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Basics of VB theory and XMVB program

Exercise 1 : HF molecule weights

The Gamess and XMVB input files for the HF molecule are provided in the Exercise folder on the tutorial machines. These are VBSCF calculations with the 6-31G(d,p) basis set, with the fragment specification in terms of symmetry-adapted orbitals (frgtyp=sao). Then these input files could serve you as templates for the next exercises.

  1. Just inspect these inputs, run the gamess-xmvb program (using : vbrun hf-scf), and analyze the outputs. Which structure(s) should be kept in further BOVB calculations ?
  2. Using VBSCF orbitals as guess orbitals :
    1. Compute a L-BOVB wave function on a selected subset of structures ;
    2. Compute a VBCISD wave function, freezing the 1s core orbital of fluorine in the VBCI calculation (NCOR=1 option), and printing only structures which have a coefficient superior to 0.01 (ctol=0.01 option) ;
    3. Compare structure weights at the VBSCF, L-BOVB and VBCI levels


>> general guidelines for BOVB calculations

Exercise 2 : F<math>{}_2</math> molecule and bond energy

To answer the first two questions below, we strongly recommend you to comment your input files as extensively as what was done in the input files of the previous exercise. This will allow you to clearly see if you understand all components of an XMVB input.

For the first question below, you can use the hf-scf.xmi input file from the previous exercise as a template.

  1. Compute a L-VBSCF wave function for the F<math>{}_2</math> molecule (all inactive orbitals localized on the fluorine atoms), using:
    • the frgtyp=sao specification ;
    • the boys keyword in the $ctrl section ;
    • automatic guess (guess=auto option) ;
  2. BOVB level :
    1. First, compute a π-D-VBSCF wave function using previous VBSCF orbitals as guess orbitals. To do that, you should allow the π inactive orbitals of fluorine to delocalize onto the two atoms, while keeping all <math>\sigma</math> (active and inactive) orbitals localized (see also : >> see "high symmetry case" in the "general guidelines for BOVB calculations")
    2. Compute then a π-D-BOVB solution for the F<math>{}_2</math> molecule, starting from previous orbitals as guess.
  3. VBCI : compute a VBCI(D,S) wave function (vbcids keyword in the $ctrl section, see pp. 11 and 44 of XMVB Manual), freezing the core orbitals of fluorine in the calculation.
  4. Deduce F<math>{}_2</math> bond energies at both the π-D-BOVB and VBCI(D,S) levels.
  5. Recompute the same L-VBSCF wave-function, this time specifying converged RHF MOs as guess orbitals, through the guess=mo option in the $ctrl section together with an extra $gus section in the input (see hints below, XMVB Manual, and/or Peifeng Su's lecture slides). This option is not necessary for this exercise, but it will prove useful for Exercise 1 of tutorial 3, and in the general case.

Exercise 3 : The lone pairs of H2O

(for further reading, see S. Shaik and P.C. Hiberty, "The Chemist's Guide to VB theory", Wiley, Hoboken, New Jersey, 2008, pp. 107-109)

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.

H2o ex1.png
<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. By convention, one may write the doubly occupied lone pair first, then the singly occupied one.
  4. The two ionized states are the symmetry-adapted combinations Ion-neg.png and Ion-pos.png. 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>.