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Semiempirical Calculations

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Abstract

Semiempirical quantum mechanical calculations are based on the Schrödinger equation. This chapter deals with SCF semiempirical methods, in which repeated (in contrast to the simple and extended Hückel methods) diagonalization of a Fock matrix refines the wavefunction and molecular energy. These calculations are much faster than ab initio ones, mainly because the number of integrals to be dealt with is greatly reduced by ignoring some and approximating others with the help of experimental quantities or with values from high-level ab initio or DFT calculations. In order of increasing sophistication, these SCF semiempirical methods have been developed: PPP (Pariser-Parr-Pople), CNDO (complete neglect of differential overlap), INDO (intermediate neglect of differential overlap), and NDDO (neglect of diatomic differential overlap). Today, the most popular SCF semiempirical methods are variations of NDDO: AM1 (Austin model 1, from Austin, Texas) and its offshoot PM3 (parametric method 3), and refinements up to PM7, which are carefully parameterized to reproduce experimental quantities, mainly heats of formation. Extensions of AM1 (RM1, Recife model 1, from Recife, Brazil) and PM3-PM7 seem to represent substantial improvements, and PM7 may be, or may become, the standard semiempirical “general SCF-type” method.

Current “ab initio” methods were limited to very inaccurate calculations for very small molecules.

M. J. S. Dewar, A Semiempirical Life, 1992

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Notes

  1. 1.

    Michael J. S. Dewar, born Ahmednagar, India, 1918. Ph.D. Oxford, 1942. Professor of chemistry at Universities of London, Chicago, Texas at Austin, and University of Florida. Died Florida, 1997.

  2. 2.

    J. J. P. Stewart, personal communication, 2015 November 19.

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  1. Department of Chemistry, Trent University, Peterborough, ON, Canada

    Errol G. Lewars

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Appendices

Easier Questions

  1. 1.

    Outline the similarities and differences between the extended Hückel method on the one hand and methods like AM1 and PM3 on the other. What advantages does the EHM have over more accurate semiempirical methods?

  2. 2.

    Outline the similarities and differences between molecular mechanics, ab initio, and semiempirical methods.

  3. 3.

    Both the simple Hückel and the PPP methods are π electron methods, but PPP is more complex. Itemize the added features of PPP.

  4. 4.

    What is the main advantage of an all-valence-electron method like, say, CNDO over a purely π electron method like PPP?

  5. 5.

    Explain the terms ZDO, CNDO, INDO, and NDDO, showing why the latter three represent a progressive conceptual improvement.

  6. 6.

    How does an AM1 or PM3 “total electron wavefunction” Ψ differ from the Ψ of an ab initio calculation?

  7. 7.

    Ab initio energies are “total dissociation” energies (dissociation to electrons and atomic nuclei) and AM1 and PM3 energies are standard heats of formation. Is one of these kinds of energy more useful? Why or why not?

  8. 8.

    For certain kinds of molecules, molecular mechanics can give better geometries and relative energies than can even sophisticated semiempirical methods. What kinds of properties can the latter calculate that MM cannot?

  9. 9.

    Why do transition metal compounds present special difficulties for AM1 and PM3?

  10. 10.

    Although both AM1 and PM3 normally give good molecular geometries, they are not too successful in dealing with geometries involving hydrogen bonds. Suggest reasons for this deficiency.

Harder Questions

  1. 1.

    Why are even very carefully parameterized semiempirical methods like AM1 and PM3 not as accurate and reliable as high-level (e.g., MP2, CI, coupled-cluster) ab initio calculations?

  2. 2.

    Molecular mechanics is essentially empirical, while methods like PPP, CNDO, and AM1/PM3 are semiempirical. What are the analogies in PPP, etc., to MM procedures of develo** and parameterizing a force field? Why are PPP, etc., only semiempirical?

  3. 3.

    What do you think are the advantages and disadvantages of parameterizing semiempirical methods with data from ab initio calculations rather than from experiment? Could an SE method parameterized using ab initio calculations logically be called semiempirical?

  4. 4.

    There is a kind of contradiction in the Dewar-type methods (AM1, etc.) in that overlap integrals are calculated and used to help evaluate the Fock matrix elements, yet the overlap matrix is taken as a unit matrix as far as diagonalization of the Fock matrix goes. Discuss.

  5. 5.

    What would be the advantages and disadvantages of using the general MNDO/AM1 parameterization procedure, but employing a minimal basis set instead of a minimal valence basis set?

  6. 6.

    In SCF semiempirical methods, major approximations lie in the calculation of the \( {H}_{rs}^{\mathrm{core}},\left( rs| tu\right), \) and (ru|ts) integrals of the Fock matrix elements Frs (Eq. (6.1)). Suggest an alternative approach to approximating one of these integrals.

  7. 7.

    Read the exchange between Dewar on the one hand and Halgren, Kleir, and Lipscomb on the other [27]. Do you agree that semiempirical methods, even when they give good results “inevitably obscure the physical bases for success (however striking) and failure alike, thereby limiting the prospects for learning why the results are as they are.” Explain your answer.

  8. 8.

    It has been said of semiempirical methods: “They will never outlive their usefulness for correlating properties across a series of molecules...I really doubt their predictive value for a one-off calculation on a small molecule on the grounds that whatever one is seeking to predict has probably already been included in with the parameters.” (A. Hinchliffe, “Ab Initio Determination of Molecular Properties”, Adam Hilger, Bristol, 1987, p. x). Do you agree with this? Why or why not? Compare the above quotation with ref. [23], pp. 133–136.

  9. 9.

    For a set of common organic molecules, Merck molecular force field geometries are nearly as good as MP(fc)/6-31G* geometries (Chap. 3, Sect. 3.4). For such molecules, single-point MP(fc)/6-31G* calculations (Chap. 5, Sect. 5.4.2), which are quite fast, on the MMFF geometries, should give energy differences comparable to those from MP(fc)/6-31G*//MP(fc)/6-31G* calculations. Example: CH2=CHOH/CH3CHO, ΔE(MP2 opt, including ZPE) = 71.6 kJ mol−1, total time 1064 s.; ΔE(MP2 single point on MMFF geometries) = 70.7 kJ mol−1, total time = 48 s. (G98 on a Pentium 3). What role does this leave for semiempirical calculations?

  10. 10.

    Semiempirical methods are untrustworthy for “exotic” molecules of theoretical interest. Give an example of such a molecule and explain why it can be considered exotic. Why can’t semiempirical methods be trusted for molecules like yours? For what other kinds of molecules might these methods fail to give good results?

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Lewars, E.G. (2024). Semiempirical Calculations. In: Computational Chemistry. Springer, Cham. https://doi.org/10.1007/978-3-031-51443-2_6

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