Ab initio configuration interaction calculations for five states of ArHe+

423

Chemical Physics 111 (1987) 423-429 North-Holland, Amsterdam

AB INITIO CONFIGURATION INTERACTION FOR FIVE STATES OF ArHe + M.Z. LIAO ‘, K. BALASUBRAMANIAN Department Received

of ChemistT,

14 February

CALCULATIONS

*, D. CHAPMAN

Arizona State University,

and S.H. LIN

Tempe, AZ 85287, USA

1986

Ab initio configuration interaction calculations have been carried out on five doublet states of ArHe+ states (X 2E+, A*TI, B2E+, C *E+, D2fI) are found to be weakly bound. The calculated results experimental data.

1. Introduction The role played by the interaction potential with regard to the energy dependence of Penning and associative ionization processes in the rare-gas dimers has been the subject of considerable investigation. While it is well-known that, regarding collisions, coupling between the discrete and continuum states is stronger for short internuclear distance, the nature of the coupling is entirely a function of the interatomic potential for a given impact parameter and collision energy. Several early attempts to understand the Penning and associative ionization processes theoretically were, in fact, hampered by a lack of reliable, detailed interaction potentials [l]. Several authors [l-4] have obtained a range of potential forms for the Ar+ + He system based upon analyses of experimental scattering data. These methods have yielded an “average” potential for Ar+ + He; this potential does not account for the two *II and *2 states which arise from the separated atomic states, Ar+(*P) + He [5]. The “average” potential thus obtained is based upon

’ Permanent address: Department of Chemistry, University, Beijing, China. * Alfred P. Sloan fellow; Camille and Henry Teacher-scholar.

Tsinghua Dreyfus

0301-0104/87/$03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)

molecular ion. Five are compared with

analysis of the energy of the associative ionization cross section. Bell et al. [6], as well as other authors [7,8] have assumed the repulsive portion of the interaction potential between metastable helium atoms and ground-state rare-gas atoms to lie at very short internuclear distances, and further assumed a c6/r6 dispersion interaction at long distances to be the principal factor in determining ionization cross sections. Hausamann and Morgner [9] have suggested a model for the determination of potential curves for the heteronuclear rare-gas ions based upon a parameterization method. Thus, until recently, many of the methods for obtaining potential curves for the rare-gas dimers, including the ArHe+ system, have been indirect, relying upon the interpretation of experimental scattering data to obtain “average” potentials. In recent years, a more direct examination of the interaction potentials for ArHe+ has come about through ab initio potential calculations by Olson and Liu [5], Sidis and co-workers [lo], and Albat and Wirsam [ll]. Sidis and co-workers [lo] calculated, at the single-configuration SCF level, “quasi-diabatic” [12] potentials for the ArHe+ system from l-5 bohr. The results are displayed only graphically, however, precluding any quantitative comparisons with regard to spectrosopic constants, etc. Further, single-configuration SCF treatment may not B.V.

424

M.Z. Lmo et ui. / Ab initio CI

be adequate for these systems. Ab initio potential calculations including configuration interaction have been performed on ArHe+ by Albat and Wirsam [ll] and, more recently, by Olson and Liu [5]. Again, the “quasidiabatic” potential curves of Albat and Wirsam were shown only graphically. From a qualitative viewpoint, these authors [ll] show the A*II, B *C+ and D *Z+ states to be primarily repulsive with shallow minima at a very short distance for the A*II and B *Z’ states. The D *Z+ state is shown to be bound with double minima around 1 bohr. At such short distances it must be noted that the electronic states would correlate into coalesced atomic states. Olson and Liu [5] carried out a subsequent CI calculation and found the A*II state to be weakly bound (0.963 mhartree) at 5.65 bohr by a 4367-configuration (*II) CI calculation. Olson and Liu, however, did not consider higher-lying states of ArHe+ in that investigation. The major interest in the interaction potentials of ArHe+ has been, until recently, somewhat qualitative in nature. Workers studying both Penning and associative ionization processes have been concerned with features such as curve crossings, etc. to aid in understanding the mechanisms of these processes. Our effort here is to present ab initio calculations for five states of ArHe+ system with the intent of these curves being of some utility in the interpretation of experimental scattering data, as well as in the understanding of the bonding on heteronuclear rare-gas cluster ions.

2. Method of calculation The calculations described here are based upon the general method of relativistic configuration interaction calculations proposed by Christiansen et al. (131. This method has been found to be quite successful in calculating the spectroscopic properties of electronic states of diatomics containing very heavy atoms. Relativistic configuration interaction (CI) calculations have been performed on a number of rare-gas diatomics and ions, including Ar,, Ar:, Kr,, Kr:, Xe,, and Xe: [13-211. Balasubramanian and Pitzer [16] have recently reviewed the field of relativistic quantum chemistry.

culcularions forfive states of ArHe ’ Table 1 shows a few low-lying electronic configurations of ArHe+ and the A-s states arising from these configurations. In that table, the la, 2a, and 3a orbitals are primarily Ar 3s He IS and Ar 3p, orbitals while the 17 orbital is Ar p orbital. Since the ionization potential of Ar is lower than He, the lowest dissociation limit would be Ar+ (3s23ps, *p) + He(ls 2, ‘S). Thus, the electronic states arising from this dissociation limit would be the lowest-lying states. These states are the *E’ and *II states arising from the la22a23a1~4 and lu22u23u21~3 electronic configurations. The valence electronic configurations shown in table 1 were obtained by promoting the u and 71electrons into the hole in the 3u orbital. The Rydberg 411 and *II states arise from the promotion of one of the IT electrons into the lowest Rydberg orbital. Table 2 shows the dissociation limits of some of the important doublet states arising from these electronic configurations. The electronic states shown in table 2 are of considerable interest since their dissociation limits act as channels through which ionization processes could occur. The self-consistent field (SCF) calculations were performed using effective core potentials for the argon atom, which were obtained from numerical ab initio calculations of the atom. Gaussian fits of relativistic numerical potentials of Ar atom have been reported by Patios and Christiansen [23]. The valence shell of argon included in the SCF calculations is 3s23p6 outer shell. We employ a quadruple-zeta ST0 basis set for the He atom augmented by a single 2p polarization function. This basis set was reported by Clementi [24]. For the Ar atom, the basis set consisted of triple-zeta s and p STOs, and a single d

Table 1 Some h-s ArHe ’

states arising from electronic configurations

Configuration a)

X-s states

la22023011?r4 lo22023021$ lo22013021n4 lo’2023021n4 lo220230’o(Ry)ln3

*x+ *rl 22 + (II) *z+ (III) ?I, 2II(2)

a) Ry denotes a Rydberg orbital.

of

M. Z. Liao et al. / Ab initio CI calculations

Table 2 Dissociation Molecular

limits for some electronic

of ArHe +

atoms

Energy of the atoms (cm-‘) this work

Ar+(*P) + He(‘S) Ar+(*P)+He(tS) Ar(tS)+He’(2S) Ar+(*S)+He(‘S) Ar+(Ry*P, 3s23p44s)+He(‘S)

X ‘x1 ArHe+ A’II ArHe+ B2X+ ArHe+ C 2X+ ArHe+ D *II ArHe+

S

in Slater-type basis functions. are shown in parentheses

Ar

He

2.3942490 (3) 0.7326645 (3) 0.9OOOOO (3)

1.4532 2.7709 4.1000 0.5948 1.3100

P

2.235520 (3) 0.916870 (3) 0.771248 (4)

d

1.700

(4)

0.00 0.00 71177 108733 138245

into Ar 4s, 4p and He 2s. Electrons were also promoted into the Ar virtual d orbitals. Extensive single and double excitations were allowed for the leading reference configuration for every state. Limited single and double excitations were allowed for other reference configurations. For the B*Z+ state we included la22a3u5ula4 and lu22a23u261a2 configurations also as reference configurations since these configurations seem to be somewhat important for this state. At long distances the 50 orbital correlates into Ar 5p,. The mixing of the lower *E+ states with the upper states was found to be rather small for the excited ‘Z+ states. Thus, smaller arrays of single and double excitations were allowed from less important reference configurations for the *Z+(II) and *x+(111) states to keep the number of configurations small and to avoid convergence difficulties in the iterative CI diagonalization. The *II states included about fifteen reference configurations with the leading configuration being 1u22u2 3u217r3. Single and double excitations were allowed from these reference configurations. The total number of configurations thus included in our CI calculations are shown in table 4.

The Table 4 Total number

(1) (1) (1) (1) (2)

expt. [22]

0.00 0.00 71239 130245 140403

polarization function. This basis set was optimized by Balasubramanian [25] for ArCl+ calculation. The exponents were optimized for the ground state of the neutral Ar atom and are shown in table 3. We carried out self-consistent field calculations followed by multireference configuration interaction. An array of important reference configurations was first generated by distributing the nine valence electrons among the valence orbitals and Rydberg orbitals. The leading configurations of the three doublet sigma states which we consider, namely, X ‘Z+, B*JS+ and C *Z+ are, la* 2a21a43a, la22ala43a2 and l~r2a*1~~3a*, respectively. All these three states are valence states. In addition, a number of Rydberg states exist, some of which can interact, especially with the B and C states. For the X *Z+ state, we generated twenty-four reference configurations by promoting the electrons into valence and Rydberg orbitals. The Rydberg orbitals, included in reference configurations, correlated at long distance

Table 3 Optimized orbital exponents principal quantum numbers

425

states of ArHe’

Dissociated

states

forfivestates

of configurations

included

State

Total number of configurations

X2x+ A2II BZZ+ c 2x+ D2II

8147 8279 4513 6359 7215

in CI calculations

M. Z. Llao et ~1. / Ah initio CI calculations

426

3. Results and comparisons with spectra The calculated and experimental properties for the five states of ArHe+ are shown in table 5, where some of our calculated spectroscopic constants are compared with the constants measured experimentally by Dabrowski et al. [26]. The low-lying X ‘Z+, A211 and B 2Zf states are fairly well characterized, both experimentally and theoretically. Our r, values of 2.47 (X 22:,,), 2.89 (A211) and 2.77 A (B*2’) compare well with the experimental values. However, our calculated D, values for X 2Z+ and A*II states are higher than the corresponding experimental values [26]. The experimental D, value is calculated by extrapolating the B(22’) bands into Ar + He’ dissociation limit. The discrepancy between our calculated and experimental values could, in part, be attributed to this extrapolation. Earlier all-electron calculations of Olson and Liu [5] yield a D, value of 55 meV which is in very good agreement with the present calculations which yield a D, value of 58 meV. There could be small differential correlation errors in our CI calculations introduced as a result of the choice of our reference configurations. Although the magnitude of the spin-orbit splitting in Ar+ ion is small (178 meV), it could alter the D, value of the ground state. We therefore performed limited relativistic CI calculations on the $ state of ArHe+ in order to ascertain the effects of the spin-orbit mixing. Calculations were performed in the basis shown in table 3. The one-electron spin-orbit operator was obtained by differencing the relativistic effective potentials with respect to spin. Configurations were generated by allowing single and double excitations from the

Table 5 Spectroscopic State

constants

for fiuestatesof ArHe +

X 2Z+ and A211 reference configurations (lo2 respectively). These 2cJ2a1114 and la22a23a21a3, excitations resulted in 7145 configurations for X ‘Z+ state. The X ‘Z+ wavefunction was found to consist of 89% 2Z+ (la22u3u1~4) and 9% 211 (lu22u23u21n3) at rmin, while at 15.0 bohr the wavefunction was composed of 84% 2Z+ and 14% ‘II from the same configurations. Thus, while mixing of states is evident, we note that the change in composition of the wavefunction with internuclear distance appears to be small from r,,,,, to the near dissociation limit. To facilitate comparison between relativistic and non-relativistic calculations, identical calculations were performed in the absence of the spin-orbit operator. The effect of spin-orbit interaction on D, is found to be only 6 meV. The spin-orbit interaction lowers the D, by 6 meV. This is anticipated based on the fact that the spin-orbit contamination is not substantially different at equilibrium and long distances. The highest-lying states in the present calculation, the C ‘Z’ and D ‘II states, have not yet been well characterized experimentally. Thus, there is more room for study in this region. Our calculated potential curves are plotted in fig. 1. As seen in fig. 1, the B*Z+ state is more strongly bound in comparison with the X 2Z+ state (ratio of their D, is 2.33). Since the B*Z+ arises from the interaction of He+ with Ar, this suggests that He+ interacts with Ar much strongly in comparison with the interaction of Art with He. Table 6 shows our calculated vertical transition energies at the equilibrium geometry of the X *Z+ state. Our calculated values compare very favora-

for ArHe+ We (cm-‘)

G(A)

T, (cm-‘)

0, ‘) (meV)

this work

expt. [26]

this work

expt. [26]

this work

expt. [26]

this work

expt. [26]

X22+ l/Z AZrI

2.47 2.89

2.585 2.872

182 140

120 _

BZZ+ c *Iz+

2.11 2.94

2.695 _

218 125

245 _

4.60

0 _ 70088 _ _

51.5 40 143 36 33

32.48 19.09 170.47

DZrI

0 321 70734 130607 140784

‘) Refers to energy

46 difference

from curve minima

to 20 bohr.

_

M.Z. Liao et al. / Ab initio CI calculations for fioe states of At-He +

421

17.44

1240

17.36 1 16.18

16.1A

16.10 1 3

0.84

3 w 8.80 -

0.76-

0.72 -

8.60 0.04

0.0

- 0.04

- 0.08 ! 2.0

4.0

Fig.

6.0

8.0

12.0 R(bohr)

10.0

14.0

16.0

18.0

12.0

Potential energy curves for five states of ArHe+.

bly with the experimental values of Tanaka et al. [27] and Dabrowski et al. [26]. In addition, we list a few transition energies which do not appear to be described elsewhere. Our CI curves for the X *E’, A*II and B*Z+ states were also fit into empirical potentials of the form cd/r4 + c6/r6 + cs/r8. These curves for the

X *Z+ and A2 II states are shown in fig. 2. From the X *Z+ curve, we obtained a polarizability for He atom (aHe) of 0.23 A33;the experimental value [28] is given as 0.2 A3. The B*Z+ curve yielded a polarizability for Ar atom (a&,) of 1.53 A3 versus the experimental value [29] of 1.64 A3. Thus, the calculated and observed polarizabilities are in very good agreement.

M.Z. Liao et al. / Ah initio CI calculutions for

428

I

of ArHe ’ I

I

I

I

I

fioesmes

0.0 i

-.-‘-

from

Cl

c,/r4+c6/r6+c8/r~

0.08

A21-I 0.06

-.-.-

from

c,/r4+c6W

0.04

+cdP

f

h

2

Cl

0.0 2

1

t;

I

0.00

3.

-.

- 0.02

\ - 0.04

// Lb

4.0

I 6.0

8.0

I

I

I

10.0

12.0

14.0

R(bohr) Fig. 2. Potential

energy curves for ArHe+

4. Nature of the electronic states of ArHe +

The X ‘IS+ and A’II electronic states of ArHe+ arise from the interaction of Ar+(2P) with He(‘S). The la orbital of ArHe+ is, in great part, Ar 3s orbital. The 2u orbital is dominantly He 1s with Ar p, making a non-negligible contribution. Con-

showing

fit of CI curves to empirical

potential.

versely, the 3a MO is dominantly Ar p, with He 1s making a small contribution. The IT orbitals are non-bonding Ar p orbitals. At long distances the la, 2a and 3a orbitals correlated into Ar 3s He 1s and Ar 2p, orbitals. The B 2Z+ state results from the interaction of Ar(3s23p6, ‘S) + He+(ls’, 2S) states. The C 2Z+

M. Z. Liao et 01. / Ab initio CI culculutions for fioe states of At-He ’

Table 6 Vertical transition

Acknowledgement energies

for ArHe+

in meV

Transition

This work

Ref. [27]

Ref. [26]

B2Z++X2Z+ A*H+X’Z+ c*x++xZ~+ D’H+X’z+ B*Z+ + A*H C ‘Z+ + A*H D’H-A*II C 2Z+ t B2Z+

8770 40 16193 17407 8729 16152 17367 7423 8638

8695 _ _

8698 _ _

8505 _ _ _

8509 _ _ _ _

D’II+B’Z+

429

state arises from the interaction of Ar+(3s’3p6, *S) + He(ls2, ‘S). The D *II (*II( is a Rydberg state. This state is weakly bound, as seen from fig. 1. It resutls from the interaction of Art Rydberg 2P state arising from the 3s23p44s configuration with He (‘S). The calculated separation at 15.0 bohr of this state with respect to X 2Zf at the same distance should correspond to the separation of Ar+(*P, 3s23ps) and Ar+(*P, 2~~3~~4s). Our calculated splitting for these two states (140403 cm-‘) is m . g oo d a g reement with the corresponding experimental value (138245 cm- ‘). The X 2Z+ ground state is 98% *Z+(1a22a23a la4) at 4.75 bohr. This makeup remains almost the same for several distances. The A’II state is dominantly (97%) 211(la22u23u21a3). The B2Zi state is predominantly ‘C+ arising from the lu22u3u21a4 configuration. At 5.25 bohr the B *Z+ is made up of 89% *Zf(1u22u3u21a4), and 1% *Z~(1u23u2u(Ry)17r4), where u(Ry) is a mixture of Ar(3s), Ar(3p,) and He(2s), and He(2pz). This makeup remains alsmost the same at intermediate distances. The C 2Z+ state is mainly 2Z+ arising from lu2u23a21a4 configuration. This is expected since this state arises from the interaction of Art(3s’3p6, *S) with He(ls*, ‘S). The D *II state is predominantly *II arising from lu22u23uu(Ry)1~3 configuration. It is actually made up of three Cartesian configurations with the total CI population being 95%. The u(Ry) is a Rydberg orbital and is predominantly Ar(4s), but mixes with Ar(3pZ), Ar(3d) and He(s). At long distances this orbital dissociates into Ar+ (3s23p44s, *P) + He(ls*, ‘S).

SHL would like to thank the National Science Foundation for a partial support of this work. KB would like to thank the National Science Foundation for the support of this work through grant no. CHE 8520556. The authors would like to thank the referees for their invaluable comments.

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