Fig. 1. Electronic orbitals structure due to: (A) Nuclei configurations away from the level crossings (non-degenerate ground state) region where the manifold of occupied (filled) orbitals is separated by an energy gap E_g from the manifold of virtual (unfilled) electronic orbitals. (B) Nuclei configurations in the vicinity of the level crossing (n-fold degenerate ground state) where a manifold of mid-gap (almost) degenerate orbitals appears.

Fig. 2. Optical field induces transitions which can be associated with the components of δρ (Eq. (25)): v describes transitions between filled and virtual orbitals, u between active space and filled/virtual orbitals, and w between the correlated singlet states formed from the active space orbitals.

Fig. 3. Illustration of differential form and classical Hamiltonian pull-back. Mapping is due to the natural embedding . Accordingly, in the associated tangent spaces the operator mapping exists. All possible pairs from the set (v,u,w) in should to be sampled, and their images in should be substituted to . The result of the calculations is identified as the form restriction ω(ρ^M). Similar procedure is used to perform the Hamiltonian pull-back.

Fig. 4. Illustration of the lift operation. Consider a fiber to the base at point ρ^a, and associated tangent spaces touching at points ρ and ρ^a, respectively. Then an arbitrary vector ζ in the total space can be decomposed into two components denoted by ζ and _ρξ. The former component belongs to the tangent space of the fiber. The latter component _ρξ is the lift of some vector ξ from the tangent space to the base at point ρ^a. In other words, ζ is fully defined by ζ and ξ.

Fig. 5. Proposed scheme of two-color pump–probe spectroscopy. (A) Photo-excitation of vibrational wave-packet at the energy ω_PM in the region away from the level crossing. In this region only transitions between occupied and virtual orbitals exist. (B) Quantum mechanical/semiclasscial propagation of photo-excited wavepacket to the region of level crossing. (C) In the level crossing region the mid-gap states appear, and associated response can be monitored by the probe pulse delayed by time τ whose energy ω_PR matches the mid-gap transitions.