5836
J . Phys. Chem. 1987, 91, 5836-5837
Theoretlcal Investigation of the Electronic States of He,' K. Balasubramanian,*+M. Z. Liao, and S. H. Lin Department of Chemistry, Arizona State University, Tempe, Arizona 85287- 1604 (Received: August 17, 1987)
Complete active space MCSCF calculations followed by second-orderconfiguration interaction calculations (SOCI) are carried out on the 2Bz and *A, electronic states of the He4+ion with a large Gaussian basis set. A number of geometries such as linear, rhombus, square, and tetrahedron are considered for these electronic states. Two nearly degenerate lowest electronic states are found. The ground state is found to be a linear 'ZU+electronic state. Another electronic state of ZBzu(Du) symmetry with an equilibrium geometry of rhombus is only 2.2 kcal/mol above the linear 'Eu+state. The acute angle of the rhombus structure is 26.6'.
1. Introduction The clusters of rare gas ions are of considerable interest in recent years.'-15 These cluster ions play an important role in interstellar clouds, nucleation, and phase transition phenomena. There are many questions related to these species. The nature of the lowlying electronic states including the ground state, their geometries, their binding energies, etc., remain uncertain for these ions. The He4+ion is found in the helium plasma even a t room temperat u r e ~ Large . ~ ~ ~clusters of He are also formed around positive ions of helium in liquid helium. It has been proposed (see ref 16) that possible formation of similar clusters accounts for some surprising results obtained during an attempt to measure the Lamb shift in muonic helium. The molecular emissions observed recently in a helium plasma are also attributed to small clusters of helium ions." To date, there is very little knowledge on the low-lying electronic states, their geometries, and binding energies of He4+. There are no ab initio calculations including correlation effects on the He4+ ion. Experimentally, there is also nothing known on the nature of the low-lying states of He4+ or their geometries. Both correlation effects and basis set sizes could play an important role for rare gas cluster ions as shown by our earlier investigation on ArHe'." The objective of this investigation is to carry out complete active space MCSCF (CASSCF) followed by second-order configuration interaction (SOCI) calculations employing a large (6slp/4slp) basis set. A number of different geometries such as linear, rhombus, square, tetrahedron, etc., are considered for the two low-lying electronic states of He4+. Section 2 describes our method of calculations while section 3 contains results and discussions. 2. Method of Investigation We start with basis set considerations. The van Duijneveldt's18 5s Gaussian basis set was augmented by another diffuse s function with an exponent of 0.068 43 1. To this we add a set of p-type polarization functions with an exponent of 0.7139. The resulting basis set can be described as a (6slp) basis set. This was contracted to a (6slp/4slp) basis set shown in Table I. We believe that the resulting basis is sufficiently large and flexible to minimize basis set superposition errors. First, multiconfiguration S C F (MCSCF) calculations are carried out to generate the orbitals for configuration interaction calculations. The MCSCF calculations carried out were complete active space MCSCF (CASSCF) calculations. In this method, the seven outer electrons are distributed in all possible ways among a chosen set of orbital space, referred to as the internal space. For He4+,the internal or active space included the strongest occupied orbitals (namely, the 1s orbitals of the He atoms) of the separated atoms. All calculations described here were carried out in the C , group. The internal space included 2 a l and 2 bz orbitals for the linear, rhombus, and square structures and 3 a l and 1 bz orbitals for the tetrahedral geometry. Configuration interaction calculations were carried out following CASSCF. The CI calculations made are second-order CI (SOCI) 'Alfred PSloan Fellow; Camille and Henry Dreyfus Teacher-Scholar.
TABLE I Gaussian Basis Set for He Atom (6slp/4slp) shell exponent contraction coeff S S
S
P
98.124267 14.768910 3.318825
0.007 576 0.054 836 0.220 765
0.874 047 0.244 564 0.068 430
1.o 1.o 1.O
0.713900
1.o
calculations. These calculations included all configurations in the CASSCF plus the configurations generated by distributing (i) six electrons in the internal and one electron in the orthogonal external spaces and (ii) five electrons in the internal and two electrons in the external spaces in all possible ways. The SOCI calculations included about 4669-6567 configurations depending on the electronic state and the orientation of the cluster. The CASSCF/SOCI calculations are carried out using the ALCHEMY II codes.19
3. Results and Discussions The results of our calculations are summarized in Table 11. As one can see from Table 11, two nearly degenerate states exist for He4+. At the CASSCF level of theory the 2Zu+linear state is only 0.41 kcal/mol below the rhombus structure. However, more accurate second-order configuration interaction calculations (SOCI) lower the 28,+state relative to the 2B2u(rhombus) state. Although our (6slp/4slp) basis set is quite large and flexible, further extension of the basis set by adding one more set of ptype functions and a set of d-type functions could improve the results
(1) Patterson, P. L. J . Chem. Phys. 1968, 48, 3265. (2) DeVries, C. P.; Oskam, H. J. Phys. Lett. A 1969, 29, 299. (3) Gusinow, M. A.; Gerber, R. A,; Gerardo, J. B. Phys. Rev. Lett. 1970, 25,'1248. (4) Johnson, A. W.; Gerardo, J. B. Phys. Reu. Lett. 1971, 27, 835. (5) Poshusta, R. D.; Zetik, D. F. J . Chem. Phys. 1968, 48, 2826. (6) Poshusta, R. D.; Haugen, J. A,; Zetik, D. F. J , Chem. Phys. 1969.51, 3343. (7) Geltman, S.Phys. Reu. 1953, 90, 808. (8) Russell, J. E. J . Chem. Phys. 1985, 83, 3363. (9) Collins, C. B.; Carroll, J. M.; Lee, F. W.; Cunningham, A. J. J . Chem. Phys. 1976, 28, 535. (10) Wadt, W. R. J . Chem. Phys. 1981, 38, 1030. (11) Liao, M. Z.: Balasubramanian. K.; Chapman, D. A,; Lin, S. H. Chem. Phys. 1987, 1 1 I, 423. (12) Vauge, C.; Whitten, J. L. Chem. Phys. Lett. 1972, 13, 541. (13) Liu, B.; McLean, A. D. J . Chem. Phys. 1973, 59, 4557. (14) Olson, R. E.; Liu, B. Chem. Phys. Lett. 1978, 5 , 537. (15) Illies, A. J.; Bowers, M. T. Org. Muss Spectrom. 1983, 18, 553. (16) Cohen, J. H. Phys. Reu. A 1982, 25, 1791. (17) Downes, L. W.; Marcum, S. D.; Wells, H. E. Phys. Rev. A 1986, 34, 401. (1 8) Van Duijneveldt, F. B. IBM Research Report 945, 1971. (19) The major authors of ALCHEMY I1 codes are B. Liu, B. Lengsfield, and M. Yoshimine.
0022-3654/87/2091-5836$01.50/00 1987 American Chemical Society
Letters
The Journal of Physical Chemistry, Vol. 91, No. 23, 1987 5831
TABLE 11: Geometries and Energies of Electronic States of He,+ at Various Levels of Calculations
state
geometry
22°C
linear
!-?-?-!
2B2u
rhombus 1
3
.:>
bond lengths,
CASSCF bond angles, deg
SOCI Ea
Hel-He2 He2-He3 He,-He2 He2-He3
2.020 1.072 2.380 1.071
Hel-He2-He,
180.0
0.0
He,-He,-He2
26.0
0.4
Hel-He2 He2-He,
1.470 2.079
Hel-He2-He,
90.0
Hel-He2 He2-He,
1.970 1.970
Hel-He2 He2-He,
2.620 3.705
bond lengths, 8, Hel-He2 He2-He3 He,-He2 He2-He,
1.650 1.130 2.380 1.095
24.3
Hel-He2 He2-Hel
1.477 2.089
37.6
Hel-He2 He2-He,
1.940 1.940
60.4
Hel-He2 He2-He,
2.560 3.620
bond angles, deg
E'
He,-He2-He3
180.0
0.0
He3-Hel-He2
26.6
2.2
Hel-He2-He,
90.0
12.1
2
4
2A,,
:qy: 3!-l
4
tetrahedral 4
/A>. 2
41.7
3
2B2u
1.-.2 square
3!-!,
Hel-He2-He4
90.0
Hel-He2-He,
90.0
56.9
'Zero energy for CASSCF is -10.640 21 1 hartree and for SOCI is -10.757 349 hartree. All other energies are in kcal/mol.
-
further since a difference of 2.21 kcal/mol is quite small. The ,Al, state with an equilibrium geometry of a square is the next state. The ,A, state with a tetrahedral structure and ,Bzu state with an equilibrium geometry of square are well above the ,I;,,+ linear state. The ,I;,,+ linear structure has a He-He central bond length of 1.13 A with the terminal bonds being much longer (1.65 A). The equilibrium bond length of He2+calculated by using the same level of theory is 1.10 A. Thus, the centeral bond length compares with the re of the 22u+ state of Hez+. The experimentalZore of the ground state of the weakly bound Hez is 2.97 A. Thus, the terminal bonds of the linear He4+could be considered as a 70% (He,') and 30% (He,) hybrid bonds while the central bond is close to a He2+ bond. The ,B2,, state with an equilibrium geometry of rhombus can be rationalized as follows. The short acute angle of the rhombus of 26.6O corresponds to one of the diagonal bond length of 1.095 A. This diagonal bond is similar to the Hez+ bond and the central bond of the linear He4+. Since the two diagonally opposite atoms form a He,+-like bond, the apex angle is quite acute to enable this interaction. The bond length of the side of the rhombus of 2.38 A corresponds to a 68% (He,) and 32% (He,+) hybrid bond. Thus, the four comer atoms are held together by both the diagonal interaction and the side interactions. For the linear structure (?Z,+), the terminal interactions are a bit stronger than the side interactions of the rhombus structure. The tetrahedral ,A, state and the square ,B2,, states are somewhat weakly bound complexes in comparison to the electronic states discussed up to now. The tetrahedral He-He bond is a 45% (He,) and 55% (He,+) hybrid while the square structure is a much weaker complex (thus higher in energy).
4. Conclusion In this investigation, we carried out complete actkve space MCSCF followed by second-order configuration interaction (SOCI) calculations of low-lying electronic states of He4+. A number of different geometries such as linear, rhombm, square, and tetrahedron were considered for the two doublet states. The ground state was found to be a linear ,Z,+ state; a rhombus ,B2,, state is only 2.2 kcal/mol above the linear state at the SOCI level of calculations. The calculated acute angle of the rhombus (26.6') is rationalized based on a short diagonal interaction which resembles a Hez+ bond. The linear structure has a shorter central bond and longer terminal bonds. The stability of the He4+ ion with respect to individual atoms and He+ is calculated to be 2.42 eV.
(20) Huber, K.;Herzberg, G. Constants of Diatomic Molecules; Von Nostrand: Princeton, NJ, 1979.
Acknowledgment. K.B. thanks the National Scienw Foundation for partial support of this work through Grant CHE8520556.
The atomization energy of the linear 22,,+ state of He4+ at the SOCI level (Le., He4+ He+(*S) 3He('S)) is calculated to be 2.42 eV, which is close to the calculated De value of He2+(2.36 eV) at the SOCI level of calculations. Thus, the stability of the lowest state of He,+ is predicted to be about 2.4-2.5 eV. It can be seen from Table I1 that electron correlation effects are quite significant for He4+. The bond lengths of the relatively weakly bound atoms ( k , He, and He2+hybrid bond lengths) are quite sensitive to electron correlation. For example, the terminal bond len th of the linear 2Zu+state dro s from a CASSCF value of 2.02 to the SOCI value of 1.65 . The bonds which are similar to the Hez+ bond are, however, not affected as much by higher order correlation effects. For example, the central bond length of the linear structure increases to 1.13 A from 1.07 A (CASSCF). The higher order correlations tend to delocalize the positive charge for the linear structure.
1
+
1
