Accessing electronic correlations by half-cycle pulses and time-resolved spectroscopy

Ultrashort nonresonant electromagnetic pulses applied to effective one-electron systems may operate on the electronic state as a position or momentum translation operator. As derived here, extension to many-body correlated systems exposes qualitatively new aspects. For instance, to the lowest order in the electric field intensity the action of the pulse is expressible in terms of the two-body reduced density matrix enabling us to probe various facets of electronic correlations. As an experimental realization we propose a pump-probe scheme in which after a weak, swift “kick” by the nonresonant pulse the survival probability for remaining in the initial state is measured. This probability we correlate to the two-body reduced density matrix. Since the strength of electronic correlation is bond-length sensitive, measuring the survival probability may allow for a direct insight into the bond-dependent two-body correlation in the ground state. As an illustration, full numerical calculations for two molecular systems are provided and different measures of electronic correlations are analyzed.

Figure 1

Potential energy surfaces for the stretched LiH molecule.

Figure 2

Full CI calculation for the stretched LiH molecule performed with the 6-311++G $\left(2d,2p\right)$ basis set, 20 molecular orbitals are included in a correlated treatment. The electron correlation increases monotonically as the bond is stretched. This is manifested in the correlational energy (a), the entropies (b), the Frobenius norm of the second cumulant matrices (c), and the expectation value of the $z\otimes z$ operator [cf. $\langle [\widehat{S}\otimes \widehat{S}]\rangle$ in Eq. (7) or ${\sum }_{ijkl}{S}_{ij}{S}_{kl}{}^2{\Delta }_{jl}^{ik}$ in Eq. (8)] (d). The entropies are given with respect to the single determinant values $S_{AA}^0$ and $S_{AB}^0$.

Figure 3

Potential energy surfaces for the symmetrically stretched ${\mathrm{H}}_6$ molecule.

Figure 4

Full CI calculation for the stretched ${\mathrm{H}}_6$ ring molecule performed with $aug\text{−}cc\text{−}pvqz$ basis set, 30 molecular orbitals included in a correlated treatment. The electron correlation increases monotonically as the bonds are stretched. This is manifested in the correlational energy (a), the entropies (b), the Frobenius norm of the second cumulant matrices (c), and the expectation value of the $z\otimes z$ operator [cf. $\langle [\widehat{S}\otimes \widehat{S}]\rangle$ in Eq. (7) or ${\sum }_{ijkl}{S}_{ij}{S}_{kl}{}^2{\Delta }_{jl}^{ik}$ in Eq. (8)] (d). The entropies are given with respect to the single determinant values $S_{AA}^0$ and $S_{AB}^0$. Thin lines with arrows denote results for $\varkappa =7$ state. Notice that this state was only computed for larger interatomic distances where it becomes indistinguishable from the ground state (see the energy plot). Blue and red shading stands for the first and second Carlen-Lieb bound [Eq. (11)]. Black and red circles denote results for the $3{\mathrm{H}}^{-}$ asymptotic state.