*From*: <BETTENS (+ at +) MPS.OHIO-STATE.EDU>*Subject*: Spin contamination & AM1 "ROHF" versus UHF*Date*: Fri, 16 Dec 1994 14:10:00 -0500 (EST)

Dear Netters, I posted a number of questions to the CCL about spin contamination on the 14th November. Below is the original questions and a summary of the responses. I also posted on the 7th of December questions relating to the 5th question below. The responses to this posting are also summarized here. ====================================================================== 14th November Posting ====================================================================== I have a number of questions regarding the effects of spin contamination on total electronic energies and structures. My understanding of spin contamination is that unrestricted Hartree-Fock (UHF) wave functions are not eigenfunctions of the total spin operator, so the electronic wave function of interest can be contaminated by functions corresponding to states of higher spin multiplicity. This brings me to several of my questions: 1. Given that, (a) we have performed an ab initio study on an open shell molecule using a single-determinant wave function (i.e., variational), (b) we have found its theoretical equilibrium geometry for the ground electronic state, and (c) have projected out ALL contaminating higher spin multiplicity states along the way to the minimum energy geometric configuration. Will the total calculated electronic energy be the lowest possible calculated energy for the given basis set and Hamiltonian? 2. What can be said about 1, above, regarding the total calculated electronic energy when a perturbation treatment to the configuration interaction is introduced, e.g., MP4(SDTQ)? 3. What can be said regarding the total calculated electronic energy, if in the case of 2, above, condition (1c) is not fully met, i.e., some spin contamination remains while going to the optimum geometry. 4. Given that different spin states correspond to different equilibrium geometries, is an optimized structure, which had significant spin contamination in its electronic wave function all the way down to its "minimum" energy geometrical configuration, some kind of mixture of structures involving the state of interest and the different contaminating higher spin multiplicity states? 5. Regarding semiempirical calculations, in the paper of Novoa et al., Inorganica Chemica Acta, Vol. 198-200 (1992) 133, the heats of formation of some very large carbon clusters were calculated. In their paper the authors state: "For an odd-membered linear C_n with 13 <= n < 20, the AM1 calculations predict the triplet state to be more stable than the singlet, due mainly to the spin contamination of the UHF calculations." The calculated stabilities of these larger linear carbon clusters are not expected based on what is known for the smaller linear clusters where, for odd n (n > 1), the singlet states are more stable than the triplet states. My question is this, is it possible that the authors are not correct regarding their reason for the for greater stabilities of the triplet states? (It is not my intention to attack the above authors work. I merely wish to evaluate the quoted heats of formation because I require some kind of estimate for the heats of formation for these species.) Regards, Ryan Bettens, OSU Physics Department, BETTENS (+ at +) MPS.OHIO-STATE.edu ________________________ ----------------------| Responses to Questions |---------------------- ------------------------ From: Christopher J. Cramer, University of Minnesota, Department of Chemistry Re your queries on spin contamination. I can not speak to all of your questions, but one point I think worth making is that the UHF SCF procedure is variational in that it produces the unique orthogonal set of MO's that give rise to the lowest energy wavefunction for whatever basis set has been chosen. However, those MO's derive from the UHF calculation itself, which includes all spin contamination. Any projection operator formalism applied thereafter can provide you some (significant) improvement in energy because of the annihilation of higher spin states, BUT that procedure does NOT reoptimize the MO's. So, even if you optimize a geometry by hand using PUHF (or PMPn) energies (since gradients for these methods are not, to my knowledge, available) you will still suffer from potentially poor wavefunctions. Alternatives include ROHF (which I personally don't like because we tend to be interested in spin-polarization, which you can't get at this level), spin-adapted MCSCF treatments, and, something you might consider, spin-polarized DFT. The latter is in essence a UHF-like calculation using an approximate scheme for exchange/correlation (as all DFT does), but it has been shown to be far less prone to spin contamination. ---------------------------------------- From: Dave Ewing, John Carroll University, See warren J. Hehre, et. al., Ab initio Molecular Orbital Theory (John wiley & Sons, 1986), pp. 203-4. My own experience has been that structures are OK as long as the spin contamination isn't too gross, e.g. no more than <S2>=1.0 for a doublet. ---------------------------------------- From: I. Shavitt, Ohio State University, Department of Chemistry 1. The wave function obtained by projecting out all spin contamination AFTER having calculated a UHF solution is not the lowest-energy solution for the spin-state in question. This is so because the orbitals have been optimized for the spin-contaminated function, before projection, not for the projected function. 2. After the addition of a correlation treatment, like MP4(SDTQ), the effects of spin contamination would usually be diminished, though not eliminated. The effect on the energy cannot be predicted with confidence, because there are diverse factors determining how the choice of orbitals will affect the correlation energy. In fact, orbitals which give the lowest Hartree-Fock energy are not guaranteed to produce the lowest correlated energy, though they usually do. Coupled cluster methods, such as CCSDT, are less sensitive to the choice of orbitals, and therefore less likely to suffer from the use of nonoptimal orbitals. 3. See 2. 4. The effect of spin contamination on optimized geometries depends both on the degree of spin contamination and on the relative energies and characters of the higher-spin states. I don't think I can give a general answer to this question. 5. Personally, I am very skeptical about the ability of methods like AM1 to give definitive answers to questions concerning the relative energy of different electronic states. But the same is true, even more strongly, for UHF calculations. For some molecules it is very difficult to determine which structure or multiplicity is lower in energy. An example is C_4, for which a linear triplet state and a cyclic singlet are almost of the same energy, and different levels of even high-quality calculations give conflicting answers. My impression was that odd-membered carbon chains have a singlet ground state, and it would take much more than AM1 calculations to convince me otherwise. ---------------------------------------- From: Ab Initio Molecular Orbital Calculations for Chemists, 2nd ed., W.G. Richards and D.L. Cooper, Clarendon Press, Oxford, 1983. I found this comment in the above book after posting the above questions. The comment essentially answers my question 4. I quote this book here (from page 59) as it may be of convenience to people. The average interaction interaction for alpha-spin electrons and beta-spin electrons may be different in open-shell systems so that it is not unreasonable that orbitals differing only in their spin quantum number should have different spatial functions. This is the origin of the unrestricted Hartree-Fock (UHF) method implemented in some programs. Unfortunately, UHF wavefunctions are not always eigenfunctions of S^2. If for example we carried out calculations on a molecule with a doublet state and a quartet state that were close in energy, then the equilibrium geometry of the doublet state would be partly characteristic of that for the quartet state. Although there are methods for projecting out spin eigenfunctions at the end of the UHF calculation, many theoreticians prefer the restricted Hartree-Fock (RHF) method and this is implemented in many open-shell SCF programs. In the RHF method, orbitals differing only in spin quantum number . . . have identical spatial parts. ====================================================================== 7th December Posting ====================================================================== Thanks to those of you who responded to my last posting on the 14th of November regarding spin contamination. Before summarizing these responses I would like to ask one further question which is related to my last question in the previous posting. I will then compile all responses and post a summary. My question concerns ROHF versus UHF calculations using the AM1 semiempirical model. Is anybody aware of any published work which recommends the use of either in calculating heats of formation, specifically for hydrocarbon open shell species? I have examined the original paper of Dewar, M.J.S.; Zoebisch, E.G.; Healy, E.F. and Stewart, J.J.P., J. Am. Chem. Soc., 107 (1985) 3902 where AM1 was introduced. In this work the heats of formation of 6 hydrocarbon radicals (all doublets) where compared with the experimental values. I could find no indication as to whether the reported calculations used the UHF or ROHF approaches. This is important because the calculated heats for each approach produces significantly different results. I consequently repeated the calculations on the 6 hydrocarbon radicals given in the above paper and found the following heats of formation (kcal mol^{-1}). I also give the values of S^2 and the reported heats as well as the current experimental values (from J. Phys. Chem. Ref. Data, 17 (1988) Sup. 1) for comparison below. AM1 AM1 Reported UHF S^2 ROHF AM1 Exp. Exp. - Calc.(ROHF) CH3 29.95 0.7613 31.25 31.25 34.8(3) 3.6 C2H3 60.46 0.8589 64.78 64.78 63.4(10) -1.4 C2H5 15.49 0.7619 18.14 15.48 28 10 CH2CHCH2 30.20 0.9300 38.58 38.58 39 0 (CH3)2CH 3.61 0.7622 6.80 10.07 22.3(6) 15.5 (CH3)3C -6.14 0.7623 -2.66 -2.66 11.0(6) 13.7 For C2H5 the authors seem to have reported the UHF value. I was not able to reproduce the reported result for (CH3)2CH with either the ROHF or the UHF result. I was able to reproduce the calculated heat of formation for this species given by Dewar, M.J.S. and Thiel, W., J. Am. Chem. Soc., 99 (1977) 4907, using the MNDO Hamiltonian. In this earlier work the authors explicitly stated that they used ROHF heats of formation calculated with MNDO. It is noted that for the remaining species the reported AM1 heats of formation also appear to be the ROHF values. While agreement with the experimental values are not terrific and appear particularly bad for the larger saturated hydrocarbon species, they are adequate for my purposes given that these errors can be taken as typical for calculated heats of formation of analogous but larger open shell hydrocarbon radicals. Does anybody know of any further studies, using the AM1 Hamiltonian, on radicals where the calculated heats of formation have been compared with experiment? It seems to me, from the above table, that the UHF results are not useful in comparison with the ROHF results, and that the extent of spin contamination tends to increase with molecular size and unsaturation, being close to 0.75 for the most saturated species. For instance, I repeated the calculation on triplet linear C19 given by Novoa et al., Inorganica Chemica Acta 198-200 (1992) 133, using UHF and ROHF. I obtained the same heat of formation as these authors with no assumed symmetry in the linear chain and using the UHF AM1 Hamiltonian. However, I obtained an S^2 of 4.356 (should be 2) and a ROHF heat of formation of 678.72 kcal mol^{-1}, which is greater than the UHF value by 24.17 kcal mol^{-1}! I would very much appreciate any helpful comments or literature references which deals with any of the above issues I've raised here about AM1 and its applicability of open shell systems. Regards, Ryan Bettens, OSU Physics Department, BETTENS (+ at +) MPS.OHIO-STATE.edu ________________________ ----------------------| Responses to Questions |---------------------- ------------------------ From: Andrew Holder, University of Missouri, Department of Chemistry You have indeed found some the reporting errors in the original AM1 paper. The problem with your larger systems is that the description of an open shell system of this size will usually require more than one determinant. (That is what the bad spin contamination values indicate.) A better way to handle this is to do everything at the same level of configuration interaction (CI). Be sure that you compare everything in a reaction profile or an analogous series computed at the same level of theory. ---------------------------------------- From: John M. McKelvey, Eastman Kodak Company, Rochester NY Computational Science Laboratory There is no proper ROHF in QCPE MOPAC or AMPAC codes! It is a half- electron method supplemented by a 3x3 CI. I THINK there is proper ROHF for AM1 and PM3 and MNDO in GAMESS. ---------------------------------------- Comment to J.M. McKelvey's response. It seems I had too loosely used the ROHF terminology in my posting. When I stated that Dewar and Thiel, ". . . explicitly stated that they used ROHF heats of formation calculated with MNDO" I ignorantly mistook the statement actually made by the authors about the method they used, i.e., the half-electron method, for ROHF (which from what I gather from various modern books on ab initio theory is Roothaan's open shell method). Therefore, in the above posting wherever ROHF appears, read: half-electron method. John McKelvey pointed me toward the publication: Dewar, M.J.S. and S. Olivella, J. Chem. Soc. Faraday Trans. II 75 (1979) 829, which compared the calculated heats of formation and geometries for the half-electron (HE) method with what is now meant by ROHF using MINDO/3. Their results showed that the ROHF heats of formation were "systematically more negative than the HE ones". They also found that for the molecules studied the r.m.s. differences in the heats of formation for open shell doublets, triplets and singlets were 2.2, 2.5 and 3.7 kcal mol^{-1} respectively. For those interested, in the ROHF method the wavefunction is (a) variationally optimized and (b) is an eigenfunction of the S^2 operator, unlike UHF wavefunctions. In the HE method the wavefunction is not variationally optimized and is the same for analogous open shell states of different multiplicity. A non-mathematical description of the HE method can be found in the introduction of the Dewar and Olivella paper, above.