J. Phys. Chem. 1990, 94, 5543-5548
5543
Ab Inltlo Study on Dlgallane(4), Ga,H, Koop Lammertsma* and Jerzy Leszczyiiski Department of Chemistry, University of Alabama at Birmingham, UAB Station 21 9 PBS, Birmingham, Alabama 35294 (Received: November 1 , 1989)
The potential energy surface of digallane(4), Ga2H4,has been studied by ab initio molecular orbital theory at the MP2 level by using a Huzinaga valence triply {basis set supplemented with d-polarization functions. Six of the eight HF/3-21G* optimized structures were characterized as minima. Their bonding properties were studied by a topological electron density analysis. Of the ionic, covalent, and p-hydrido bridged species the tri- and bidentate Ga+[GaH4]- forms are the most stable isomers. However, the energy differences with the other isomers are less than 15 kcal/mol. The covalent Du isomer 5 is 3.5 kcal/mol less stable than 1. The head and tail complexes of :GaH with GaH3 (3 and 7), which are of similar energy, are ca. 13 kcal/mol higher in energy than 1. The potential energy surface of Ga2H4is similar to that of Al2H4 but differences are noted.
Introduction The recent experimental observation of Ga2H6' and Ga2H4Cl,Z illustrates the accessibility of small gallanes. Such species are of theoretical interest3because little is known about their structural properties and stabilities. In this paper we evaluate the still unknown Ga2H4by a b initio M O theory and make comparisons with the previously studied dialane(4), A12H4.4 Several experimental studies have reported on the tetrahalides of digallane and suggest these to be of a saltlike Ga+[GaX4]- (X = Br, C1, I) compo~ition.~Raman spectra of In2X4and AlInX4 also suggest ionic character? A recent study indicates A1CI2 to be a dimer in the condensed state but not under matrix isolation conditions.' MNDO calculations on AI2C147and ab initio studies on A12H44give an energetic preference for the tridentate saltlike structure which relates to conventional ionic molecules as Li+[CH,]? M+[AlH4]-, and M+[BH4]-(M = Li, Na).9 In contrast, the diborane(4) tetrahalides B2X4 (X = F, C1, Br) possess a classical covalent structure.1° Various theoretical calculations" (1) Downs, A. J.; Goode, M. J.; Pulham, C. R. J. Am. Chem. Soc. 1989, I l l , 1936. (2) Goode, M. J.; Downs, A. J.; Pulham. C. R.; Rankin, D. W. H.; Rob ertson, H. E. J. Chem. Soc., Chem. Commun. 1988, 768. (3) (a) Lammertsma, K.; Leszczyllski, J., submitted for publication. (b) Lammertsma, K.; Lesmyfiski,J. J. Chem. Soc., Chem. Commun. 1989,1005. (c) Liang, C.; Davy, R.D.; Schaefer H. F., 111. Chem. fhys. Lett. 1989, 159, 393. (4) (a) Lammertsma, K.; Giiner, 0. F.; Drewes, R. M.; Reed, A. E.; Schleyer, P. v. R. Inorg. Chem. 1989, 28, 313. (b) Zakzhevskii, V. G.; Charkin, 0.P. Chem. fhys. fen. 1982.90, 117. Charkin, 0.P. The Stability and Structure of the Gaseous Inorganic Molecules, Radicals and Ions; Nauka: Moscow, 1980. (5) (a) Honk, W.; Simon, A.; Gerlach, G. Z. Naturforsch. 1987,426,546. (b) For thermodynamicsof Ga$I ( x = 1,2; y = 1-6) systems, see: Bernard, C.; Chatillion, C.; Ait-Hou, A.; killel, R.; Monteil, Y.; Bouix, J. J . Chem. Thermodyn. 1988, 20, 129. (c) Beamish, J. C.; Wilkinson, M.; Worrall, I. J. Inorg. Chem. 1978, 17, 2026. (d) Beamish, J. C.; Boardman, A,; Small, R. W. H.; Worrall, I. J. Polyhedron 1985, 4, 983. (e) Hillel, R.;Ait-Hou, A,; Berthet, M. P.; Bouix, J. J . Raman Spectrosc. 1987, 18, 265. (f) Garton, G.; Powel. H. M. J. Inorg. Nud. Chem. 1957, 4, 84. (6) Randloff, P. L.; Papathdorou, G. N. J . Chem. fhys. 1980,72,992. (7) Olah, G. A.; Faroog, 0.; Farnia, M.; Bruce, M. R.; Clouet, F. L.; Morton, P. R.;Rakash, G. K. S.;Stevens, R. C.; Bau, R.;Lammertsma, K.; Suzer, S.;Andrews. L. J. Am. Chem. Soc. 1988, 110, 3231. (81 Schlever. P. v. R.: Tidor. B.: Jemmis. E. D.: Chandrasekhar. J.: Wurihwcin,*E. U.; Kos, A. J.; Luke, B. T.; Pople, J. A. J . Am. Chem. Soc: 1983. 105.484. (9) Bonaccorsi, R.;Scrwco, E.; Tomasi, J. Theor. Chim. Acta 1979, 52, 113. (IO) (a) Danielron, D. D.; Hedberg, K. J . Am. Chem. Soc. 1979, 101, 3199. Odom, J. D.; Saunders, J. E.; During, J. R. J. Chem. fhys. 1972.56, 1643. (b) Atoji, M.; Wheatley, P. J.; Lipscomb, W.N. J. Chem. fhys. 1957, 27, 196. (c) Ryan, R. R.;Hedberg, K. J . Chem. fhys. 1%9,50,4986. (d) Trafanos, L.; Lipscomb, W. N. J. Chem. Phys. 1958, 28, 54. (e) Durig, J. R.; Thompson, J. W.; Witt, J. D.; Odom, J. D. J. Chem. fhys. 1973, 58, 5339. (f) Danielson, D. D.; Patton, J. V.; Hedberg. K. J. Am. Chem. Soc. 1977, 99, 6484. (1 1) Vincent, M. A.; Schaefer 111, H. F. J. Am. Chem. Soc. 1981, 103, 5677. McKee, M. L.; Lipscomb, W. N. Ibid. 1981, 103, 4673. Dill, J. D.; Schleyer, P. v. R.;Pople, J. A. Ibid. 1975, 97, 3402.
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0022-3654/90/2094-5543$02.50/0
have shown a preference for a perpendicular H2B-BH2 (Du) but a recent study12 found a doubly H-bridged structure (C,) to be energetically competitive. Digallane tetrahalides (like the diindanes and dialanes) can be generated by Ga metal reduction of GaX3.5 These products are in equilibrium with GaX and GaX3. Digallane(6), the dimer of GaH,, has been made under matrix isolation condition by the LiAlH4 reduction of Ga2C12H4,'whereas the LiAlH4 reduction of AICl, (at higher temperatures) gives the a-AlH3 phase of alane.13 These synthetic approaches call attention to the structural and stability aspects of digallanes. The objectives of this study are to explore the potential energy surface of Ga2H4,to obtain more insight in the bonding characteristics of its various isomers, and to provide structural, energetic, and spectroscopic data.
Methods The geometries of the digallanes 1-8 (Figure 1) were optimizedI4within the indicated symmetries at the split valence 3-21G(*) basis set,15 which includes polarization functions for the gallium atoms, using Pople's GAUSSIAN 88 series of programs.I6 The structures were characterized by calculation of the harmonic vibrational frequencies. All positive eigenvalues of the 3-21G( *) force constant matrix indicates a minimum energy structure (Le. 1-4, 6, and 7), whereas one imaginary frequency characterizes a transition structure (i.e., 5); isomer 8 is a saddle point of second order. Single-point calculations were performed with the partially uncontracted [433111/43111/4*] Huzinaga basis set" supplemented with d-polarization functions for gallium and Pople's 6-31G** for hydrogen. The effects of all-electron correlation were estimated with Mdle~Plessetperturbation theory at m n d order, MPZ(ful1). Our final set of relative energies are corrected for scaled18 zero-point-energy differences. These energies are used in the text unless indicated otherwise. The HF/3-21G(*) geometrical parameters for 1-8 are given in Table I; the numbering (12) Mohr, R. R.;Lipscomb, W. N. Inorg. Chem. 1986, 25, 1053. (13) Finholt, A. E.; Bond, A. C.; Schlesinger, H. I. J. Am. Chem. Soc. 1947,69, 1199. (14) For an introduction to the methods employed see: Hehre, W. J.; Radom, L.; Schleyer, P. v. R.; Pople, J. A. Ab Initio Molecular Orbital Theory; Wiley: New York, 1986. (15) Dobbs, K. D.; Hehre, W. J. J. Compur. Chem. 1986, 7, 359. (16) Frisch, M. J.; Head-Gordon, M.; Schlegel, H. B.; Raghavachari, K.; Binkley, J. S.;Gonzalez, C.; DeFrees, D. J.; Fox, D. J.; Whiteside, R. A.; Secger, R.; Melius, C. F.; Baker, J.; Martin, R.; Kahn, L. R.;Stewart, J. J. P:; Fluder, E. M.; Topiol, S.;Pople, J. A. GAUSSIAN 88; Gaussian, Inc.: Pittsburg, PA, 1988. (17) Huzinaga, S.;Andzelm, J.; Klobukowski, M.; Radzic-Andzelm, E.; Sakai, Y.; Tatewski, H.Gaussian Basis Sets for Molecular Calculations; Elsevier: New York, 1984. (18) Pople, J. A,; Schlegel, H. B.; Krishnan, R.; DeFrcts, D. J.; Binkley, J. S.;Frisch, M. J.; Whiteaide, R. A. Int.J. Quantum Chem.. Quat" Chem. Symp. 1981, 15, 269. DeFrees, D. J.; McLean, A. D. J . Chem. Phys. 1985, 82, 333.
0 1990 American Chemical Society
Lammertsma and Leszczyfiski
5544 The Journal of Physical Chemistry, Vol. 94, No. 14, 1990 TABLE I: HF/3-21C(*) Optimized Ca,H, Ceo”‘es geometry 1, HGa-H,-Ga, C,,
Ga-Ga 2.5863
2, H2Ga-H2-Ga, C,
2.8470
3, H,Ga-H-Ga, C3,
3.5994
4, H2Ga-GaH2, DW 5, H2Ga-GaH2, D u 6, H,Ga+aH, C3, 7, H2Ga-H-GaH, C,
2.5167 2.5423 2.7074 2.5566
8, HGa-H2-GaH, C ,
2.2041
bond distances. Gal-H 1.6959 (GalH3) 1.5743 (GalH4) 1.7499 (GalH3) 1.5913 (GalH4) 1.8708 (GalH3) 1.6056 (GalH4) 1.6022 (GalH3) 1.6022 (GalH3) 1.6378 (GalH4) 1.6820 (GalH3) 1.5860 (GalH5) 1.8730 (GalH4)
TABLE 11: Total (-au) and Rtlative (kcal/mol) Energies of GIz& Isomers’ HF/3-21G//HF/3-21G structures total re1 NIMAG ZPE 16.94 1, c3u 3830.293 82 0.0 (0) 17.26 2, c , 3830.295 96 -1.3 (0) 16.26 39 c3u 3830.278 23 9.8 (0) 4, Dw 3830.296 39 -1.6 (0) 16.54 16.42 5, 0 2 6 3830.29295 0.6 (1) 16.30 6, c,u 3830.278 13 9.9 (0) 7, c, 3830.276 06 1 1.1 (0) 16.23 8, Ca 3830.24746 29.1 (2) 16.00 GaH, C, 1914.55027 GaH,, D3h 1915.705 19 Ga2H6,D2h 383 1.445 98
A Ga2-H 2.1447 (Ga2H,)
bond angles, deg
1.9536 (Ga2H,)
124.86 (H4GalH5)
1.7286 (Ga2H3)
1.6053 (Ga2H,) 2.3006 (Ga2H3) 1.6247 (Ga2H4) 1.5565 (Ga2H,)
HF/Huzinaga basis total re1 3845.18781 0.0 3845.19255 -3.0 3845.18002 4.9 3845.19482 -4.4 3845.191 71 -2.5 3845.177 08 6.7 3845.17342 9.0 3845.14083 29.5 1922.000 88 1923.16339 3846.34995
97.45 (H4GalGa2) 122.26 (GalGa2H3) 122.32 (GalGa2H3) 97.96 (GalGa2H3) 141.54 (GalGa2H4) 125.41 (H,Ga,H,) 164.30 (XGalGa2) 169.27 (GalGa2H,) 97.49 (HIXHS)
MPZ/Huzinaga basis total re1 3845.432 37 0.0 3845.43070 1.1 15.3 3845.401 97 3845.42628 3.8 3845.42231 6.3 3845.41007 14.0 3845.41 1 35 13.2 3845.391 33 25.8 1922.09608 1923.286 15 3846.567 4ob
final 0.0 1.3 14.7 3.5 5.8 13.4 12.6 24.9
‘The column NIMAG lists the number of imaginary frequencies; ZPE is the zero-point energy; final are the final relative energies corrected for zero-point energies, scaled by 0.9. Frozen core.
TABLE III: HF/3-21G(*) Vibrational H a d c Frequencies (cm-’) for GatHl Isomers‘ geometries frequencies 1,
c 3 u
2,
c,
3, c3u 4, D u
5,
D2h
6, C,”
7,
8,
c,
228 (e, 27); 246 (al, 48); 819 (e, 219); 856 (e, 93); 871 (al, 1410); 1592 (e, 72) 1734 (al, 250); 2012 (al, 269) 152 (bl, 12); 209 (a, 14); 438 (b2, 1); 723 (a2, 0);765 (al, 490); 806 (bl, 122) 1055 (b2, 464); 1145 (al, 1536); 1291 (b2. 0); 1569 (a,, 627); 1957 (bl, 242); 1963 (al, 122) 30 (e, IO); 137 (al, I); 658 (e, 10); 719 (al, 1034); 813 (e, 158); 1775 (al, 3363) 1900 (e, 315); 1939 (al, 64) 167 (bl, 0); 245 (al, 0);327 (e, 52); 575 (e, 57); 793 (b2. 613); 862 (al, 22) 1915 (b2, 464); 1921 (e, 309); 1939 (al, 0) 165[i] (a,,); 243 (ar, 0); 306 (bhr 32); 483 (b,,,, 201); 490 (b3,, 0);595 (ba, 0) 801 (blu, 660); 863 (ag, 0); 1918 (b,,,, 488); 1920 (b,,, 0); 1925 (bZu,645); 1941 (a8, 0) 142 (al, 7); 206 (e, 6); 481 (e, I ) ; 718 (al, 661); 810 (e, 129); 1828 (al, 804) 1897 (e, 268); 1925 (al, 15) 118 (a’, 16); 2249 (a”, IO); 271 (am, 188); 454 (a’, 3); 456 (a”, 3); 720 (a’, 695) 796 (a”, 78); 837 (a’, 294); 1643 (a’, 173); 1858 (a’, 609); 1972 (a”, 222); 1981 (a’, 81) 34O[i] (b2); 237[i] (a2); 366 (al, 0);555 (al, 14); 574 (bl, 32); 776 (a2, 0) 937 (b2, 470); 938 (al, 175); 1451 (al, 301); 1452 (b,, 479); 2066 (b2, 261); 2075 (al, 0)
‘The number in parentheses indicates the IR intensity in kM/mol.
scheme refers to the structure displayed in Figure 1. Throughout the text we refer to this numbering scheme, but we also label bridging hydrogens as Hb and terminal hydrogens as H,. The absolute and relative Ga2H, energies are listed in Table 11. Their 3-21G(*) harmonic frequencies are given in Table 111. The bonding properties of the HF/3-21G(*) optimized digallane(4) geometries 1-8 were investigated with Bader’s t o p logical electron density analysis.1g The one-electron density distribution p(r) was analyzed with the aid of its gradient vector field Vp(r). The zero-flux lines (Vp(r)-n(r) = 0) define the boundaries of the atomic basins in the molecular space. Extrema or critical points in p where p(r) = 0 are classified according to rank (A, # 0, Le., the three nonzero eigenvalues Ai(r) of the Hessian matrix p(r)) and signature or curvature (excess X i > 0 (19) (a) Bader, R. F. W. Acc. Chem. Res. 1985,IB. 9. (b) Bsder, R. F. W.; Npyen-Dang, T. T.Ado. Quantum Chem. 1981, I4,63. (c) Bader, R. F. W.; Nguyen-Dang, T.T.:Tal, Y . Rep. Prog. Phys. 1981, 44, 893.
over A, < 0). In a basin the electron density is a maximum at the nuclear position and is characterized as a (3, -3) critical point. Of particular importance are the (3, -1) or bond critical points rb (also termed saddle points) which have a minimum value in p(r) along the maximum electron density path connecting two nuclei (positive curvature) and two minimum values in p(r) (negative curvatures) in the two orthogonal directions. The (3, +1) or ring and (3, +3) or cage critical points coincide with the centers of rings and cages, respectively. The u and r character of a bond can be estimated from its elipticity t, which is defined as t = AI/A2 - 1; At and Xz are the two negative curvatures of the bond critical point. The sign of the Laplacian VZp(r) provides information where the electronic charge is locally concentrated (positive) or depleted (negative). The local total energy density H ( r ) characterizes a bond as covalent (H(r) < 0) or ionic (H(r) > 0). The characteristics of the electron density analysis for 1-8 are listed in Table IV. The contour maps of the Laplacian for the illustrative isomers 1-4,
The Journal of Physical Chemistry, Vol. 94, No. 14, 1990 5545
Potential Energy Surface of Digallane
TABLE I V Bood Properties for Ca2H4Isomers at HF/I21G(*)O struct
crit pt GalH4
A, -0.150 ).098 -0.024 0.001 -0.143 -0.047 -0.083 -0,027 -0.045 -0.137 -0.094 -0.140 -0.044 -0.044 -0.140 -0.044 -0.044 -0.130 -0.136 -0.020 -0.105 -0.144 -0.132 -0.029 -0.049 -0.158 -0.035
tYPe
GalH3
GaZH3
Cage GalH5 GalH,
GaZH3 Ring GalH, GaZH3 GaZH3 GalPs max GaZH3
GalPs max GalH4
GaZH3
GalGaz
Gal% GaZH4
GalGaz GalH, GazH3 GalGaz 'A's are in e.A-$; p(r) is in
V2p(r)
A1
A3
z
-0.150 -0.092 -0.013 0.001 -0.142 -0.035 -0.075 -0,029 -0.045 -0.134 -0.094 -0.136 -0.044 -0.044 -0.136 -0.043 -0.043 -0,130 -0.134 -0.020 -0.098 -0.144 -0.131 -0.008 -0.0 19 -0.157 -0.028
0.583 0.483 0.122 0.061 0.579 0.225 0.434 0.041 0.296 0.574 0.446 0.564 0.003 -0.001 0.567 0.005 -0.002 0.519 0.569 0.112 0.496 0.581 0.530 0.075 0.237 0.580 0.084
O.OO0
0.072 0.797 O.OO0
0.012 0.325 0.101 0.000 O.OO0
0.019 O.OO0 0.03 1
0.010 0.000 0.028 0.009 0.012 0.000 0.01 1 0.000 0.073 0.006 0.010 2.440 1.575 0.010 0.225
dr) 0.684 0.509 0.223 0.208 0.658 0.293 0.434 0.224 0.267 0.635 0.469 0.647 0.446 0.446 0.649 0.436 0.436 0.6 19 0.633 0.226 0.524 0.666 0.628 0.280 0.376 0.708 0.445
V2p(r) 6.821 7.050 2.040 1.523 7.104 3.442 6.653 1.025 4.951 7.290 6.207 6.962 -2.062 -2.156 6.998 -1.985 -2.139 6.297 7.229 1.723 7.071 7.051 6.43 1 0.913 4.062 6.390 0.513
H(r) -0.227 -0,076 -0.001 -0,029 -0,202 0.005 -0.025 -0.035 0.042 -0.180 -0.056 -0.194 -0.183 -0.185 -0.195 -0.177 -0.180 -0.169 -0.180 -0.013 -0.088 -0.210 -0.178 -0,060 -0,030 -0.252 -0.138
is in e.A-$; H(r) is in hartree.A-3. analysis lend support for these closed-shell interactions. The face complexed structure 1 has a surprisingly short Ga-Ga separation of only 2.586 A, but does not contain a bond path between these elements (Figure 2 and Table IV), even thou h it is of similar length as the covalent Ga-Ga bond of 2.517 in 4 and that of 2.442 A in metallic gallium.20 As expected, the Ga-Ga distance in the edge complexed structure 2 is larger and amounts to 2.847 A, again without a Ga-Ga bond path. In both isomers, but also in 3, the terminal Ga (or "Ga+") interacts only with the bridging hydrogens Hb of which there are three in 1, two in 2, and one in 3. In the case of the face- and edge-complexed structures these hydrogens are more strongly bonded to the tetracoordinated Gal atom. However, this distance increases from 1.969 A for 1 to 1.750 A for 2 to 1.871 A for 3, while simultaneously the Ga2-Hb distance decreases from 2.145 A for 1 to 1.954 A for 2 to 1.729 A for 3. In each case the distance between Gal and the terminal hydrogens H, is virtually constant with variations between 1.574 A for 1 and 1.606 A for 3. These geometrical parameters suggest (a) that structure 1 is a tight complex between a Ga+ and the face of a tetrahedral G a H i anion, (b) that the complexation of Ga+ to an edge of GaHC is less tight, but that the ionic strength between Ga+ and the H i s in 2 increases, and (c) that the bridging hydrogen in the corner complexed structure 3 *transfers" from the tetracoordinate to the terminal gallium. This analysis is similar to that presented earlier for the related dialane(4) which was supported by charge distributions and bonding characteristics (hybridization) obtained with Reed and Weinhold's natural population analysis.21 In the present case we employ Bader's topological electron density analysisI9to evaluate the bonding characteristics of the Ga+[GaH4]-complexes. Analysis of the Ga-Hb bonds clearly illustrates the closed-shell interactions in these complexes. This is evidenced by the values of the local energy density, H(r), for the Ga2-Hb bond critical points which are -0.001 for 1 and +0.005 for 2. These values sharply contrast the negative values for the Gal-Hb bonds of -0.076 for 1 and -0.025 au-A-3 for 2, which are indicative of covalent bonding. For comparison, the
1
3 G u
x
.I-
4
&d
4
HI
7
c.
8
G"
Figure 1. Digallane(4) GazH4structures 1-8. 6, and 7 are displayed in Figure 2.
Results and Discussion The various digallane(4) structures are grouped in three sets of isomers. The first set (1-3) will be discussed in terms of Ga+[GaH4]- complexes, the second set (4-6) are covalent structures, and in the final set we consider the hydrido-bridged structures 7 and 8. Ga+[GaH4]-Complexes. Conceptually, the structures 1-3 can be considered as composed of tight interactions between a Ga+ cation and the face, edge, and vertex of a tetrahedral GaH4- anion, respectively. The structural features and the electron density
(20) West. R. C., Ed. Handbook of Chemistry and Physics, 64th 4.; CRC: Boca Raton, FL, 1983. (21) (a) Reed, A. E.;Curtiss, L. A,; Weinhold, F. Chem. Reu. 1988,88, 899. (b) Foster, J. P.; Weinhold, F. J. Am. Chem. Soc. 1980.102,7211. (c) Reed, A. E.; Weinstock, R. B.;Weinhold, F. J . Chem. Phys. 1985.83, 7 3 5 . (d) Reed, A. E.; Weinhold, F. Ibid. 1985, 83, 1736.
Lammertsma and LeszczyAski
5546 The Journal of Physical Chemistry, Vol. 94, No. 14, 1990
I
U''
Figure 2. Display of the contour map of the Laplacian concentration of the HF/3-21G(*) charge density V2p (a) of 1 for the plane containing Gal, Gal, H3,and H,;(b) of 2 for the plane containing Gal, Gal, and the bridging hydrogens H,; (c) of 3 for the plane containing Gal, Ga,, H3, and H,; (d) of 4 for the plane containing Gal, Gal, and the terminal hydrogens H,; (e) of 6 for the plane containing Gal, Ga,, H3, and H,; (f') of 7 for the plane containing Gal, Gal, H,, and H,. Dashed lines in denote negative values of V z p and indicate regions where electronic charge is concentrated. The Laplacian concentrations are overlaid with the molecular graphs in the projected plane. Bond paths are indicated by heavy lines.
total energy densities for the critical points of the terminal hydro ens bonded to Gal vary between -0.227 for 1 and -0.202 au. -3 for 2. Therefore, the two sets of H(r) values indicate a decrease in the covalent character of the Gal-Hb bonds and a slight increase in the ionicity of the Ga2-Hb bonds for the structures 1 and 2, respectively. Similar trends in the smaller ellipticities for the Gal-Hb critical points versus those of the Ga2-Hb bonds point in the same direction. The higher energy isomer 3 deviates from 1 and 2 with its Gal-Hb bond of 1.871 A being slightly longer than the terminal Ga2-Hb distance of 1.729 A. The electron density analysis confirms the opposing character and shows the shorter Ga2-Hb bond to be of covalent character, while the bridging hydrogen is ionicly complexed to GaH3, Le., H ( r ) = +0.042 Accordingly, structure 3 would be a likely intermediate for the transfer of hydrogen from :GaH to GaH3 to form a saltlike bi- or tridentate digallane(4). Interestingly, all three Ga+[GaH4]-complexes are minima on the 3-21G(*) potential energy surface. This is in contrast with the related dialane(4) structures that showed the comer complexed isomer (3-like) to be a 6-31G* transition structure of second order.& However, because the smallest harmonic 3-21G(*) frequency is calculated to be only 30 cm-I, the identity of 3 may well change at higher levels of theory. Coualent Srrucrures. Similar to dib0rane(4),~IJ~ dialane(4): and the ethylene C2H42+dication,22also the most stable digallane(4) structure with a covalent Ga-Ga bond is of D u symmetry. This equilibrium structure 4 has a Ga-Ga bond distance of 2.5 17 A. Rotation around this bond to give the planar transition
w
structure 5 (D2h)results in a bond lengthening of 0.025 A which is caused by the repulsion between the two in-plane empty Ga p-orbitals. Evidently, the cross-hyperconjugative effect23in 4 is weak probably because of the diffuse character of the Ga 4porbitals. The electron density p at the center of the Ga-Ga bond (seealso below) amounts to 0.446 but the negative Vp value of -2.062 indicates a relative depletion of electronic charge at this bond critical point. This may be caused by a strong polarization of the gallium atoms toward the hydrogens which would explain a diminished cross-hyperconjugativeeffect. The effect is similar in magnitude to that in the corresponding dialane(4) which shows a lengthening of 0.016 A upon rotation of the 2.613 A AI-AI bond (6-31G*).4a The D r D Z hbond elongations in the tighter first-row isomers B2H4 and C2H42+are much larger and amount to 0.089 A (MP2/6-31G**)I2 and 0.155 A (HF/6-31G*),22respectively. Despite the relative depletion of electron density at the midpoint of the Ga-Ga bond in 5, its H(r) value is negative (-0.185 and indicates covalent bonding. Interestingly, this critical point in 4 is a maximum in p. Such maxima at nonnuclear positions are termed "pseudoatoms" or nonnuclear attractors.% However, the properties of this maximum and the two bond critical points that connect this nonnuclear attractor in 4 with the Ga nuclei are virtually the same; e.g., both have p values of 0.446 e-A-3, and their separation is only 0.048
A.
(23) This is defined as hyperconjugation occurring in two orthogonal planes, see: Lammertsma, K.; Schleyer, P. v. R.; Schwarz, H. Angew. Chem., Int. Ed. Engl. 1989, 28, 1321. (24) (a) Goner, 0. F.; Lammertsma, K. J . Am. Chem. Soc., in press. (b) Gatti, C.; Fantucci, P.; Pacchioni, G. Theor. Chim. Actu 1987,72,433. (c) Cao, W. L.; Gatti, C.; MacDougall, P. J.; Bader, R. F. W. Chem. Phys. Lett. 1987, 141, 380. (d) Bader, R. F. W.; Nguyen-Dang, T. T. Adu. Quuntum Chem. 1981, 14,63.
Potential Energy Surface of Digallane The Ga-Ga bond in the C3, structure 6 of 2.707 A is significantly longer (0.190 A) than that of 4 and even 0.121 A longer than the Ga-Ga separation in the face-complex4 structure 1. This suggests that the weak Ga-Ga interaction in the equilibrium structure 6 may be viewed to result from the ylide-like complexation of a :GaH Lewis base with a GaH, Lewis acid. This is supported by GaGaH bond angle of 9 7 9 , which shows that the GaH3 fragment is only 7.5O distorted from planarity. The ylide-like character of 6 is also evident from the electron density analysis. The bond critical point for the 2.707 A long Ga-Ga bond is displaced by 0.274 A toward the GaH, unit and the total energy density H(r) at this critical point has the low value of -0.013 a d q 3 . Evidently, even though the Ga-Ga bond in 6 is covalent, it is strongly polarized and supports the ylide-type donor-acceptor interaction. pHydrido Structures. The two digallane(4) structures 7 and 8 contain covalent Ga-Ga bonds and bridging hydrogens. Conceptually, the mono-bridged equilibrium structure 7 can be considered as a complex of :GaH and GaH,, but unlike 6,7 allows for back-donation from the Ga-H bonds of the GaH3 unit into an empty :Ga-H p-orbital. This is reflected in the bonding characteristics. The 0.140-A shorter Ga-Ga distance of 2.557 A in 7 has at its bond critical point also higher values in p and H(r) than for 6. The Ga distances to the bridging hydrogen H b of 1.682 and 2.301 A differ significantly. The latter Ga-Hb distance being indicative of electronic stabilization but too long for bonding is supported by the lack of a bond path between the ‘:GaH” gallium atom and Hk The shorter Ga-Hb bond of 1.682 A in 7 is similar in length to the 1.696 A Ga-Hb bonds in the ionic structure 1. The character of their bond critical points ( p , t , and H(r)) is remarkably similar and suggests an electronic interaction with the terminal gallium. Also the GaGaH bending of the terminal Ga-H bond by 38S0 from linearity is consistent with the electronic bonding model. A similar interpretation was used previously for the Clike dialane(4), which is also an equilibrium structure (HF/6-3 lG*). The increased atom polarizabilities and electropositivities of these heavier elements (A1 and Ga) may explain the absencet2of the analogous diborane(4) isomer. Of all digallane(4) isomers, the doubly hydrogen bridged structure 8 (C,) has by far the shortest Ga-Ga bond with a bond length of 2.204 A. While the covalent bond in the related isomer 4 is 0.313 A longer, both 4 and 8 have similar p and H(r) values for their nonnuclear attractor and Ga-Ga bond critical points, respectively. However, their V2p values are of opposite sign with for 8. This is expected since 8 r e g a positive value 0.5 13 resents the structure with maximum cross hyperconjugation-the bridging hydrogens are shared by both galliums. These hydrogens have bond paths to both Ga nuclei with bond critical points that have small negative H(r) values of -0.030 a d - , . This indicates that the bridging hydrogens have covalent, but very lar, bonds to the gallium atoms; the Ga-Hb distance is 1.873 Isomer 8 with all its hydrogens on one side of a plane through both gallium atoms is, however, a HF/3-21G(*) transition structure of second order. This is in contrast with the analogous dialane(4) isomer, which is a HF/6-3 lG* transition structure,&and the di-H-bridged B2H4 isomer, which is an equilibrium structure only 1.5 kcal/mol (MP2/6-31G**) less stable than the global minimum D2,, structure.12 Energies and Stabilities. The global digallane(4) minimum is the saltlike tridentate structure 1 with the bidentate isomer 2 being only 1.3 kcal/mol less stable. At the same correlated level of theory (MP2 with Huzinaga’s triple {basis set) the preference of 1 over the covalent isomer 4 is also small and amounts to 3.5 kcal/mol. In fact, at the H F level not only covalent 4 but also 2 are energetically favored over isomer 1. From Figure 3 is it clear that the influence of electron correlation is most pronounced for the energy separation between the bi- and tridentate complexes and the other structures. However, since the energy differences between the isomer 1-7 are within 15 kcal/mol, it is also evident that these energy variations are small, in particular when compared with the dialane(4) isomers where the energy differences between the same isomeric structures are within 25 kcal/mol.& For ex-
The Journal of Physical Chemistry, Vol. 94, No. 14, 1990 5547 SO1
15. 10.
I-
O.
-4 Figure 3. Plot of the relative energies (kcal/mol) vs basis set. All energies are relative to 1. Tot means the MP2/Huzinaga basis set adjusted by 0.9 scaled zero-point energies.
ample, in the latter case the ionic tridentate isomer (cf. 1) is 2 6 form (cf. 5) by 10.5 kcal/mol energetically favored over the 0 (MP4/6-31G** + ZPE). Also the stability difference between the two GaH, :GaH complexes 6 and 7 of 0.8 is smaller than in dialane(4) where the Clike form is less stable by 4.7 kcal/mol. The D w D z r Ga-Ga bond rotation in 4 (to 5) has a small barrier of 2.3 kcal/mol, which is slightly larger than the 1.5 kcal/mol (MP4/6-31GS* ZPE) AI-A1 barrier in the corresponding dialane(4).& For the related diborane(4) the barrier for the B-B bond rotation amounts to 21.6 kcal/mol (MP2/63 1G**).I2 The dissociation energy for 1to GaH and GaH3 is endothermic by 14.8 kcal/mol at the H F level (Huzinaga basis set) and 31.6 kcal/mol at the MPZ(ful1) level (same basis). For the related dialane(4) A12H4this dissociation energy at the H F and MP4/ 6-31G** levels is 23.8 and 36.2 kcal/mol, respectively. The hydrogenation energy for 1 to the diborane-like structure Ga2H6 amounts to 19.3 and 16.0 kcal/mol at H F and MP2 (frozen core) levels (Huzinaga basis set), re~pectively.’~The corresponding hydrogenation energy for A12H4 (cf. 1) to dialane(6) AI2H6 amounts to 39.2 and 24.9 kcal/mol at the H F and MP4/6-31G** levels of theory, respecti~ely.~~
+
Conclusions The important points from the present study can be summarized as follows: 1. The global Ga2H4 minimum is the saltlike tridentate structure 1. 2. The potential energy surface for digallane(4) is rather flat with all isomers (except 8) within 15 kcal/mol of the global minimum 1. 3. The relative stability order of digallane(4) isomers resembles that of the dialane(4) isomers, but the differences in the nature of the stationary points for 3 and 8 are noted. 4. The electron density analysis gives a clear description of the ionic, covalent, and hydrido-bridged species. 5. The hydrogenation and dissociation (to GaH and GaH3) energies are smaller for Ga2H4than they are for dialane(4). Since GaH,25GaH3,%and Ga2H6have been observed, also Ga2H4must (25) (a) Ginter, M. L.; Innes, K. K. J . Mol. Spcrrosc. 1961, 7,64. (b) Breisacher, P.; Siege], B. J. J . Am. Chem. Soc. 1965, 87, 4255. (26) (a) Wiberg, E.; Johannsen, T. Natunvfssenschaften 1941, 29, 320; Chemie 1942, 55, 38. (b) Wiberg, E.; Schmidt, M. Z . Naturforsch. 1952, IB, 517.
J . Phys. Chem. 1990, 94, 5548-5551
5548
be considered an experimentally viable species. Our structural, vibrational, and energetic data should assist experimental studies directed toward the search for digallane(4).
Acknowledgment. This work was in part supported by the
United States Air Force Astronautics Laboratory under Contract F046 1 1-86-K-0073. The Alabama Supercomputer Center is acknowledged for the generous allotment of computer time. Registry No. Ga2H,, 127065-46-7.
Methods for Flm)tnIl Untestrlcted Hartree-Fock Solutions and Multiple Solutions Peter Pulay* and Rui-feng Liu Department of Chemistry and Biochemistry, The University of Arkansas, Fayetteville, Arkansas 72701 (Received: November 1 , 1989)
The unrestricted Hartree-Fock (UHF) wave function is fundamental for the description of open-shell systems. However, finding solutions to the UHF equations is often not trivial, particularly in systems with an even number of electrons. In addition, the existence of multiple solutions is more widespread than assumed hitherto. On the basis of the theory of triplet instability, we describe methods that automatically find the existing UHF solutions. We also analyze the nature of multiple solutions. The method is illustrated with a number of examples: ozone, the nitrite anion, stretched LiH, dilithium, dioxygen difluoride, and nitrogen oxide.
I. Introduction There has been much recent interest in the unrestricted Hartree-Fock (UHF) theory.' UHF is the simplest theoretical model which is able to describe open-shell systems and systems with broken or partially broken bonds. Unfortunately, the UHF model frequently fails to describe a pure spin state and this may lead to artifacts. Spin projection applied to the U H F wave function is simple, but it gives rather poor results in the intermediate region (neither fully paired nor completely separated spins)? However, optimization of orbitals in the spin-projected UHF wave function, Le., the extended Hartree-Fock (EHF) model,3 describes nondynamical correlation effects very weK4 There has been a resurgence of interest in E H F t h e ~ r ybecause ~ . ~ of technical improvements which make the once formidable optimization problem'tractable.6 Similarly, there has been much recent interest in projected unrestricted Mlaller-Plessett theory.sv7-'0 Our interest in U H F theory derives from the unrestricted natural orbital-complete active space ( U N W A S ) method."J* This is full configuration interaction in the space of the fractionally occupied U H F natural orbitals. UNO-CAS is a low-cost alternative to the CAS-SCF method; the results are in general very close to CAS-SCF. A further advantage of the method is its "black box" character: it replaces the arbitrary selection of the active space in CAS-SCF theory by a well-defined scheme. In our opinion, it has several advantages over the extended HartretFock (EHF) model: it includes all spin couplings, and it does not suffer from the arbitrary inclusion of a small fraction of dynamical correlation" among the essentially doubly occupied orbitals. UNO-CAS gradients can be readily formulated12and efficiently evaluated.13 ( I ) Pople, J. A.; Nabct, R. K. J. Chem. Phys. 1951.22, 571. (2) Hehre. W. J.; Radom, L.; Schleyer, P. v. R.; Pople, J. A. Ab Initio Molecufor Ofbtral Theory; Wiley: New York, 1986; pp 65, 203. (3) Uwdin. P. 0.In QuwUum Thcory of Atoms, Mdmclcs, und the Solid State; &din, P. 0.. Ed.; Academic: New York, 1966; p 601. (4) Maya, I. Adv. Quantum Chem. 1980, 12, 189. (5) Yamaguchi, K.;Takahara, Y.; Fueno, T.; Houk. K. N. Theor. Chim. Acro 1988, 73, 337. (6) Handy, N. C.; Rice, J. E. J. Chem. Phys., in press. (7) Schlegel, H. B. J . Chem. Phys. 1986,844530. (8) Schlegel, H. B. J. Phys. Chem. 1988, 92, 3075. (9) Knowles, P. J.; Handy, N. C. J. Phys. Chem. 1988, 92,3097. (IO) Knowles, P. J.; Handy, N. C. J . Chem. Phys. 1988,88, 6991. (1 1) Pulay, P.; Hamilton, T. P. J . Chem. Phys. 1988.88, 4926. (12) Bofill, J. M.; Pulay, P. J . Chem. Phys. 1989, 90,3637. I
,
,
As discussed above, most black box methods for dealing with strongly correlated or open-shell systems rely on the U H F wave function. It is usually assumed that the latter can be simply obtained and that there is a unique solution for the ground state. Unfortunately, in many cases this is not true. In systems with an even number of electrons, there may not be a U H F wave function different from the restricted Hartree-Fock (RHF) one. Even if a UHF solution exists, it may be difficult to find it. The S C F iterative process has a tendency to collapse to the R H F solution: this behavior is difficult to distinguish from the absence of a true UHF solution. For instance, previous attempts to obtain a UHF solution for F202were reported to be unsu~cessful'~ in spite of the existence of solutions that are much lower in energy than the R H F one (see below).l3 The difficulties of finding UHF solutions detract significantly from the value of these supposedly "black box" methods. We have therefore developed a method that allows a more automatic determination of UHF wave functions and comes closer to the "black box" ideal. 11. Method
The techniques recommended by YamaguchiIs and Dewar et a1.I6 are useful to locate UHF wave functions, but they still require substantial experimentation. The essence of these methods (in the case of an even number of electrons) is to start the iteration with a wave function in which the highest S C F orbital li) is replaced by different orbitals for a and @ spins: $a
= cli)
+ sla)
48 = cli) - sla)
(1)
c = cos 6, s = sin 6
This introduces the necessary asymmetry in the starting wave function and is usually a good choice in small molecules. The recommended mixing angleI5J60 is s / 4 though this is often too large and may cause convergence to an excited tripletlike state. (!3) Liu, R.;Bofill, J. M.; Pulay, P. J . Chem. Phys., to be. submitted for oublication. (14) Newton, M. D.;Lathan, W. A.; Hehre, W. J.; Pople, J. A. J . Chem.
Phys. 1970, 52, 4064. (15) Yamaguchi, K. Chem. Phys. Lerf. 1975, 33, 330. (16) Dewar, M. J. S.; Olivella, S.; Rzepa, H.S.Chem. Phys. Lett. 1977, 47, 80.
0 1990 American Chemical Society
