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
Downloaded by UNIV OF MANITOBA on September 7, 2015 | http://pubs.acs.org Publication Date: July 1, 1990 | doi: 10.1021/j100377a026
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.' U H F is the simplest theoretical model which is able to describe open-shell systems and systems with broken or partially broken bonds. Unfortunately, the U H F 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 y because ~ . ~ 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 U H F 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
Methods for Finding UHF Solutions
The Journal of Physical Chemistry, Vol. 94, No. 14, 1990 5549
Downloaded by UNIV OF MANITOBA on September 7, 2015 | http://pubs.acs.org Publication Date: July 1, 1990 | doi: 10.1021/j100377a026
In larger molecules, or in more difficult cases, this technique frequently fails. The reason for this is that the orbitals to be mixed in eq 1 should be fair approximations to the fractionally occupied UHF natural orbitals which are not known in advance. In small molecules, they are often close to the HOMO and the LUMO, but in large molecules the latter may have incorrect symmetry or spatial form. This is particularly true for the virtual orbital (LUMO) which is often far too diffuse if large basis sets are used. Even though the active orbitals (which correspond closely to the fractionally occupied UHF natural orbitals in the final wave function) are not necessarily the HOMO and LUMO, they are generally close to the Fermi surface, say within a few tenths of a hartree (1 hartree N 4.359814 aJ). The theory of triplet instability"-22 can be used to develop a more satisfactory method for generating a good UHF starting wave function. The full theory" is too demanding computationally for generating a starting guess. A simplified version, restricted to a single orbital mixing, can be summarized as follows. The energy of the partial U H F wave function (with n - 1 doubly occupied orbitals, plus one orbital with CYspin and one with /3 spin), resulting from the orbital mixing in eq 1, can be expressed as
i 2 ~ 2 f u+ c4Jii + 2C2s2(Ji, - 2Kia) + s4J, EUHF 2 ~ 3 +
(2)
Here f denotes the Fock operator of the 2n - 2 core electrons, and J,, and K,, are standard Coulomb and exchdnge integrals. Introducing the notation x = s2, the minimum of the energy is obtained for x
-(fa,
-Ai - Jii + Ji, - 2Ki,)/(Jii + J, - Win + 4Ki,)-' (3)
Introducing the Fock operator of the closed-shell wave function
F = f + 251 - Ki
(4)
eq 3 takes the form
In order to get a real solution, x = sin2 6 must be positive. The denominator of eq 5 is always positive (see,e.g., ref 12). Therefore, triplet instability (Le., energy lowering relative to the RHF wave function) will only appear if (F, - Fii - JL- Kh) is negative. The UHF energy lowering corresponding to the mixing ratio in eq 5 is
The best method for the generation of a good starting UHF wave function would be the minimization of this expression with respect to both orbitals li) and la). This is, however, too expensive for the present purpose. We therefore recommend the minimization of only the numerator in eq 5 with respect to the virtual orbital la) and repeating this for a few higher occupied MOs li), say, for all li) within 0.3 au of the HOMO. The justification of this is that the denominator in eq 6 is a much weaker function of the virtual orbital la) than the numerator. It is easy to prove that both (Jii + J , - 2Jia) and Ki, are always positive,12 and therefore no cancellation occurs in the denominator. Moreover, an appropriate virtual orbital la) must have large differential overlap with li), Le., a large exchange interaction Kk This will be then approximately constant for all appropriate orbitals. In contrast, there is strong cancellation in the numerator. It is less easy to defend the choice of a fixed canonical MO for the HOMO-like orbital li). and indeed it seems that canonical MOs are usually too delocalized for our purpose. The advantage (17) Cizek, J.; Paldus, J. J . Chcm. Phys. 1967, 47, 3976. (18) Ostlund, N. S.J . Chcm. Phys. 1972.57, 2994. (19) Jordan, K. D.; Silky, R. S. Chcm. Phys. k z r . 1973, 18,27. (20) Yamaguchi, K.;Fuego, T.; Fukutome, H. Chcm. Phys. Lcrr. 1973, 22, 461. (21) BonadoKwkcky,V.;Koutscky, J. Thaw. Chim. Acfa 1975,36,149. (22) Yamaguchi, K. Chcm. Phys. k r r . 1975, 33, 330; 1975, 35, 230.
of working with a f m d orbital li) is that the minimization problem is equivalent to the determination of the lowest eigenvalue of the operator F - Ji - Ki - ciI in the virtual space. Here ci = Fii is the orbital energy of li), and I is the unit operator. The occupied orbital part of the spectrum is best eliminated by level-shifting them to a high positive value; this is better than projecting them out since projection introduces spurious zero eigenvalues. In A 0 basis, we have to solve the eigenvalue equation
AiC, = SC,X,
(7)
with
Ai = F - Ji - Ki - 91
+ dSDS
(8)
where C, represents the orbital coefficients of la), S is the A 0 overlap matrix, F is the Fock matrix of the closed-shell wave function, Ji and Ki are the Coulomb and exchange operators for the trial orbital li), D is the SCF density matrix in A 0 basis, and the level shift parameter d can be set equal to the lowest SCF orbital energy plus a large constant, say 10 hartrees. If there is only one occupied orbital, the above criterion is exact: the presence of a negative eigenvalue X signals that the SCF wave function is triplet unstable. In the usual many-electron case, a negative eigenvalue of Ai is a sufficient but not necessary condition of triplet instability. If, for a given orbital li), the matrix Ai is positive definite, this only shows that no energy lowering can be obtained by mixing li) with virtual orbitals. The wave function may be triplet unstable even if the instability test fails for all orbitals li) since there may be a particular linear combination of occupied orbitals for which the matrix Ai does possess a negative eigenvalue. Nevertheless, the matrix Ai in these cases will have low (
