An ab initio study of the electronic spectrum of Zn2 including spin–orbit coupling

Chemical Physics 311 (2005) 35–44 www.elsevier.com/locate/chemphys

An ab initio study of the electronic spectrum of Zn2 including spin–orbit coupling K. Ellingsen a, T. Saue a, C. Pouchan b, O. Gropen a b

a,*

Department of Chemistry, Faculty of Science, University of Tromsø, N-9037 Tromsø, Norway Laboratoire de Chimie Structurale, UMR 5624, I.F.R. rue Jules Ferry, F-64000 Pau, France Received 3 May 2004; accepted 30 September 2004 Available online 11 November 2004

Abstract The ground state as well as low-lying excited states of the zinc-dimer are studied using ab initio calculations. Spectroscopic constants and potential curves from all-electron, multi-configurational second-order perturbation calculations are compared to coupled cluster including triples corrections and averaged coupled pair functional results as well as available experimental data. Scalar relativistic effects are included through the use of the one-electron Douglas–Kroll operator. Spin–orbit coupling is accounted for perturbatively using the atomic mean-field spin–orbit operator. Both scalar relativistic and spin–orbit transition dipole moments are presented. The importance of correlation of 3d-orbitals is demonstrated, and recommended values of spectroscopic constants for the ground state are provided. Ó 2004 Elsevier B.V. All rights reserved.

1. Introduction The zinc dimer has been studied both experimentally [1–9] and theoretically [10–16]. Relevant reviews have been provided by Morse [17] and Koperski [18]. The interest is in part due to the possibility of laser applications in analogy with the rare gas dimers. Zn2 is an excimer with a shallow van der Waals ground state and low-lying bound excited states. Zn2 is also interesting from a theoretical point of view due to the different character of the ground and excited states and consequently the different methodological demands in the accurate theoretical description of the spectrum. The covalent contributions to the ground state bonding in the group 12 dimers has been investigated in [11,19]. It was concluded that the bond is a mixture of 3/4 van der Waals and 1/4 covalent interactions. Furthermore, *

Corresponding author. Tel.: +00 47 77 6447 82; fax: +00 47 77 6440 65. E-mail address: [email protected] (O. Gropen). 0301-0104/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.chemphys.2004.09.038

[11] provides spectroscopic constants at the CCSD(T) level correlating the 3spd4s electrons. A small-core pseudopotential and a large valence basis was used and the importance of triple excitations and the use of BSSEcorrection for describing the weakly interacting ground state potential was emphasised. Correlation of 3d-electrons has not been accounted for in most of the calculations of the low-lying excited states [10,14–16] which are studied at the multi-reference CI level. Scalar relativistic effects were included through the use of a large-core pseudopotential and core-valence effects were included by the means of a semi-empirical core-polarization potential in [13]. In this study our aim is to perform accurate all-electron calculations on the ground state as well as the eight lowest-lying excited states. We present non-relativistic as well as spin-free Douglas–Kroll results. The ground state is studied at several levels of theory: ACPF, CCSD(T) and CASPT2. The excited states are multiconfigurational and are studied at the MR-ACPF and CASPT2 level. We correlate 24 electrons including the

36

K. Ellingsen et al. / Chemical Physics 311 (2005) 35–44

3d-electrons and examine the effect of BSSE. Spin–orbit coupling is accounted for perturbatively diagonalizing a small matrix over spin-free CASSCF eigenstates. Assuming that the coupling between correlation and the spin–orbit interaction is negligible, correlation was added as a shift of the diagonal elements of the matrix. This procedure is reasonable for a first row transition element where spin–orbit coupling is expected to be weak. Spectroscopic constants are presented and compared to earlier works and available experimental data. CASPT2 excitation energies as well as spin-free and spin–orbit transition dipole moments are calculated.

2. Computational details Non-relativistic (NR), spin-free relativistic Douglas– Kroll (DK) as well as spin–orbit (SO) calculations were 2 2 performed on Zn2 in its 1 Rþ g ½7rg 7ru
ground state and the lower lying excited states using the MOLCAS-4 [20] and MOLCAS-5 [21] program packages. Calculations were carried out in D2h symmetry with the molecule aligned along the z-axis. Basis sets of the atomic natural orbitals (ANO) type from the MOLCAS library were employed in the non-relativistic calculations, i.e. a primitive (17s12p9d4f) basis contracted to [8s8p8d4f] [22] (ANO-S) and a (21s15p10d6f4g) primitive basis contracted to [9s8p7d5f3g] [23] (ANO-L). The primitive ANO-L basis was recontracted within the DK-scheme for use in the relativistic calculations. Correlation of the ground-state was handled through averaged coupled pair functional (ACPF) and coupled cluster including triples corrections (CCSD(T)) calculations correlating up to 24 electrons corresponding to the 3d and 4s electrons in the atomic limit. Due to the open-shell, multireference character of the excited states, the complete active space self-consistent field (CASSCF) approach was used to describe the ground state and all excited states on the same footing. We carried out CASSCF-calculations with 4 active electrons in 10 orbitals (7rg, 7ru, 8rg, 8ru, 4pgx, 4pgy, 4pux, 4puy, 9rg and 9ru) corresponding to the 4s, 4p and 5s orbitals for large internuclear separations (CAS3110). The core-electrons (1s–3d) were treated as inactive. In addition, calculations were performed using an active space augmented by the 10rg/u and 5pg/u orbitals corresponding to atomic 5p orbitals for large internuclear separations (CAS4220) as well. Dynamical correlation was included through multi-configurational second-order perturbation theory (the CASPT2-method [24]) and multi-reference ACPF calculations correlating 24 electrons. Spin–orbit integrals were calculated within the atomic mean-field approach [25]. The spin–orbit coupling was included perturbatively by diagonalizing a small matrix over spin-free CASSCF eigenstates using

the restricted active space state interaction (RASSI) program [26,27] in the MOLCAS-5 program package. The influence of higher lying states arising from the 3 P + 3P asymptote were investigated. The CASPT2 correlation energy was added as an energy shift to the diagonal elements of the Hamiltonian improving the description of the spin-free eigenstates. The influence of second order spin–orbit effects was studied employing the restricted active space configuration interaction (RASCI) program LUCIA [28] modified to include spin–orbit (SO) coupling [29]. A common set of average SCF spin-free orbitals were used for all states. A singles CI (CIS) calculation was performed in order to obtain individual spin-free states, and spin–orbit coupling were introduced perturbatively or variationally through spin– orbit CIS calculations. Spectroscopic constants were obtained using the MOLCAS program VIBROT fitting the potentials to an analytical form and solving the vibrational Schro¨dinger equation numerically. The RASSI program was employed to calculate dipole transition moments over the individually optimised DK-CASSCF-states using the CASPT2 correlation as an energy shift to the diagonal elements of the matrix. Transition dipole moments over the spin–orbit wavefunctions were calculated as well. Basis set superposition errors (BSSE) were estimated using the counterpoise approach [30].

3. General features The ground state of the Zn-dimer has a ½7r2g 7r2u
electron configuration essentially arising from the interaction of atomic 4s orbitals. The potential curve displays a shallow van der Waals type of minimum. Exciting electrons from 7rg and 7ru to the set of molecular orbitals spanned by the atomic 4p-orbitals gives rise to a manifold of states of which some are strongly covalent bound. An overview of the resulting spectrum was obtained performing an average SCF-calculation on the 4s4p-manifold and then resolving the individual electronic states by a complete open shell CI (COSCI) calculation. In D2h symmetry there is a total of 784 CSFÕs distributed on singlets, triplets and quintets; the states arising from the four lowest dissociation limits are displayed in Fig. 1. In this work, we concentrate on the nine states (ground state included) dissociating to the 1 3 3 lowest three asymptotes ð1 S þ 1 S : 1 Rþ Pg ; g; S þ P : 3 3 þ 3 þ 1 1 1 1 1 þ 1 þ Pu ; Rg and Ru ; S þ P : Pg ; Pu ; Rg and Ru Þ. Our COSCI-calculations show that the 1Pu state and the upper 1 Rþ g state are perturbed by interaction with corresponding states from the close-lying 3P + 3P asymptote; the resulting avoided crossings are displayed in Figs. 2 and 3. Analysing the excited states at the CASSCF level, we find that the Pg states are built from two main determi-

K. Ellingsen et al. / Chemical Physics 311 (2005) 35–44

37

3588.1

-3588.1

3588.15 3 P+3 P

Energy (a.u.)

-3588.3

Energy (a.u.)

1S+1 P

-3588.2

1 S+3 P

3P+ 3P

3588.2

1

S+ 1P

3588.25 -3588.4

3588.3

1

2

3

4

5

o

1 S+ 1S

-3588.5

1

2

3 o Bond-length (A)

4

5

Fig. 1. Potential curves for all electronic states of Zn2 dissociating to the four lowest asymptotes. All singlet states are given with solid lines, all triplet states with dotted lines and all quintet states with longdashed lines.

3588.18

Energy (a.u.)

3P+ 3P

1S+ 1P

3588.23

3588.28

1

2

3

4

5

o

Bond-length (A) Fig. 2. Avoided crossing between the two 1 Rþ g states at the COSCIlevel.

nants: 7r2g 7r1u 4p1u and 7r1g 7r2u 4p1g . Both states are bound with the former determinant dominating at shorter bond-lengths. The Rþ u states are bound as well with main contributions from 7r2g 7r1u 8r1g at shorter bond-lengths, while 7r1g 7r2u 8r1u is added to the description at longer bond-lengths. The 3Pu state is repulsive and built from the two main determinants 7r2g 7r1u 4p1g and 7r1g 7r2u 4p1u . The former determinant dominates at very short bond-lengths; at longer bond-

Bond-length (A) Fig. 3. Avoided crossing between the two 1Pu states at the COSCIlevel.

lengths the two determinants contributes with equal weight. Including dynamic correlation the potential curve possesses a shallow minimum. The 1Pu state has contributions from these two determinants as well, but has also important contributions from 7r2g 8r1g 4p1u . The 1 Pu is clearly bound and has furthermore a barrier of ˚ at the DK-CASPT2(3110) level. The 0.70 eV at 4.0 A 3 þ Rg is repulsive with contributions from 7r2g 7r1u 8r1u and 7r1g 7r2u 8r1g . The corresponding singlet ˚ which is bound and has a barrier of 0.53 eV at 4.3 A at the DK-CASPT2(3110) level has additional contributions from 7r2g 8r2g and 7r2g 4p2u . The contributions of the latter determinant are due to the interaction with its homologue from the 3P + 3P asymptote and is causing ˚ at the the hump in the potential curve at about 2.38 A DK-CASPT2(3110) level (see Fig. 4). There is also a small hump in the ground state potential curve at 3 ˚ , but this appears to be an artefact. A States dissociating to Zn+(2S) + Zn(2P) have the same symmetries as the four singlet and triplet states corresponding to the 1S + 1,3P asymptotes. Hence, all the 1S + 1,3P states may formally incorporate some degree of ionic character due to coupling to the Zn+Zn ion pair states. It should be remarked, though, that the Zn anion is not stable. The unexpected attractive wells of 1Pu and the upper 1 Rþ g has been explained by avoided curve crossings with the ion pair states [14]. The 7r2g 7r1u 4p1g and 7r1g 7r2u 4p1u determinants of the 1 Pu state can be assigned to both the 1S + 1P and the Zn+Zn dissociation limits whereas the 7r2g 8r1g 4p1u state arises from the covalent 3P + 3P asymptote. Using a simple bond-order argument we conclude that the 1Pu being bound cannot be explained by contributions from the ion pair states; the first two determinants are

38

K. Ellingsen et al. / Chemical Physics 311 (2005) 35–44

3589.2

3589.3 1Π u

Energy (a.u.)

1 + Σg 1 + Σu

3589.4

3

3 + Σg

Πu 1

Πg 3Π g

3589.5

3 + Σu 1 + Σg

3589.6 1.5

2.5

3.5

4.5

5.5

o

Bond-length (A) Fig. 4. Douglas–Kroll CASPT2 potential curves for the three lowest asymptotes.

non-bonding with bond-order zero while the determinant arising from the 3P + 3P asymptote is bonding with bond-order one. For the 1 Rþ the 7r2g 7r1u 8r1u g and 7r1g 7r2u 8r1g determinants have bond-order zero and can be assigned to both the 1S + 1P and the Zn+Zndissociation limits; the 7r2g 8r2g determinant has bond-order two and corresponds to the 3P + 3P asymptote. Accordingly, 1Pu and the upper 1 Rþ g being attractive is rather explained by the interaction with their 3 P + 3P homologues. 4. Spectroscopic constants 4.1. The ground state Several theoretical studies of the ground state of the zinc dimer exist in the literature [10–16]. A bound ground state was obtained only in [11–13]. The most accurate values of the spectroscopic constants are probably the CCSD(T) result of Yu and Dolg [11]. The authors correlated the 3spd4s electrons and reports the ˚ , De = 0.024 eV and xe = 22 cm1. values re = 3.959 A The authors furthermore analyse bonding in the ground state of Zn2 based on the concept of charge fluctuations and conclude that the bonding is mostly of van der Waals type, but with a 25% covalent contribution.

Experimental studies tend to derive spectroscopic constants under the assumption of pure van der Waals bonding. In one experiment spectroscopic constants 3 were obtained from an analysis of the 0þ u ð Pu Þ 1 þ X0þ ð R Þ transitions of Zn excited in crossed molecu2 g g lar and laser beams [4]. The bond-length was estimated ˚ using the Morse approximation and to be 4.8 ± 0.07 A ˚ using the London dispersion relation. Recently 4.67 A the spectrum was reanalysed [6]. The values for the dissociation energy and vibrational constants were slightly modified to De = 0.035 eV and xe = 25.9 ± 0.2 cm1. However, it is pointed out in the new analysis that the Morse approximation gives too long bond-lengths and ˚ . The the authors provide a new estimate re = 4.19 A new value is estimated using the expression De ð1Þ= De ð2Þ ¼ C 6 ð1Þ=C 6 ð2Þ  R6e ð2Þ=R6e ð1Þ using accurate data from a reference molecule. This procedure does not only depend on the quality of the reference data, but also assumes the potential to be pure van der Waals. The covalent contribution to the bond is not accounted for, and the new estimate is therefore likely to be too long as well. An earlier experiment based on the statistical analysis of the line broadening in atomic Zn absorption spectra due to the interaction with surrounding Zn atoms gives a significantly higher dissociation energy De = 0.056 eV,

K. Ellingsen et al. / Chemical Physics 311 (2005) 35–44

˚ [8]. In the fitting procedure and a bond-length of 4.0 A the pairwise interaction potential was assumed to be of the Lennard-Jones type. However, the data from corresponding experiments on Cd2 were more extensive and the fitting could be done assuming no functional form of the potential. The analysis was then repeated assuming a Lennard-Jones potential and yielded identical results. As the bond in Cd2 is more covalent than in Zn2 [11,19], the use of a Lennard-Jones potential should be appropriate. Spectroscopic constants calculated at different levels of theory are displayed in Table 1. The potential curve is repulsive at the SCF-level, and inclusion of dynamical correlation is necessary to obtain the shallow minimum. BSSE is large using the ANO-S basis; correcting for BSSE increases the bond-length by approximately 10% and decreases the dissociation energy by more than 50%. The ANO-L basis set contains more diffuse functions as well as g-functions which reduces the error significantly, i.e. the BSSE-correction augments the bond-length by 1% and reduces the dissociation energy by 7%. Furthermore, the large basis set provides a stronger and shorter bond. The CCSD(T) calculations correlating 24 electrons ˚ , De = 0.030 eV and xe = 22.5 (3d4s) gives re = 3.96 A 1 cm . These values are in good agreement with the results of Yu and Dolg. Moreover, De and xe are close to the experimental values of [6]. The experimental dissociation energy of [8] is larger, but the bond-length is in better agreement with our results. It should be emphasised that the triples correction has a rather large effect on the coupled-cluster results, decreasing the ˚ (9%), increasing the vibrational bond-length by 0.37 A constant by 5.4 cm1 (24%) and the dissociation energy by 0.012 eV (40%) bringing the results closer to the experimental De and xe of [6]. It has previously been shown that triple excitations are important for weakly interacting systems as they couple inter- and intrasystem correlation [31]. Correlation of the 3d-electrons is important. Excluding the 3d-electrons from the correlation treatment gives a 7% longer and 37% weaker

39

bond at the CCSD(T) level. The results from the ACPF calculation gives a too long and too weak bond compared to CCSD(T). The T1-diagnostic from the coupled cluster calculations with s1 = 0.021 validates the use of single-determinant methods describing the ground-state. However, a multi-reference approach was employed to be able to describe the ground state and the excited states at the same footing. The CASPT2 method is performing rather well compared to CCSD(T). The bond-length decreases by ˚ and is probably too short. The vibrational con0.2 A stant increases with 1.6 cm1 and the dissociation energy increases by 0.004 eV compared to CCSD(T) and the De and xe values are well in line with experiment [6] as well. The multi-configurational treatment seems to be more flexible and able to include excitations which are important for the description of the van der Waals interaction. Furthermore, covalent contributions to the bond are described at the CASSCF-level by excitations from s- to ptype orbitals. Scalar relativistic effects as described by the Douglas– Kroll method are of minor importance. Spectroscopic constants changes insignificantly compared to corresponding non-relativistic values. Due to the disagreement of the experimental and theoretical bond-length, a more direct comparison with the þ experimental 3 Pu ð0þ X1 Rþ uÞ g ð0g Þ excitation spectrum [6] is useful. The DK-CASPT2 potential energy curves are used to calculate the experimentally observed transitions between the first (m = 0) and second (m = 1) vibrational levels of the ground state and ten vibrational levels of 3Pu. Transition energies are presented along with Franck–Condon transition rates in Table 2. The deviation between experimental and calculated energies is only 1.4–1.5%. This confirms that the disagreement between the theoretical and experimental bond-length could be due to the methods used in derivation of the experimental value. The potential energy curve being so flat makes it particularly sensitive to the chosen method. Based on this ˚ , De = 0.025–0.030 discussion, we consider Re = 4.00 A eV and xe = 22 cm1 to be our best estimate.

Table 1 BSSE-corrected spectroscopic constants for the ground state, uncorrected values in parentheses ˚) Basis Corr. electrons re (A

xe (cm1)

De (eV)

NR-CCSD NR-CCSD(T) NR-CCSD NR-CCSD(T) NR-CCSD NR-CCSD(T) NR-ACPF NR-CASPT2(3110) NR-CASPT2(3110) DK-CASPT2(3110) Experimental

14.2 (16.0) 16.6 (24.6) 13.8 (13.8) 18.5 (18.5) 17.1 (17.7) 22.5 (22.5) 18.0 (18.6) 15.0 (15.0) 24.1 (24.6) 24.3 (24.6) 25.9 ± 0.2 [6]

0.009 0.015 0.010 0.019 0.018 0.030 0.018 0.013 0.034 0.034 0.035

ANO-S ANO-S ANO-L ANO-L ANO-L ANO-L ANO-L ANO-L ANO-L ANO-L

3d4s 3d4s 4s 4s 3d4s 3d4s 3d4s 4s 3d4s 3d4s

4.02 4.09 4.53 4.23 4.33 3.96 4.13 4.42 3.77 3.77 4.19

(4.39) (3.64) (4.58) (4.23) (4.29) (3.93) (4.09) (4.42) (3.75) (3.75) [6], 4.00 [8]

(0.020) (0.032) (0.010) (0.019) (0.019) (0.032) (0.019) (0,013) (0.037) (0.036) [6], 0.056 [8]

40

K. Ellingsen et al. / Chemical Physics 311 (2005) 35–44

Table 2 Rates and energies DE(m00  m 0 ) for vibrational 3 Pu Transition

Rþ g transitions calculated on the DK-CASPT2 level along with DG values

DK-CASPT2

Experimental [6] 00

0

1

0

m

Transition rate

DE(m  m ) (cm )

0

1

0.4290

32053.2

m

00

1

1

DE(m00  m 0 ) (cm1)

DG (cm )

DG (cm1)

32533.0 24.0

0

0

0.4821

32077.2

1

0

0.2737

32098.7

2

0

0.1291

32119.0

3

0

0.0590

32138.0

4

0

0.0274

32155.8

5

0

0.0132

32172.2

6

0

0.0066

32187.3

7

0

0.0035

32201.1

8

0

0.0029

32213.5

9

0

0.0011

32224.6

24.3 32557.3

21.5

18.9 32576.2

20.3

18.1 32594.3

19.0

17.0 32611.3

17.8

16.0 32627.4

16.4

15.0 32642.4

15.1

14.0 32656.4

13.8

13.1 32669.5

12.4

12.4 32681.6

11.1

11.0 32692.6

Experimental results are given as well.

3–10 cm1 higher at the DK level and dissociation energies deviates by 0.012–0.073 eV. For the 3Pu state which is essentially of repulsive character, the bond-length is slightly longer and the vibrational constant slightly lower at the DK level. Augmenting the active space in the CASSCF calculation (from CAS3110 to CAS4220) only small changes are seen in the spectroscopic constants and appear to result from more dynamical correlation. Comparing the CASSCF and CASPT2 results, we find that inclusion of dynamic correlation is crucial. Correlating 24 ˚ shorter, electrons, the CASPT2 bond-lengths are 0.2 A 1 the vibrational constants, 20–53 cm higher and the dissociation energies 0.06–0.6 eV higher than the corresponding CASSCF results. As for the ground state, the

4.2. The excited states Calculated spectroscopic constants are displayed along with results from other theoretical works as well as available experimental values in Tables 3–5. The displayed spectroscopic constants are calculated at the Douglas–Kroll level and thereby includes scalar relativistic effects. The innermost part of the 1 Rþ g curve was discarded in the determination of the spectroscopic constants due to the strong interaction with its homologue of the 3P + 3P asymptote. DK-CASPT2 potential curves are displayed in Fig. 4. Comparing DK and nonrelativistic CASPT2(3110) calculations correlating 24 electrons we find that bond-lengths are relativistically ˚ , vibrational constants are contracted by 0.02–0.04 A Table 3 ˚ for the lower-lying excited states Bond-lengths in A

DK-CASSCF(3110) DK-CASSCF(4220) NR-CASPT2(3110) DK-CASPT2(3110) DK-CASPT2(3110) DK-CASPT2(3110) + BSSE DK-MRACPF Ref. [14] Ref. [15] Ref. [13] Ref. [10] Experimental

Corr. electrons

3

– – 3d4s 4s 3d4s 3d4s 3d4s

2.53 2.51 2.36 2.49 2.32 2.33 2.35 2.56 2.53 2.38 2.41

Pg

3

Rþ u

2.66 2.67 2.51 2.67 2.47 2.48 2.50 2.74 2.70 2.59 2.70

1

3

2.47 2.46 2.33 2.45 2.30 2.30 2.33 2.51 2.48 2.38 2.33

Diss. 5.25 3.97 4.56 4.01 3.99 4.11 Diss. Diss. 4.36 Diss. 4.49 [6]

Pg

Pu

3

Rþ g

Diss. Diss. Diss. Diss. Diss. Diss. Diss. – Diss. Diss. Diss.

1

Rþ u

2.90 2.86 2.64 2.88 2.63 2.64 2.69 2.97 2.92 2.64 3.22 3.30 [8] ˚ [6] 3.0 A

1

Pu

2.58 2.46 2.42 2.57 2.39 2.40 2.42 2.64 2.64 ca. 2.65 2.40

1

Rþ g

3.03 2.99 2.75 3.00 2.73 2.74 2.92 3.07 – ca. 2.65 3.05

K. Ellingsen et al. / Chemical Physics 311 (2005) 35–44

41

Table 4 Vibrational constants in cm1 for the lower-lying excited states

DK-CASSCF(3110) DK-CASSCF(4220) NR-CASPT2(3110) DK-CASPT2(3110) DK-CASPT2(3110) DK-CASPT2(3110) + BSSE DK-MRACPF Ref. [14] Ref. [15] Ref. [10] Experimental

Rþ u

Corr. electrons

3

3

– – 3d4s 4s 3d4s 3d4s 3d4s

180 189 225 222 233 231 220 175 192 211 223 ± 5 [1]

169 166 193 196 197 200 208 150 175 169 161 ± 5 [3]

Pg

1

3

Pg

207 213 244 212 253 250 244 202 210 212

3

Pu

Diss. 7.53 23.5 24.5 22.9 22.6 32.4 Diss. – Diss. 20.3 ± 0.2 [6]

Rþ g

1

Diss. Diss. Diss. Diss. Diss. Diss. Diss. Diss. Diss. Diss.

Rþ u

112 122 147 121 137 131 121 107 134 77 122 ± 10 [7]

Rþ g

1

1

177 182 204 256 214 211 205 166 178 175 148 ± 6 [2]

123 116 113 118 116 57.8 104 104 – 112

Pu

Table 5 Dissociation energies in eV for the lower-lying excited states

DK-CASSCF(3110) DK-CASSCF(4220) NR-CASPT2(3110) DK-CASPT2(3110) DK-CASPT2(3110) DK-CASPT2(3110) + BSSE DK-MRACPF Ref. [14] Ref. [15] Ref. [13] Ref. [10] Experimental

Rþ u

Corr. electrons

3

3

– – 3d4s 4s 3d4s 3d4s 3d4s

0.902 1.026 1.479 1.178 1.521 1.502 1.457 1.05 1.10 1.21 0.91

0.881 1.073 1.228 1.008 1.240 1.225 1.204 0.87 0.98 0.95 0.90

Pg

BSSE-correction brings about minor changes. We performed MR-ACPF calculations using the two-three most important CASSCF-determinants as references for each state. These calculations give spectroscopic constants in overall good agreement with CASPT2. For the weakly bound 3Pu state, the ACPF bond is longer, but stronger. Earlier theoretical works [10,13–16] show spectroscopic constants calculated at the MRCI-level correlating four valence electrons. A large-core relativistic pseudopotential including core-polarisation through a core-polarisation potential was employed in [13], while [10,14–16] used all-electron basis-sets. The bonds are generally shorter and weaker corresponding to our CASSCF results, indicating a deficient correlation description. Excluding the 3d-electrons from the CASPT2-calculations, we arrive at longer bond-lengths, lower vibrational constants and dissociation energies in agreement with [10,13–16] clearly demonstrating the need of explicit correlation of 3d-electrons. As discussed for the ground state, the data from the þ experimental 3 Pu ð0þ X1 Rþ uÞ g ð0g Þ excitation spectrum [4] has been reanalysed [6]. The difference in bond-length between the two states involved in the excitation spec˚ , yielding an expertrum are predicted to be 0.3 ± 0.03 A ˚ for 3Pu. As the value is imental bond-length of 4.49 A

Rþ g

1

3

3

2.684 2.632 2.680 2.572 2.744 2.713 2.694 2.42 2.43 2.26 2.35

Diss. 0.002 0.003 0.012 0.029 0.026 0.110 Diss. – 0.016 Diss. 0.027 [4]

Diss. Diss. Diss. Diss. Diss. Diss. Diss. Diss. Diss. Diss. Diss.

Pg

Pu

1

Rþ u

1.180 1.176 1.256 1.123 1.210 1.189 1.292 1.06 1.13 1.12 0.71 1.117 ± 0.025 [7] 1.300 [8]

1

Pu

1.044 1.020 0.830 0.888 0.763 0.734 0.718 0.83 0.66 0.63

1

Rþ g

0.604 0.427 0.253 0.416 0.180 0.60 0.204 0.44 – 0.32

based on the experimental ground state bond-length ˚ larger we expect it to be too large; it is 0.38 and 0.5 A than the MRACPF and the best CASPT2 results, respectively. However, the experimental difference in bond-length between the two states is in excellent agreement with the theoretical result. The experimental dissociation energy of 0.027 eV and the vibrational constant of 20.3 ± 0.2 cm1 is deviating from the CASPT2 results by 15% and 10%. ˚ The bond-length of 1 Rþ u was determined to 3.3 A and the dissociation energy to 1.3 eV from absorption ˚ longer and 0.09 eV higher spectroscopy [8]; this is 0.58 A than the CASPT2-result. The dissociation energy and the vibrational constant was also determined from a 1 þ bound-free emission spectrum ðX1 Rþ Ru Þ to g 1 1.117 ± 0.025 eV and 122 ± 10 cm [7]. The theoretical spectroscopic constants are in agreement with these results, the vibrational constant is within the range of the experimental value and the dissociation energy deviates by 0.05 eV. The difference in bond-length between the two states in the experiment is determined to ˚ in agreement with the theoretical 1.13 A ˚ 1.19 ± 0.02 A (CASPT2). Based on the experimental value for the ˚ is ground-state an experimental bond-length of 3.0 A obtained [6]. The result is as expected longer than the theoretical value.

42

K. Ellingsen et al. / Chemical Physics 311 (2005) 35–44

Experimental vibrational constants for 3Pg, 3 Rþ u and Pu are available from excitation and fluorescence spectra [1–3]. The 3Pg vibrational constant is in very good 1 agreement with experiment whereas for 3 Rþ u and Pu the deviation between theory and experiment is considerable i.e. 17% and 30% (best CASPT2 result compared to experiment), respectively. We want to emphasise that the coupling of the 1Pu and the 1 Rþ g with higher lying states leads to unregular potential curves, a situation where the Born–Oppenheimer approximation breaks down, and the determination of spectroscopic constants is questionable.

changes the excitation energies by 0.02–0.34 eV. Scalar relativistic effects are as important as correlation, the Douglas–Kroll results are 0.15–0.35 eV higher than the corresponding non-relativistic energies. The results are generally higher than earlier theoretical works, but in better correspondence with available experimental data [4,9]. It was suggested in [9] that the state found at 3.36 eV dissociate into 1S + 3P1. It has later been assigned to 3 Rþ u , but this is in poor agreement with our results as well as other theoretical results. The transition moments were computed using CASSCF wavefunctions shifting the diagonal elements by the CASPT2 correlation energy. Transition moments are not so sensitive to inclusion of dynamic correlation as transition energies and CASSCF values are usually good [21]. The shift only changes the transition mo˚ with 0–3%. ments at 2.65 A

1

5. Excitation energies and dipole transition moments Vertical excitation energies and dipole transition moments are given in Tables 6 and 7. The dipole transition moments are given for all transitions involving the ground state as well as the lowest lying excited states 3 Pg and 3 Rþ u . In possible laser applications as well as emission spectra interesting transitions would be from the covalently bound excited states to the ground state. Furthermore, due to the repulsive character of the ground state the excited states are often reached by excitations from the lowest-lying excited states. These states may also be reached via emission from higher excited ˚ as an estimate for the bondstates. We chose 2.65 A length for the ensemble of excited states and calculated the excitation energies at this bond-length. Correlation

6. Spin–orbit effects Spin–orbit coupled potential curves were obtained from calculations where spin–orbit coupling is introduced perturbatively over the DK-CASSCF(3110) optimised using the CASPT2 correlation energy as an energy shift to the diagonal elements of the SO-matrix. The effects of spin–orbit coupling are small, the spin–orbit curves are in general found to be parallel to the unperturbed curves with the exception of two spin–orbit induced avoided crossings.

Table 6 Vertical excitation energies at 5 a.u. State

Non-relativistic CAS(3110)

CASPT2(3110)

CAS(3110)

CASPT2(3110)

Rþ g 3 Pg 3 þ Ru 1 Pg 3 Pu 3 þ Rg 1 þ Ru 1 Pu 1 þ Rg

0.00 2.21 2.16 3.13 3.76 5.05 4.31 4.64 4.99

0.00 2.33 2.45 3.08 3.88 5.22 4.26 4.78 5.28

0.00 2.37 2.34 3.34 3.93 5.20 4.63 4.93 5.25

0.00 2.54 2.66 3.32 4.09 5.39 4.56 5.13 5.59

1

Douglas–Kroll

Experimental reference [10]

0.0 [9] 2.79 [9] 3.36 [9]

0.0 2.04 2.18 3.08

4.03 [4] 4.86 [9]

3.78 4.68 4.73

Table 7 Dipole-transition moments calculated using DK-CAS(3110) wave-functions ˚) Transition Bond-length (A h Pu jrjX Rþ gi 1 þ h1 Rþ u jrjX Rg i 3 h3 Rþ u jrj Pg i h3Pujrj3Pgi 3 þ h3 Rþ g jrj Ru i 3 jrj Pu i h3 Rþ g 1

1

1.59

2.12

2.38

2.65

2.91

3.18

0.513 2.780 0.472 1.505 2.571 0.055

0.627 3.072 0.487 0.997 2.385 0.682

0.603 3.195 0.432 0.950 0.281 0.358

0.533 3.258 0.358 0.838 0.535 0.293

0.433 3.262 0.278 0.698 0.558 0.254

0.326 3.237 0.204 0.547 0.533 0.217

K. Ellingsen et al. / Chemical Physics 311 (2005) 35–44 Table 8 Mixing of unperturbed states for the highest 1u state ˚) Bond-length (A 1u

0g 1g 2g 0+ g

589.26

589.31

33% 1Pu + 66% 3Pu 8% 1Pu + 92% 3Pu 93% 1Pu + 7% 3Pu(2) 94% 1Pu + 6% 3Pu(2) 89% 1Pu + 11% 3Pu(2) 64% 1Pu + 36% 3Pu(2) 21% 1Pu + 79% 3Pu(2) 7% 1Pu + 93% 3Pu(2) 86% 1Pu + 14% 3Pu(2) 100% 1Pu

1.59 1.85 2.12 2.25 2.65 2.91 3.18 3.70 3.97 4.23

589.36

Energy (a.u.)

43

589.41

589.46

589.51

589.56

589.61 1.5

2

2.5

3

3.5

4

Bond-length (Å) Fig. 5. Spin–orbit coupled gerade potential curves.

The fine-structure splitting of the 3Pg state into þ  0 g ; 1g and 2g can be seen in Fig. 5. 0g and 0g are þ 1 practically degenerate, 0g are 151–183 cm lower than the 1g state depending on the bond-length. The 1g–2g splitting is slightly larger, 155–191 cm1. The fine-structure splittings were determined experimentally to 130 and 170 cm1 for the 0g–1g and 1g–2g splittings, respectively [5]. A large avoided crossing is induced between the attractive 1g(1Pg) and the repulsive 1g ð3 Rþ g Þ. The potential well of the former state is reduced compared to the unperturbed case, and the latter potential becomes bound. The 3Pu state is split by spin–orbit coupling into four  states 0þ u ; 0u ; 1u and 2u displaying parallel curves. The 0þ g;

1 and the 1u–2u splitting 0þ g –1u splitting is 190–214 cm 1 180–172 cm depending on the bond-length. Furthermore, an avoided crossing is induced between the repulsive 1u(3Pu) component and the attractive 1u(1Pu) state at short internuclear distances (between 1.85 and 2.12 ˚ ). The 1u(1Pu) state couples to a higher but close-lying A 1u(3Pu dissociating to (3P + 3P) denoted 3Pu(2)) going towards longer internuclear distances as shown in Table þ 3 1 þ 8. A weak coupling of the 0þ u ð Ru Þ and 0u ð Pu Þ are ob˚ ˚ the lowest served between 2.12 and 2.25 A; e.g. at 2.12 A þ 3 1 þ 0u state has 94% Pu and 6% Ru character. Second-order spin–orbit effects were investigated by comparing results from the perturbative approach with results from single-CI calculations. In single-point calculations, the difference in excitation energies between the two approaches was for the gerade states 0.02–0.05 eV and for the ungerade states 0.01–0.02 eV. The spin–orbit transition moments are presented in Table 9 and are generally straightforward compared to the parent KR state values. However, spin–orbit coupling induces some weak singlet-triplet transitions. The xx coupled states have been assigned to the KR state

Table 9 Dipole-transition moments calculated using spin–orbit wave-functions ˚) Transition Bond-length (A 1 þ h1u ð1 Pu ÞjrjX0þ g ð Rg Þi 1 1 þ þ 1 þ h0u ð Ru ÞjrjX0g ð Rg Þi 1 þ h1u ð3 Pu ÞjrjX0þ g ð Rg Þi þ 3 þ 1 þ h0u ð Pu ÞjrjX0g ð Rg Þi  3 3 þ h0 u ð Ru Þjrj0g ð Pg Þi  3 þ h0u ð Ru Þjrj1g ð3 Pg Þi 3 þ 3 h0 u ð Ru Þjrj2g ð Pg Þi 3 þ 3 h1u ð Ru Þjrj2g ð Pg Þi h0uþ= ð3 Pu Þjrj0gþ= ð3 Pg Þi h1u(3Pu)jrj1g (3Pg)i h2u(3Pu)jrj2g (3Pg)i þ 3 1 þ h0þ u ð Ru Þjrj0g ð Pg Þi h1u(1Pu)jrj1g (3Pg)i  3 þ 3 þ h0 g ð Rg Þjrj0u ð Ru Þi 3 þ 3 þ ð R Þjrj1 h0 u ð Ru Þi g g

1.59

2.12

2.38

2.65

2.91

3.18

0.498 2.772 0.143 1.120 0.335 0.472 0.335 0.335 1.512 1.489 1.509 0.180 0.313 2.571 2.571

0.616 3.066 0.167 0.002 0.382 0.540 0.382 0.382 1.070 1.061 1.055 0.115 0.203 2.391 2.391

0.596 3.192 0.122 0.001 0.346 0.490 0.346 0.346 0.969 0.967 0.967 0.062 0,066 0.289 0.289

0.520 3.256 0.090 – 0.253 0.358 0.253 0.253 0.828 0.828 0.825 0.030 0.057 0.535 0.535

0.384 3.261 0.070 – 0.196 0.277 0.196 0.196 0.683 0.683 0.683 0.009 0.357 0.558 0.558

0.287 3.237 0.054 – 0.154 0.218 0.154 0.154 0.532 0.531 0.531 0.004 0.511 0.523 0.523

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K. Ellingsen et al. / Chemical Physics 311 (2005) 35–44

with highest weight. It should be noted that some of the states are considerably mixed with other KR states.

7. Discussion In this paper we have studied the nine lower states of the zinc dimer corresponding to the 1S + 1S, 1S + 3P and 1S + 1P atomic asymptotes. We find that the CASPT2 method performs well for both the ground state and the excited states. The ground state dissociation energy and vibrational constant are in good agreement with the CCSD(T) and experimental values. The best bond-lengths are considered to be the CCSD(T) value as well as the experimental value in [8]. The experimental bond-length derived in [6] is probably too long. The CASPT2 bond-length is somewhat shorter than the CCSD(T) value. The deviation between MRACPF and CASPT2 spectroscopic constants are small for the excited states. We obtain vibrational constants in very good agreement with experiment for 3Pg, 3Pu and 1Ru. The 3Pu and 1Ru bond-lengths are shorter than the experimental values. The experimental bonds are probably too long as they are calculated using the ground state bond length of [6] as a reference value. Available experimental dissociation energies are in accordance with the theoretical values. Furthermore, we want to emphasise the importance of correlating the 3d-electrons; excluding the 3d-electrons gives longer bonds. Correlation and scalar relativistic effects are equally important for the description of the excitation energies. The CASPT2 results are in overall agreement with available experiments. Spin–orbit effects are of minor importance. The spin–orbit potential curves are generally parallel to the spin-free curves. However, spin–orbit coupling induces two avoided crossings. One large avoided crossing is observed for between the attractive 1g(1Pg) and the repul3 sive 1g ð3 Rþ g Þ and another between the repulsive 1u( Pu) 1 component and the attractive 1u( Pu) state. CASSCF dipole transition moments are presented; spin–orbit coupling induces singlet-triplet transitions with small transition moments.

Acknowledgement K. Ellingsen acknowledges the award of a grant from Elf Aquitaine and is grateful to participate in the program of ‘‘co-tutelle de The`se’’ financed by the French government. This work has received support from The Research Council of Norway (Programme for Supercomputing) through a grant of computing time.

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