering and Physical Sciences Research Council for an allocation of time on the Columbus computer at the CLRC Rutherford Appleton laboratory, (iii) the British Mass Spectrometry Society for its financial contribution to the development of the programs used in this work.

References [I] N. Jeffreys, I.W. Griffths, D.E. Pan-y. F.M. Harris, Chem. Phys. Lett. 266 (1997) 537. [2] E. Wigner, Nachr. Akad. Wiss. GGttingen. Math. Phys. Kl., 2A: Math. Phys. Chem. Abt. (1927) 325.

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