Figures index : Visualizing the Origin of the Exchange Energy : Science and Education Publishing

Figures index

World Journal of Chemical Education. 2023, 11(4), 141-148 doi:10.12691/wjce-11-4-3

Figure 1. Energetic consequence of the exchange interaction illustrated for a molecule with a singlet ground state S₀ (left). This state features doubly occupied orbitals. The one highest in energy is denoted with ψ₁. The unoccupied orbital lowest in energy is denoted ψ₂. Excited states are formally constructed by promoting one electron from the orbital ψ₁to the orbital ψ₂. The excitation can result in a singlet state S₁ (center) and a triplet state T₁(right). The two excited states are energetically separated by twice the exchange energy, 2K.

Figure 2. Correspondence between the signs in the singlet and triplet wavefunctions (eq. (4) and (5)) and the symmetry properties of the respective diagrams (for a description of these diagrams see Figure 4). The plus sign in the spatial part of the singlet wavefunction corresponds to a symmetric behavior of the respective diagram upon rotation around the diagonal. For the triplet state the minus sign corresponds to an anti-symmetric behavior

Figure 3. Lowest two (n=1,2) PIB wavefunctions (dash-dotted lines) and their absolute squares (solid lines). The wavefunctions and their squares have been shifted vertically according to their energy

Figure 4. Contour representation of the spatial part of the two-electron wavefunction considered here (left) and their absolute square (right). The top row refers to the singlet state, the lower one to the triplet one

Figure 5. One dimensional cuts along the contour representations plotted in Figure 4. The graphs on the left refer to the diagonal (x₁=x₂), the ones on the right the “anti-diagonal” (x₂=L-x₁). Dash-dotted lines represent wavefunctions, the solid ones their absolute square