Double core–hole correlation satellite spectra of N2 and CO molecules

Chemical Physics Letters 521 (2012) 45–51

Contents lists available at SciVerse ScienceDirect

Chemical Physics Letters journal homepage: www.elsevier.com/locate/cplett

Double core–hole correlation satellite spectra of N2 and CO molecules Motomichi Tashiro a, Kiyoshi Ueda b, Masahiro Ehara a,⇑ a b

Research Center for Computational Science and Institute for Molecular Science, Myodai-ji Nishigo-naka 38, Okazaki 444-8585, Japan Institute of Multidisciplinary Research for Advanced Materials, Tohoku University, Sendai 980-8577, Japan

a r t i c l e

i n f o

Article history: Received 8 October 2011 In final form 23 November 2011 Available online 29 November 2011

a b s t r a c t We have theoretically investigated the correlation satellite spectra associated with the single-site (ss) and two-site (ts) double core–hole (DCH) states. The calculated ssDCH satellite spectra of N2 (N 1s2) and CO (O 1s2) reproduced the observed experimental spectra satisfactorily and provided the first detailed assignments. DCH spectroscopy using X-ray two-photon photoelectron spectroscopy (XTPPS) was examined taking account of the single core–hole (SCH) and DCH satellite spectra. The results demonstrate that in XTPPS the ssDCH satellite spectrum is separated from other SCH/tsDCH spectra, while the tsDCH states and their satellite states may be overlapped with the SCH satellite spectrum. Ó 2011 Elsevier B.V. All rights reserved.

1. Introduction Molecular double core–hole (DCH) states, in particular the Kshell DCH states, were first investigated by Cerderbaum et al. theoretically in 1980’s [1–3] after some fundamental investigations of atomic DCH states. They proposed the essential issues of molecular DCH states, in particular the orbital relaxation effects, via the DCH creation with respect to chemical environment around core–holes. These pioneering works were followed by some theoretical works [4,5], which extended the concept to other types of core–holes such as K1L1 and L2 DCH states. The observation of the molecular Kshell DCH states has, however, been so limited until very recently because of their low cross sections in most of the experiments. The related works are on the core-core-core type Auger processes such as KLL states, and these Auger final DCH states have been measured for L-shell DCH states only indirectly [6–9]. Recently, molecular double-core hole (DCH) states have been receiving significantly renewed interest. In 2009, Santra et al. [10] suggested that DCH spectroscopy could be realized by X-ray two-photon photoelectron spectroscopy (XTPPS) using an intense short-wavelength free-electron laser source. Following this suggestion, Cryan et al. [11] and Fang et al. [12] indeed observed the DCH states of N2 molecule and subsequent Auger decay, using XTPPS with Linac Coherent Light Source (LCLS) in USA. At nearly the same time, Eland et al. [13] and Lablanquie et al. [14] observed the DCH states of many other small molecules and subsequent Auger decay using multi-electron-coincidence (MECO) techniques with conventional synchrotron radiation (SR) sources. There are two types of molecular DCH states; one is the singlesite (ss) DCH state, where two core electrons are ionized from a

⇑ Corresponding author. Fax: +81 564 55 7025. E-mail address: [email protected] (M. Ehara). 0009-2614/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2011.11.062

single atom, and the other is the two-site (ts) DCH state, where two core electrons are ionized from the different atoms [1]. However, the experimental observations listed above [11–14] were as yet limited to the ssDCH states. The reasons were twofold: As for MECO with SR, the cross section for single-photon tsDCH creation was much smaller than ssDCH creation, though it was non-zero. In case of XTPPS, the tsDCH spectrum could not be identified because it was overlapped with SCH satellites [11,12]. Despite these difficulties, Lablanquie et al. [15] recently identified the tsDCH of C2H2 using MECO by improved statistics, while Berrah et al. [16] and Salen et al. [17] found the tsDCH states of CO, N2, CO2 and N2O using XTPPS by subtracting the contribution from single-photon SCH spectra. We have undertaken theoretical studies of DCH spectroscopy using the complete active space self-consistent filed (CASSCF) calculations [18,19] in collaboration with some experimental groups [16,17]. We suggested that the relaxation energy and interatomic relaxation energy that represent chemical environment of core– holes can be extracted from XTPPS. This theoretical DCH spectroscopy has also been extended to the K1L1 and L2 DCH states including second-row atoms [20,21] and applied to the biochemical molecules, namely, formamide and nucleobases [22]. We note that Lablanquie et al. [14] observed not only the DCH main line but also DCH correlation satellites, though they gave no assignments of the observed spectra of satellites. In view of the DCH satellite spectrum, theoretical calculations were applied to SiH4, SiF4 [4] and aminophenol [10] to our knowledge. It is of particular interest to study the correlation satellites, because they provide information about electron correlation effects in the DCH states. Understanding of the satellite spectra would also be helpful to design spectroscopic measurements of the DCH states. The correlation satellite states are described by multi-electronic processes [23,24]. They are associated with the main line (primary state) and caused by the breakdown of one-electron picture. The

46

M. Tashiro et al. / Chemical Physics Letters 521 (2012) 45–51

correlation satellite states have been investigated for both valence and core-electronic single ionizations. For valence satellites, various kinds of spectroscopy such as synchrotron radiation PES [24], electron momentum spectroscopy (EMS) [25], and Penning electron ionization spectroscopy [26] have been applied. High-resolution soft X-ray spectroscopy has also been applied to investigate the satellite spectrum of core-electron processes. For theoretical studies of correlation satellite states, high-level ab initio methods are necessary to describe the electron correlations of multi-electron processes satisfactorily. As for core-electronic processes, orbital relaxation is also significant so that the theory should be able to describe both electron correlations and relaxation precisely. In our previous studies on the single core– hole (SCH) correlation satellites [27–31], we found that triple or higher electronic processes could significantly contribute to the correlation satellite states even in the low-energy region. Therefore, we expected that a method describing highly correlated systems is necessary to investigate the DCH correlation satellite states. In the present work, we have investigated the DCH correlation satellite spectra of N2 and CO molecules theoretically. The satellite states accompanied by both ssDCH and tsDCH states were calculated by the restricted active space configuration interaction (RASCI) method based on the state-averaged CASSCF [32] references for the low-lying satellite states. The calculated ssDCH satellite spectra are compared with the observed spectra by MECO and detailed assignments of the peaks are proposed. The energy region of the SCH and DCH satellite spectra is discussed in view of the DCH spectroscopy by XTPPS. 2. Computational details Ab initio calculations of the vertical ionizations related to the double core–vacancy satellite states of N2 and CO were performed. The bond distances of N2 and CO were set to 1.09 768 and 1.1283 Å, respectively, taken from the experimental values [33]. For calculating the DCH states and their satellite states in a wide energy region, as many as 50–100 solutions should be achieved. For this purpose, the complete active space self-consistent field (CASSCF) method [32] followed by the restricted active space configuration interaction (RASCI) was performed. CASSCF calculations were made in the state-averaged scheme for low-lying satellite states. The configurations in CASSCF and RASCI were restricted to those having two holes in K-shell orbitals. In CASSCF, the active space was adopted as the orbitals from 2s to p⁄ and r⁄. The RASCI space consists of all the CASCI configurations and the restricted singly excited configurations from all the CASCI space. The correlation consistent polarized valence triple zeta basis sets augmented with the diffuse functions (d-aug-cc-pVTZ) [34] were adopted to describe the satellite states. It is shown in Ref. [18] that the cc-pVTZ basis sets describe the relaxation of the valence orbitals caused by the DCH ionizations. The present basis sets can describe the satellite states with valence excitations and some transitions to diffuse orbitals. The scalar relativistic effect, which is not so significant in the case of K-shell ionization of first-row atoms [18], was disregarded. The CASSCF and RASCI calculations were performed using Molpro2008 [35] and congen and scatci modules in the UK R-matrix codes [36], respectively. 3. The DCH satellite spectrum in MECO First, we discuss the DCH satellite spectrum observed by MECO. Experimentally, the X-ray SR MECO enables the single photon double ionization, while the measurement with the XTPPS by XFEL

achieves the two-photon double ionization. The former MECO measures the kinetic energies of two ionized electrons. Therefore, the double ionization energy (DIE) of the DCH satellite state as well as the primary DCH state is given by

DIE ¼ hv  KE1  KE2

ð1Þ

where KE1 and KE2 are the kinetic energies of two ionized electrons. In this work, the DIEs of primary and satellite states were calculated by the RASCI method. However, the cross section of double photoionization, which is another relevant quantity, is not straightforward because other cooperative processes participate. The cross section of single-photon double ionization for He atom was discussed in detail using the many-body perturbation theory including mechanisms such as knockout and shake-off processes [37,38]. In SR MECO, various processes may take place in double ionization. However, we adopted the sudden approximation for simplicity and calculated the intensity with the approximation analogous to the SCH pole strength by

D   E2    I ¼  UN0 ayi ayj WN2 RASCI 

ð2Þ

where UN0 and WN2 RASCI represent the ground-state Hartree–Fock and DCH RASCI wave functions, respectively, with i and j being the core orbitals. Although simple approximation is utilized, the calculated intensities agree reasonably well with those in experimental spectra, as shown below. 3.1. N2 The MECO spectrum of the ssDCH and its DCH satellite states of N2 was observed up to 970 eV. The present RASCI calculations were performed to solve 100 solutions in order to cover the energy range of 900–940 eV. In the calculations of N2 DCH states, the localized/delocalized picture always matters. In the present work, localized picture of the ssDCH state was adopted. The calculated spectrum is compared with the experimental spectrum by Lablanquie et al. [14] in Figure 1. It was reported that the relative intensities of satellites compared to main peak are higher in the DCH state than those in SCH, because the relaxation of valence shell is much significant in the DCH state. The results of ssDCH (Na 1s2) and tsDCH (Na 1s1 Nb 1s1) satellite states with the pole strength higher than 0.004(0.003) are summarized in Tables 1 and 2, respectively. The satellite states in R symmetry were mainly calculated for both ssDCH and tsDCH states, because they have dominant intensity in the spectrum interacting with the primary states. The low-lying P states for the ssDCH states were also calculated to interpret the low-lying peaks observed in MECO. First, we compare the calculated ssDCH satellite spectrum with the experimental spectrum [14]. The lowest satellite state was calculated at about 910.9 eV in P symmetry, which is assigned as the rp⁄ transition accompanied by the ssDCH ionization. A small structure exists in this energy region of the experimental spectrum. This state might have been observed more clearly in the condition that the conjugate process is enhanced like using angle-resolved spectroscopy, as seen in the SCH states [39–41]. Another low-lying state was also calculated at 912.7 eV with weak intensity. This state is characterized as the ssDCH pp⁄ transition. A strong peak was observed around 918 eV [14]. This peak is attributed to the satellite state calculated at 919.3 eV, which is characterized as the linear combination of pp⁄ and rr⁄ transitions. A small shoulder, which seems to exist in the lower energy side of this peak, may be attributed to the second P state calculated at 915.9 eV. The third P state was obtained at 919.9 eV which overlaps with the prominent R state at 919.3 eV. In the higher energy side of this R state, the quadruple electron process, (Na 1s)2(5r)2(2p)2, was obtained at 921.4 eV. The experimental spectrum seems to have a shoulder in

M. Tashiro et al. / Chemical Physics Letters 521 (2012) 45–51

47

For tsDCH satellite states of N2, singlet and triplet spin states are possible as shown in Table 2. The primary tsDCH states of N2 were obtained at 836 eV with the singlet–triplet splitting of 0.5 eV. There are numerous tsDCH satellite states in contrast to the ssDCH satellite states. About 100 states were solved to cover the range of 835–870 eV with the present basis set and configurational space. The lowest singlet and triplet satellite states with considerable intensities were obtained at about 11 and 7 eV, respectively, above the primary tsDCH states. They are characterized as pure pp⁄ transition and have large pole strength unlike the ssDCH state. It may be of interest to note that the spin coupling schemes of the core electrons in these singlet and triplet DCH satellite states are triplet and singlet, respectively. From the low-energy region, multiple electron processes with considerable intensity were calculated; for example, quadruple electronic process [core](1p)2(2p)2 was calculated around 854 eV. The intensities of the DCH satellite states were observed to be higher than those of the SCH ones [14]. This is partly because the final ionic-state correlation and relaxation of the DCH states are stronger than those of the SCH states and therefore, the intensities of the primary DCH states are more significantly redistributed to the satellite states.

3.2. CO

Figure 1. N 1s2 DCH satellite spectrum of N2: Expt. [14] and RASCI calculation. Binding energy is derived from sum of the two photoelectron energies in MECO [14]. In MECO, data points with intensities multiplied by factor 3 are also shown in the range of 907–913 eV.

Table 1 DIE (eV), intensity (I) and character of ssDCH satellite states of N2. State No.a 1 2 3 5 7

Expt.b

Present

DIE

DIE

I

Characterc

902.55 ± 0.5

901.5 912.7 919.3 921.4 925.7

0.8502 0.0 0.0534 0.0048 0.0108

930.1

0.0067

935.3

0.0063

(Na 1s)2 (1p)1(2p)1 (1p)–1(2p)1/(5r)1(6r)1 (5r)2(2p)2/(4r)1(5r)1(2p)2 (4r)1(5r)1(2p)2/ (5r)2(1p)1(2p)3 (5r)1(6r)1/(5r)1(8r)1/ (5r)1(6r)1(1p)1(2p)1 (4r)1(6r)1/(4r)1(7r)1/ (4r)1(8r)1

910.9 915.9 919.9

– – –

919 Shoulder 929

10 22

934

P 1 2 3 a b c

(5r)1(2p)1 (4r)1(2p)1 (5r)1(1p)1(2p)2

States with pole strength larger than 0.004 are listed. Ref. [14]; DIEs of satellite states are estimated from the experimental spectrum. Character is shown relative to the N 1s2 state.

this region. A broad band was observed in the range of 925–940 eV [14], for which continuous satellite states were obtained in the present calculation. The satellite states with large pole strengths were obtained at 925.7, 930.1, and 935.3 eV which are characterized as multiple electron processes: (4r)1(5r)1(2p)2, (5r)1(6r)1, and (4r)1(6r)1, respectively. Besides these states, numerous states were also calculated in the range of 930–940 eV as shown in Figure 1. These states are responsible for the broad band observed in the spectrum.

The results of the satellite states associated with the ssDCH C 1s2 and O 1s2 states of CO are summarized in Table 3. The DCH R satellite states that have dominant intensities in the sudden limit were mainly calculated. Previously, the O 1s2 satellite spectrum of CO was experimentally observed [14], although the uncertainty in the spectrum seems to be larger than that of N2. The results of tsDCH satellite states are also summarized in Table 4. The calculated O 1s2 satellite spectrum is compared with the experimental one in Figure 2. We first discuss the O 1s2 satellite spectrum. It is generally recognized that an accurate calculation of the satellite spectrum of the O 1s1 state is more difficult than that of the C 1s1 state, because the charge reorganization is more extensive in the O 1s1 satellites [28]. This trend must be the same in the O 1s2 satellite spectrum. The O 1s DCH state was calculated at 1177.1 eV, in reasonable agreement with the experimental value, 1178.0 ± 0.8 eV [14]. The satellite spectrum seems to have a small structure around 1185 eV, which may be attributed to the P state. The lowest P satellite state was calculated to be 1185.4 eV, although this state has no pole strength within the sudden approximation. Unlike N2, other P satellite states are energetically higher, as calculated at 1199.7 and 1201.3 eV. A broad continuous band was observed in the range of 1190–1210 eV. The satellite state with a large pole strength of 0.04 was calculated at 1193.3 eV. This state was characterized as (5r)2(2p)2 and attributed to the structure around 1193 eV in the experimental spectrum. It is of interest that the low-lying state has the nature of quadruple electron processes. This is similar to the fact that the lowest O 1s1 SCH satellite of CO is significantly contributed from the triple electron process [28]. The calculated satellite states at 1199.8 and 1200.6 eV also have considerable intensities, to which the peak around 1200 eV in the MECO spectrum is attributed. The structure observed around 1205 eV in the experimental spectrum can be ascribed to the state calculated at 1206.9 eV. The C 1s2 satellite spectrum of CO appears in the range of 680– 700 eV. The DIE of the primary C 1s2 ssDCH state was calculated at 664.5 eV. The satellite state calculated at 679.6 eV has large pole strength of 0.02. This state has a character of quadruple electron process, (C1s)2(1p)2(2p)2. It is of interest that hexaple electron process exist in the high-energy region around 695 eV, (C1s)2(1p)4(2p)4. The P satellite states were also obtained in

48

M. Tashiro et al. / Chemical Physics Letters 521 (2012) 45–51

Table 2 DIE (eV), intensity (I) and character of tsDCH satellite states of N2. a

No.

a b

b

DIE

I

Character

Singlet 1 2 7 30 52 54

836.0 847.1 854.2 868.4 872.7 873.0

0.7803 0.1040 0.0035 0.0096 0.0054 0.0056

(Na 1s)1(Nb 1s)1 [core T] (1p) 1(2p)1 [core S] (1p)2(2p)2/[core S] (1p)2(2p)2 [core S] (1p)2(2p)2/[core S] (1p) 4(2p)4/[core T] (1p)3(2p)3 [core T] (5r)1(8r)1/[core S] (4r)1(8r)1/[core T] (1p)3(2p)3 [core T] (5r)1(8r)1/[core S] (4r)1(8r)1/[core S] (1p)2(2p)2/[core T] (1p)3(2p)3

Triplet 1 2 9 39 77 83

836.5 843.6 853.7 865.4 871.0 871.9

0.7970 0.0782 0.0133 0.0045 0.0088 0.0103

(Na 1s)1(Nb 1s)1 [core S] (1p)1(2p)1 [core T] (1p)2(2p)2 [core T] (4r)2(2p)2/[core T] (4r)1(6r)1/[core T] (5r)2(2p)2 [core T] (1p)1(3p)1/[core T] (1p)2(2p)2/[core T] (1p)4(2p)4 [core T] (1p)2(2p)2/[core T] (1p)1(3p)1/[core T] (1p) 4(2p)4

States with pole strength larger than 0.003 are listed. [core S] and [core T] denote the (Na 1s)1(Nb 1s)1 configuration with singlet and triplet coupling, respectively.

Table 3 DIE (eV), intensity (I) and character of ssDCH satellite states of CO. State No.a

Expt.b

Present

DIE

DIE

I

Characterc

664.5 674.8 679.6 681.8 689.1 693.9 695.1 695.8

0.8554 0.0 0.0200 0.0036 0.0059 0.0017 0.0026 0.0208

(C 1s)2 (1p)1(2p)1 (1p)2(2p)2 (1p)1(2p)1/(5r)1(6r)1 (5r)1(6r)1/(4r)1(5r)1(2p)2 (4r)1(6r)1/(4r)1(7r)1/(5r)1(8r)1 (1p)2(2p)2/(1p)4(2p)4/(5r)1(6r)1(1p)1(2p)1 (1p)4(2p)4/(4r)1(5r)1(1p)1(2p)3/(1p)2(2p)2 /(4r)2(2p)2

672.1 676.6 681.8

– – –

(5r)1(2p)1 (4r)1(2p)1 (5r)1(1p)1(2p)2

1177.1 1191.1 1193.3 1199.8 1200.6 1206.9

0.8972 0.0 0.0400 0.0049 0.0102 0.0081

(O 1s)2 (5r)2(2p)2 (5r)2(2p)2/(5r)1(6r)1 (5r)1(7r)1/(1p)1(2p)1 (5r)1(6r)1/(1p)1(2p)1/(5r)1(8r)1 (4r)1(5r)1(2p)2/(4r)1(6r)1

1185.4 1199.7 1201.3

– – –

(5r)1(2p)1 (4r)1(2p)1 (5r)1(3p)1

2

C 1s 1 2 3 6 12 17 20 22

P 1 2 3 O 1s2 1 2 3 6 7 12

1178.0 ± 0.8 1193 1200

P 1 2 3 a b c

1185

States with pole strength larger than 0.003 are listed. Ref. [14]; DIEs of satellite states are estimated from experimental spectrum. Character is shown relative to the C 1s2 or O 1s2 state.

the low-energy region as 672.1 and 676.6 eV, which are characterized as rp⁄ transitions accompanied by ssDCH ionization. Let us turn to the satellite spectrum associated with the tsDCH states of CO, namely, C 1s1 O 1s1 states. The DIEs of the singlet and triplet primary tsDCH states were calculated to be 855.0 and 855.4 eV, respectively. The triplet DCH state is slightly higher than the singlet DCH state, which was also obtained in other cases [5]. Very recently, these tsDCH states of CO were first observed by the XTPPS using the XFEL at LCLS facility [16]. Both singlet and triplet tsDCH satellite spectra were calculated as in the case of N2. A tsDCH satellite spectrum, to be shown later, may be overlapped with the SCH satellite spectrum in XTPPS. Therefore, a careful experimental measurement is necessary for this spectrum. The lowest satellite states are located at 8.5–9.0 eV above the tsDCH states, as listed in Table 4. The second lowest states are the pp⁄ transitions that have considerable intensity, I = 0.065 and 0.032,

for the singlet and triplet states, respectively. The energy separation between the singlet and triplet spin states of the DCH satellites is larger than that between the primary peaks.

4. DCH satellite spectrum in XTPPS 4.1. Relaxation energy and cross section in XTPPS Next, we discuss DCH spectroscopy using XTPPS with respect to the positions of the DCH satellite spectra. It was shown by DCH spectroscopy that the generalized relaxation energy and interatomic relaxation energy can be extracted from the differences be1 tween the ionization energies (IEs) and DIEs, DEðS1 i ; Sj Þ [18,19]: these relaxation energies are given in Eqs. (6) and (9) in Ref. [19]. 1 This DEðS1 i ; Sj Þ can be measured by XTPPS as follows:

49

M. Tashiro et al. / Chemical Physics Letters 521 (2012) 45–51 Table 4 DIE (eV), intensity (I) and character of tsDCH satellite states of CO.

a b

No.a

DIE

I

Characterb

Singlet 1 2 3 6 7 9 10 13

855.0 864.3 867.3 871.5 872.3 875.6 878.2 880.6

0.7945 – 0.0653 0.0030 0.0260 0.0161 0.0039 0.0026

(C 1s)1(O 1s)1 [core T] (1p)1(2p)1 [core T] (1p)1(2p)1 [core S] (5r)2(2p)2 [core S] (1p)1(2p)1/[core S] (5r)1(6r)1 [core S] (4r)1(5r)1(2p)2/[core T] (4r)1(5r)1(2p)2 [core S] (1p)2(2p)2/[core S] (5r)1(7r)1 [core S] (5r)1(6r)1/[core S] (5r)1(7r)1/[core T] (5r)1(6r)1/[core S] (5r)1(8r)1

Triplet 1 2 3 4 9 11 14 27 28

855.4 864.0 864.1 865.8 872.3 874.9 876.5 880.9 881.9

0.8193 – 0.0317 0.0033 0.0371 0.0090 0.0049 0.0049 0.0032

(C 1s)1(O 1s)1 [core T] (1p)1(2p)1 [core S] (1p)1(2p)1 [core T] (1p)1(2p)1 [core T] (1p)1(2p)1/[core T] (5r)2(2p)2 [core S] (4r)1(5r)1(2p)2/[core S] (5r)1(6r)1 [core S] (5r)1(6r)1/[core S] (5r)1(7r)1/[core S] (4r)1(5r)1(2p)2 [core T] (5r)1(6r)1/[core S] (5r)1(6r)1/[core T] (5r)1(6r)1/[core T] (5r)1(7r)1 [core T] (5r)1(8r)1/[core T] (5r)1(6r)1/[core T] (5r)1(7r)1/[core T] (5r)1(6r)1/[core S] (5r)1(8r)1

States with pole strength larger than 0.002 are listed. [core S] and [core T] denote the (C 1s)1(O 1s)1 configuration with singlet and triplet coupling, respectively.

Figure 3. Possible routes to create DCH satellite state in XTPPS.

ionic molecule with S1 hole, respectively, in XTPPS; namely, these j kinetic energies with the photon energy of hv are given by 1 KEðS1 i Þ ¼ hv  IEðSi Þ

Figure 2. O 1s2 DCH satellite spectrum of CO: Expt. [14] and RASCI calculation. Binding energy is derived from sum of the two photoelectron energies in MECO [14].

1 1 1 DE ¼ DIEðS1 j ; Si Þ  IEðSi Þ  IEðSj Þ 1 1 ¼ KEðS1 i Þ  KEðSj ; Si Þ

ð3Þ

1 1 where KEðS1 i Þ and KEðSj ; Si Þ are the kinetic energies of emitted 1 electrons by creating Si hole in neutral molecule and S1 hole in i

1 KEðS1 j ; Si Þ

¼ hv 

1 ½DIEðS1 j ; Si Þ

ð4aÞ 

IEðS1 j Þ: KEðS1 i Þ

ð4bÞ 1 KEðS1 j ; Si Þ

Therefore, it is relevant to measure and of the respective SCH and ss/tsDCH states clearly. Other factors like satellite spectra should be carefully taken into account in the spectrum of XTPPS. In the present study, the XTPPS spectra were examined in comparison with the calculated SCH and DCH satellite spectra. The cross section of the DCH states in XTPPS can be approximately evaluated in a similar manner to the single ionization [24,39,42]. We formulated the pole strength in XTPPS as in the single ionizations described in Ref. [24]. In XTPPS with sufficient

50

M. Tashiro et al. / Chemical Physics Letters 521 (2012) 45–51

Figure 4. The calculated satellite spectra of (a) C 1s2 DCH, (b) C 1s1 SCH (taken from Ref. [28], Intensity of the main line is normalized.), (c) C 1s1 O 1s1 DCH (singlet), and (d) C 1s1 O 1s1 DCH (triplet) for CO molecule. The XTPPS with photon energy being hv = 1 keV is simulated.

photon energy, sudden approximation may be valid and the contribution from conjugated process that is relevant only near the threshold can be regarded as small [39,41]. In XTPPS, the final DCH state can in principle be generated from numerous intermediate SCH states. Let the wavefunction of the J-th DCH state, WN2 , and the I-th J SCH state, WN1 , which correspond to the primary or the satellite I state in the representation of configuration interaction (CI) respectively, be denoted as

WN2 ¼ J

X

C pJ UN2 p

ð5aÞ

BqI UN1 q

ð5bÞ

p

WN1 ¼ I

X q

where UN2 and UN1 are N  2 and N  1 electron configurations, p q respectively, and C pJ and BqI are CI coefficients. Then, the pole strength for the J-th DCH state generated from the I-th SCH state is given by

2   2 X    C pJ BqI spq   ¼   p;q N2

D E  S2JI ¼  WN2 jWN1 J I

ð6Þ

where the subscript of the bracket represents the integration over N  2 electrons and

D E spq ¼ UN2 jUN1 p q

N2

ð7Þ

:

Being similar to the single ionization Dyson orbital, the orbital of the J-th DCH state generated from the I-th SCH state is defined as

/JID

D E /JID ¼ S1 WN2 jWN1 JI J I

N2

¼ S1 JI

X

C pJ BqI spq :

ð8Þ

p;q

Then, the transition moment mJI from the I-th SCH state to J-th DCH state is given by

    E E D D   ¼ vJI ðk2 Þl/JID SJI mJI ¼ WN2 vJI ðk2 ÞlWN1 J I

ð9Þ

where vJI ðk2 Þ represents the wavefunction of the second outgoing electron in the generation of the J-th DCH state from the I-th SCH state. Let us focus on the pole strength of the DCH satellites in XTPPS. In the two photon DCH ionization, many possible routes exist for creating the DCH satellite states, as shown in Figure 3. Namely, from the primary SCH state jWc i, the electronic process described   by two-hole and one-particle generates the satellite state Wacci ,  a  while from the SCH state Wci , ionization from a core orbi satellite  tal also generates Wacci . The probability of generating the J-th DCH state is assumed to be proportional to the product of the probability of the first and second ionizations as ðN!N2Þ

PJI

ðN!N1Þ

 P0I

ðN1!N2Þ

 PIJ

 m2JI m02 I :

ð10Þ

where m0I is the transition moment of the ionization from the neutral ground state to the I-th SCH state. In XTPPS, the kinetic energy of outgoing second electron represented by vJI ðk2 Þ depends on the initial I-th SCH state so that the peak of the second electron appears in all possible routes. Here, we consider the contribution only from the P0 -th SCH primary state. Then, the intensity ratio of the J-th DCH satellite peak to the P-th DCH primary peak is given by

m2JP0 m02 m2JP0 IðJÞ P0  2 02 ¼ 2 : IðPÞ mPP0 mP0 mPP0

ð11Þ

  E   The approximation that the vJI ðk2 Þl/JID values for the primary and satellite states are close to each other leads to the intensity ratio,

S2JP0 S02 S2JP0 IðJÞ P0  2 02 ¼ 2 ; IðPÞ SPP0 SP SPP0

D

ð12Þ

M. Tashiro et al. / Chemical Physics Letters 521 (2012) 45–51

where S0P0 is the pole strength of the P0 -th SCH primary state,

D E  N1 N S02 P 0 ¼  WP 0 jW0

N1

2  

51

Observation of the tsDCH states using XTPPS should take proper account of this issue.

ð13Þ

and WN0 is the neutral ground state wavefunction. Note that the DCH primary state can be generated only from the SCH primary state provided that the de-excitation does not occur. In the present calculation, the pole strength of the DCH satellite spectrum in XTPPS was calculated by Eqs. (12) and (13). The final ionic-state correlation was considered, but the initial-state correlation was not included. 4.2. DCH spectroscopy in XTPPS and satellite spectrum Theoretical examination of DCH spectroscopy in XTPPS is performed for the C 1s2 and C 1s1 O 1s1 states of CO molecule. In Figure 4, the DCH satellite spectra associated with the C 1s2 and C 1s1 O 1s1 (singlet and triplet) states are compared with the SCH satellite spectrum in the case of XTPPS with the photon energy assumed to be 1 keV. The kinetic energies of the first and second electrons in XTPPS were calculated by Eq. (4). Theoretical SCH satellite spectrum was taken from our previous study [28]. Note that the intensities of the previous SCH satellite spectrum cannot be directly compared with those of the present DCH satellite spectra, because the previous calculations were performed with including both initial-state and final ionic-state correlations. The experimental XTPPS are available in Refs. [16,17] and one can interpret/compare them with the present results. As discussed in Ref. [18], the XTPPS measurement of the energy difference between SCH and DCH kinetic energies can provide the relaxation energy and the interatomic relaxation energy in DCH spectroscopy. In this regard, the ssDCH primary and satellite spectrum is located far from other spectra, which means that the measurement of the energy difference DEðS2 i Þ is possible by XTPPS. This indicates that both XTPPS and X-ray SR MECO can be utilized for extracting the relaxation energy. Actually, the relaxation energy was obtained experimentally for NH3 [13] in excellent agreement with the theoretical value. The energy region of the tsDCH primary peaks and its satellite spectra, on the other hand, partly overlap with the SCH satellite spectrum as seen in Figure 4. This implies that in a usual experiment an X-ray two-photon photoelectron spectrum is overlapped with a single-photon photoelectron satellite spectrum, so that the single-photon spectrum must be subtracted to obtain a reliable two-photon spectrum. This can be achieved, for example, by subtraction of a spectrum recorded at lower X-ray power density from the spectrum recorded at higher power density, as demonstrated in Refs. [16,17]. 5. Summary We have investigated the DCH satellite spectra of N2 and CO molecules theoretically. Both ssDCH and tsDCH satellite spectra are calculated by the RASCI method with the state-averaged CASSCF references. The calculated ssDCH satellite spectra of N2 (N 1s2) and CO (C 1s2) have satisfactorily reproduced the observed singlephoton satellite spectra and provided the first detailed assignment. DCH spectroscopy is also examined by taking account of the SCH and DCH satellite spectra for CO. The results demonstrate that in XTPPS the satellite spectrum associated with the C1s2 ssDCH state is located separately, while the tsDCH states and their satellite states may be overlapped with the SCH satellite spectrum.

Acknowledgements The authors thank Prof. P. Lablanquie for providing their experimental spectra in a numerical form and for helpful discussion. M. E. acknowledges the support from JST-CREST and a Grant-in-Aid for Scientific Research from the Japan Society for the Promotion of Science (JSPS). K. U. acknowledges the support for the X-ray Free Electron Laser Utilization Research Project of Ministry of Education, Culture, Sports, Science and Technology of Japan (MEXT). The computations were partly performed using Research Center for Computational Science, Okazaki, Japan. References [1] L.S. Cederbaum, F. Tarantelli, A. Sgamellotti, J. Schirmer, J. Chem. Phys. 85 (1986) 6513. [2] L.S. Cederbaum, F. Tarantelli, A. Sgamellotti, J. Schirmer, J. Chem. Phys. 86 (1987) 2168. [3] L.S. Cederbaum, Phys. Rev. A 35 (1987) 622. [4] E.M.-L. Ohrendorf, L.S. Cederbaum, F. Tarantelli, Phys. Rev. A 44 (1991) 205. [5] H. Ågren, H.J.A. Jensen, Chem. Phys. 172 (1993) 45. [6] D.A. Shirley, Phys. Rev. A 9 (1974) 1549. [7] N.D. Lang, A.R. Williams, Phys. Rev. B 20 (1979) 1369. [8] T.D. Thomas, J. Electron Spectrosc. Relat. Phenom. 20 (1980) 117. [9] R.G. Cavell, R.N.S. Sodhi, J. Electron Spectrosc. Relat. Phenom. 41 (1986) 25. [10] R. Santra, N. Kryzhevoi, L.S. Cederbaum, Phys. Rev. Lett. 103 (2009) 013002. [11] J.P. Cryan et al., Phys. Rev. Lett. 105 (2010) 083004. [12] L. Fang et al., Phys. Rev. Lett. 105 (2010) 083005. [13] J.H.D. Eland, M. Tashiro, P. Linusson, M. Ehara, K. Ueda, R. Feifel, Phys. Rev. Lett. 105 (2010) 213005. [14] P. Lablanquie et al., Phys. Rev. Lett. 106 (2011) 063003. [15] P. Lablanquie et al., Rhys. Rev. Lett. 107 (2011) 193004. [16] N. Berrah et al., Proc. Natl. Acad. Sci. USA 108 (2011) 16912. [17] P. Salén, P.v.d. Meulen, R.D. Thomas, H.T. Schmidt, M. Larsson, R. Feifel, M.N. Piancastelli, L. Fang, T. Osipov, B. Murphy, P. Juranic, N. Berrah, E. Kukk, K. Ueda, R. Richter, K.C. Prince, J.D. Bozek, C. Bostedt, S. Wada, M. Tashiro, M. Ehara, F. Tarantelli, J. Phys: Conf. Ser. In press. [18] M. Tashiro, M. Ehara, H. Fukuzawa, K. Ueda, C. Buth, N.V. Kryzhevoi, L.S. Cederbaum, J. Chem. Phys. 132 (2010) 183402. [19] M. Tashiro, M. Ehara, K. Ueda, Chem. Phys. Lett. 496 (2010) 217. [20] P. Linusson, O. Takahashi, K. Ueda, J.H.D. Eland, Phys. Rev. A 83 (2011) 022506. [21] O. Takahashi, M. Tashiro, M. Ehara, K. Yamasaki, K. Ueda, Chem. Phys. 384 (2011) 28. [22] O. Takahashi, M. Tashiro, M. Ehara, K. Yamasaki, K. Ueda, J. Phys. Chem. A 115 (2011) 022139. [23] L.S. Cederbaum, W. Domcke, J. Schirmer, W.V. Niessen, Adv. Chem. Phys. 65 (1986) 115. [24] A.D.O. Bawagan, E.R. Davidson, Adv. Chem. Phys. 110 (1999) 215. [25] M. Takahasi, K. Otsuka, Y. Udagawa, Chem. Phys. 227 (1998) 375. [26] N. Kishimoto, H. Yamakado, K. Ohno, J. Phys. Chem. 100 (1996) 8204. [27] K. Ueda et al., Phys. Rev. Lett. 94 (2005) 243004. [28] M. Ehara et al., J. Chem. Phys. 125 (2006) 114304. [29] M. Ehara et al., J. Chem. Phys. 124 (2006) 124311. [30] K. Kuramoto, M. Ehara, H. Nakatsuji, J. Chem. Phys. 122 (2005) 014304. [31] R. Sankari et al., Chem. Phys. Lett. 422 (2006) 51. [32] H.-J. Werner, P.J. Knowles, J. Chem. Phys. 82 (1985) 5053. [33] K.P. Huber, G. Herzberg, Molecular spectra and molecular structure, IV constants of diatomic molecules, Van Nostrand, New York, 1979. [34] T.H. Dunning Jr., J. Chem. Phys. 90 (1989) 1007. [35] H.-J. Werner, et al., MOLPRO, version 2008, a package of ab initio programs. [36] L.A. Morgan, J. Tennyson, C.J. Gillan, Comput. Phys. Commun. 114 (1998) 120. [37] T. Ishihara, K. Hino, J.H. McGuire, Phys. Rev. A 44 (1991) 6980. [38] K. Hino, T. Ishihara, F. Shimizu, N. Toshima, J.H. McGuire, Phys. Rev. A 48 (1993) 1271. [39] R.I. Martin, D.A. Shirley, J. Chem. Phys. 64 (1976) 3685. [40] L. Ungier, T.D. Thomas, Phys. Rev. Lett. 53 (1984) 435. [41] J. Schirmer, M. Baunstein, V. Mckoy, Phys. Rev. A 44 (1991) 5762. [42] T. Åberg, Phys. Rev. 156 (1967) 35.