VBTutorial1

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Basics of VB theory and XMVB program

Remarks

>> Initial remarks for tutors

Exercices

Exercise 1 (paper exercise) : The lone pairs of H<math>{}_2</math>O

This exercise aims at comparing two descriptions of the lone pairs of H<math>{}_2</math>O : (i) the MO description in terms of non-equivalent canonical MOs and (ii) the « rabbit-ear » VB description in terms of two equivalent hybrid orbitals.

<math>\Psi_{\textrm{MO}}</math>   <math>\Psi_{\textrm{VB}}</math>

1. Focusing on the lone pairs only, write the four-electron single-determinants <math>\Psi_{\textrm{MO}} </math> and <math>\Psi_{\textrm{VB}} </math> .
2. Expand <math>\Psi_{\textrm{VB}} </math> into elementary determinants containing only <math>n</math> and <math>p</math> orbitals, eliminate determinants having two identical spinorbitals, and show the equivalence between <math>\Psi_{\textrm{VB}}</math> and <math>\Psi_{\textrm{MO}}</math>.
3. We now remove one electron from H<math>{}_2</math>O. Write the two possible VB structures <math>\Phi_1</math> and <math>\Phi_2</math> in the VB framework.
4. The two ionized states are the symmetry-adapted combinations <math>\frac{1}{2}\left(\Phi_1-\Phi_2\right)</math> and <math>\frac{1}{2}\left(\Phi_1+\Phi_2\right)</math>. From the sign of the hamiltonian matrix element <math>\langle \Phi_1 \vert \hat{H} \vert \Phi_2 \rangle</math>, give the energy ordering of the two ionized states.
5. By expanding the two ionized states into elementary determinants (dropping the normalization constants), show that they are equivalent, respectively, to the MO configurations <math>\vert nn\bar{p}\vert</math> and <math>\vert pp\bar{n}\vert</math>.

Appendix

Hamiltonian matrix element between determinants differing by one spin-orbital :

<math>\langle \cdots i \cdots \vert \hat{H} \vert \cdots j \cdots\rangle = \beta_{ij}</math>

>> Answer

Exercice 2 : Simple diatomics molecules

Subject

First contact with XMVB on simple diatomics. Examination of the effect of correlation on weights, and bond energies. Calculation of a pure covalent state and (charge-shift) resonance energy.

To do

• Compute of H2 at the VBSCF level.
• Compute HF at the VBSCF, VBCI, and D-BOVB levels. Compute bond energy. Compute a single covalent structure, and deduce the charge-shift resonance energy.
• Same question for F2

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Exercice 3 : Dissociation of C(Me)3-Cl

Subject

First calculation beyond diatomics (fragment C(Me)3 and Cl). VB(PCM) method.

To do

• Compute of C(Me)3-Cl at equilibrium distance at the VBSCF and D-BOVB levels.
• Compute C(Me)3-Cl at large inter fragment distance (5Å ?), at the D-BOVB level.
• Redo previous questions using the VB(PCM) option.

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