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Dissociation of H 2

Dissociation of H 2. Do HF calculations for different values of the H-H internuclear distance (this distance is fixed since we are in the Born-Oppenheimer picture). Fig. 3.5: RHF potential curve for STO-3G (ζ = 1.24) H 2 compared with the accurate results of Kolos and Wolniewicz.

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Dissociation of H 2

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  1. Dissociation of H2 Do HF calculations for different values of the H-H internuclear distance (this distance is fixed since we are in the Born-Oppenheimer picture). Fig. 3.5: RHF potential curve for STO-3G (ζ = 1.24) H2 compared with the accurate results of Kolos and Wolniewicz. The minimal basis RHF calculation does not go to the limit of two hydrogen atoms as the H-H distance goes to infinity. This totally incorrect behavior is not specific for H2: if one stretches any bond for which the correct products of dissociation must be represented by open-shell wavefunctions, then closed-shell RHF will go badly wrong.

  2. For H2, the products of dissociation are two localized H atoms; that is, one electron is localized near one of the protons and the other electron is localized near the other, distant, proton. In RHF, however, both electrons are forced to occupy the same spatial molecular orbital. Therefore, independent of the bond length, both electrons are described by exactly the same spatial wavefunction and have the same probability distribution function in 3-dimensional space. Such a description is inappropriate for two separated H atoms. Closed-shell RUF, which restricts electrons to occupy molecular orbitals in pairs, cannot, therefore, properly describe dissociation unless the products of dissociation are both closed-shells. The problem can be examined analytically: the dissociated H2 molecule keeps a term because, since both electrons occupy the same spatial orbital, there remains even at infinity some electron-electron repulsion. RUF

  3. UHF gives a correct picture of the dissociation of H2 Fig. 3.19: 6-31G** potential energy curves for H2

  4. Does full CI get the correct dissociation picture for H2 ? Yes: the full CI energy in minimal basis H2 is In the limit and the –K12 term exactly cancels the J11 term. Fig. 4.3: 6-31G** potential energy curves for H2

  5. CI and the size-consistency problem Suppose you want to compute the energy difference between reactants and products: where A, B are smaller than C (for example H + H  H2) To get a reasonable answer, we need a method which works “equally well” for molecules with different numbers of electrons. One simple criterion for this is that the energy difference between (H+H) and (H2) should be zero in the limit of infinite separation (non-interacting limit). Such a method is called “size-consistent”. HF is size-consistent if the monomers have a closed shell. Full CI is also size-consistent (since it is an exact theory). However, for all but the smallest molecules, even with a minimal basis set, full CI is computationally impractical. With a one-electron basis of moderate size, there are so many possible configurations that the full CI matrix becomes impossibly large (greater than 109 x 109).

  6. CI and the size-consistency problem To obtain a computationally viable scheme we must truncate the full CI matrix (the CI expansion for the wave function). A systematic procedure for doing this is to consider only those configurations which differ from the HF ground state by no more than m spin orbitals. For example, m = 4 gives single, double, triple, and quadruple excitations in the trial function. m = 2 gives SDCI: singly + doubly excited CI. This choice is commonly used. HOWEVER truncated CI is not size-consistent !

  7. CI and the size-consistency problem Truncated CI (e.g. SDCI) is not size-consistent ! A + A A2 Why? Very simple: SDCI of each of the A monomers contains double excitations within the monomer. BUT the (non-interacting) dimer cannot have both monomers doubly-excited, because this would correspond to a quadruple excitation of A2. Hence the dimer wavefunction does not have enough flexibility to yield twice the monomer energy. (see page 264 of Szabo) Are there schemes to compute the correlation energy which are size-consistent? YES: Møller-Plesset (e.g. MP2) perturbation theory

  8. Møller-Plesset perturbation theory (chapter 6 of Szabo) Very useful, the “method of choice” if you need correlation. Example: calculation of pi-pi interactions (dispersion) requires electron correlation

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