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Ab Initio Calculations

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Abstract

Ab initio calculations rest on solving the Schrödinger equation; the nature of the necessary approximations determines the level of the calculation. In the simplest approach, the Hartree-Fock method, the total molecular wavefunction Ψ is approximated as a Slater determinant composed of occupied spin orbitals. To use these in practical calculations, the spatial part of the spin orbitals is approximated as a linear combination (a weighted sum) of basis functions. Electron correlation methods are also discussed. The main uses of the ab initio method are calculating molecular geometries, energies, vibrational frequencies, spectra, ionization energies and electron affinities, and properties like dipole moments which are connected with electron distribution. These calculations find theoretical and practical applications, since, for example, enzyme-substrate interactions depend on shapes and charge distributions and reaction equilibria and rates depend on energy differences, and spectroscopy plays an important role in identifying and understanding novel molecules. The visualization of calculated phenomena can be very important in interpreting results.

“I could have done it in a much more complicated way,” said the Red Queen, immensely proud.

Attributed, probably apocryphally, to Lewis Carroll

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Notes

  1. 1.

    Douglas Hartree, born in Cambridge, England, 1897. Ph.D. Cambridge, 1926. Professor applied mathematics, theoretical physics, Manchester, Cambridge. Died Cambridge, 1958.

  2. 2.

    John Slater, born in Oak Park Illinois, 1900. Ph.D. Harvard, 1923. Professor of physics, Harvard, 1924–1930; MIT 1930–1966; University of Florida at Gainesville, 1966–1976. Author of 14 textbooks, contributed to solid-state physics and quantum chemistry, developed X-alpha method (early density functional theory method). Died Sanibel Island, Florida, 1976.

  3. 3.

    John Pople, born in Burnham-on-Sea, Somerset, England, 1925. Ph.D. (Mathematics) Cambridge, 1951. Professor, Carnegie-Mellon University, 1960–1986, Northwestern University (Evanston, Illinois) 1986–2004. Nobel Prize in Chemistry in 1998 (with Walter Kohn, Chapter 5, Section 7.1). Died in Chicago, 2004.

  4. 4.

    Møller-Plesset: the Danish-Norwegian letter ø is pronounced like French eu or German ö.

  5. 5.

    F. Neese, personal communication, 2022, July 22.

  6. 6.

    Richard Bader, born in Kitchener, Ontario, Canada, 1931. Ph.D. Massachusetts Institute of Technology, 1958. Professor, University of Ottawa, 1959–1963, McMaster University, 1963–2012. Died in Burlington, Ontario, 2012.

  7. 7.

    R. Hoffmann, personal communication, 2009 August 12.

References

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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.

    In the term Hartree-Fock, what, essentially, were the contributions of each of these two people?

  2. 2.

    What is a spin orbital? A spatial orbital?

  3. 3.

    At which step in the derivation of the Hartree-Fock energy does the assumption that each electron sees an “average electron cloud” appear?

  4. 4.

    For a closed-shell molecule in the ground electronic state, the number of occupied molecular orbitals is half the number of electrons, but there is no limit to the number of virtual orbitals. Explain.

  5. 5.

    In the simple Hückel method, csi denotes the basis function coefficient for the contribution of atom number s (in whatever numbering scheme we choose) to MO number i. In the ab initio method, csi still refers to MO number i, but the s does not necessarily denote atom number s. Explain.

  6. 6.

    The derivation of the Roothaan-hall equations involves some key concepts: Slater determinant, Schrödinger equation, explicit Hamiltonian operator, energy minimization, and LCAO. Using these, summarize the steps leading to the Roothaan-Hall equations FC = Scε.

  7. 7.

    What are the similarities and the differences between the basis set of the extended Hückel method and the ab initio STO-3G basis set?

  8. 8.

    In the simple and extended Hückel methods, the molecular orbitals are calculated and then filled from the bottom-up with the available electrons. However, in ab initio calculations, the occupancy of the orbitals is taken into account as they are being calculated. Explain.

    Hint: look at the expression for the Fock matrix elements in terms of the density matrix.

  9. 9.

    Isodesmic reactions have been used to investigate aromatic stabilization, but there is not a unique isodesmic reaction for each problem. Write two isodesmic reactions for the ring-opening of benzene, both of which have on each side of the equation the same number of each kind of bond. Have you any reason to prefer one of the equations to the other?

  10. 10.

    List the strengths and weaknesses of ab initio calculations compared to molecular mechanics and extended Hückel calculations. State the molecular features that can be calculated by each method.

Harder Questions

  1. 1.

    Does the term ab initio imply that such calculations are “exact”? In what sense might ab initio calculations be said to be semiempirical – or at least not fully a priori?

  2. 2.

    Can the Schrödinger equation be solved exactly for a species with two protons and one electron? Why or why not?

  3. 3.

    The input for an ab initio calculation (or a semiempirical calculation of the type discussed in Chap. 6, or a DFT calculation – Chap. 7) on a molecule is usually just the cartesian coordinates of the atoms (plus the charge and multiplicity). So how does the program know where the bonds are, that is, what the structural formula of the molecule is?

  4. 4.

    Why is it that (in the usual treatment) the calculation of the internuclear repulsion energy term is easy, in contrast to the electronic energy term?

  5. 5.

    In an ab initio calculation on H2 or HHe+, one kind of interelectronic interaction does not arise; what is it and why?

  6. 6.

    Why are basis functions not necessarily the same as atomic orbitals?

  7. 7.

    One desirable feature of a basis set is that it should be “balanced.” How might a basis set be unbalanced?

  8. 8.

    In a Hartree-Fock calculation, you can always get a lower energy (a “better” energy, in the sense that it is closer to the true energy) for a molecule by using a bigger basis set, as long as the HF limit has not been reached. Yet a bigger basis set does not necessarily give better geometries and better relative (i.e., activation and reaction) energies. Why is this so?

  9. 9.

    Why is size consistency in an ab initio method considered more important than variational behavior (MP2 is size consistent but not variational)?

  10. 10.

    A common alternative to writing a Hartree-Fock wavefunction as an explicit Slater determinant is to express it using a permutation operator \( \hat{P} \) which permutes (switches) electrons around in MOs. Examine the Slater determinant for a two-electron closed-shell molecule, and then try to rewrite the wavefunction using. \( \hat{P}. \)

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Lewars, E.G. (2024). Ab Initio Calculations. In: Computational Chemistry. Springer, Cham. https://doi.org/10.1007/978-3-031-51443-2_5

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