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Appendix B: Adiabatic excited state dynamics

In the previous sections, we discussed the application of the Born-Oppenheimer approximation to ground state dynamics. Under certain circumstances -- well-separated electronic states, low nuclear velocities, or excitation into special symmetry states -- a similar adiabatic approximation can be made to excited state dynamics as well. We explain below how this can be the case.

Consider the stationary states of a molecule, the solutions of the time-independent Schrodinger equation $ \mathcal{H}\Psi_{i} = E_{i}\Psi_{i}$. Usually we solve this equation approximately by varying parameters of a trial function. For the lowest energy or ground state, we have the variational principle

$\displaystyle E_{0} = \braket{\Phi_{0}\vert\mathcal{H}\vert\Phi_{0}} \geq E_{0}(\mathrm{exact})$ (3.28)

so that we obtain a best estimate for $ \Psi_{0}$ by varying $ \Phi_{0}$ to minimize $ E_{0}$.

The most common trial function is an antisymmetrized product of one electron orbitals, called a Slater determinant [40]. Slater determinants are the basis of the Hartree-Fock method, and have well-understood limitations -- they do not properly describe covalent bond breaking, certain atomic symmetries, and instantaneous correlation of electron motions due to Coulomb repulsion. We can account for some of these effects using a density-dependent exchange-correlation functional (DFT) or by adding more determinants corresponding to the excitation of electrons into virtual orbitals (configuration interaction).

For excited states, we can apply a generalized variational principle to obtain an approximate wavefunction solution:

$\displaystyle E_{i} = \braket{\Phi_{i}\vert\mathcal{H}\vert\Phi_{i}} \geq E_{i}...
...ct}) \ \textrm{if} \ \braket{\Phi_{i}\vert\Phi_{j}}=0 \ \textrm{for all} \ j<i.$ (3.29)

If the excited state has a different symmetry than the ground state, we can write the trial wavefunction as single determinant and simply apply the ground-state optimization procedures to obtain an excited state solution. As long as the trial function is restricted to a symmetry different from the ground state (and the lower excited states), the orthogonality to the ground state is maintained automatically, and the solution is valid [41].

However, if the excited state has the same symmetry as the ground state, the orthogonality needs to be maintained some other way, which poses technical challenges, often overcome through use of a multi-determinant wavefunction [42]. Also, some excited states, such as the open shell $ 2s^{2} 2p^{2}$ carbon atom, require multiple determinants to describe, which is expensive for large systems, and not compatible with default Kohn-Sham density functional theory [43].

Time-dependent methods like TDDFT, described in the earlier sections, can be used to extract excited state energies properties as well, and are gaining popularity because (1) time-dependent functionals can be based on ground-state functionals, (2) an entire excitation spectra can be obtained from one calculation, and (3) we are not restricted to calculating excited states of different symmetry than the ground state. All in all, though, it is not yet possible to find excited stationary states with the same ease, accuracy, or generality as ground states.

Suppose we excite a system to a single stationary state, for instance with a long duration monochromatic pulse. What happens to a system in such a state? We have prepared a stationary state of the electrons, not necessarily the nuclei, so usually the nuclei move. As this happens, the system begins to include contributions from other stationary states, and a superposition of states evolves [28]:

    $\displaystyle \Psi(\mathbf{R}_{elec}, t) = \sum_{j} c_{j}(t) \Psi_{j}(\mathbf{R}_{elec}; \mathbf{R}_{nuc}(t))$ (3.30)
    $\displaystyle i \hbar \dot{c}_{k} = \sum_{j} c_{j}\left( E_{k} \delta_{kj} - i \hbar \dot{\mathbf{R}}_{nuc} \cdot \mathbf{d}_{jk}\right)$ (3.31)


$\displaystyle \mathbf{d}_{jk} = \braket{\Psi_{k}\vert\nabla_{R_{nuc}}\Psi_{j}} ...
...ert\nabla_{R_{nuc}}\mathcal{H}(\mathbf{R}_{nuc})\vert\Psi_{j}} } {E_{j}-E_{k}}.$ (3.32)

The non-adiabatic coupling vector $ d_{jk}$ couples electron and nuclear motions, and is responsible for the mixing of stationary states as they approach each other in energy. The final simplification to write $ \mathbf{d}_{jk}$ in terms of the operator $ \nabla \mathcal{H}$ comes from $ \nabla_{R_{nuc}}\braket{\Psi_{k}\vert\mathcal{H}\vert\Psi_{j}} = \nabla_{R}E_{k}\delta_{kj} = 0$.

If $ d_{kj}$ is small, the system will evolve adibatically along a single stationary state, and the Born-Oppenheimer approximation applies. For $ d_{kj}$ to be large, and the states to mix, certain conditions must hold. First, the states need to be similar in energy. Second, they need to be the same symmetry, so that $ \braket{\Psi_{k}\vert\nabla_{R_{nuc}}\mathcal{H}(\mathbf{R}_{nuc})\vert\Psi_{j}} \neq 0$. And third, even if the states are of similar energy and matching symmetry, when two eigenvalues of an N dimensional Hermitian matrix become the same, the degeneracy spans a N-2 dimensional space called a conical intersection [44] (Figure 3.8). At conical intersections, the Born-Oppenheimer approximation breaks down, since electrons flow from one state to another over a small variation in nuclear position. In the special case of a diatomic molecule, $ N=1$ and the conical intersection becomes an avoided crossing where the curves cannot intersect.

Figure 3.8: In sparse systems, electrons move mostly along adiabatic paths, with crossings limited to conical intersections with reduced dimensionality.

For a two-state avoided crossing, a transition from one adiabatic state to another is probable when the Massey parameter is greater than one [28]:

$\displaystyle \xi = \left\vert\frac{\hbar \dot{\mathbf{R}}_{nuc} \cdot \mathbf{d}_{jk}}{E_{j}-E_{k}}\right\vert \geq 1.$ (3.33)

Hence hopping is favorable when the energy gap is small, the nuclear velocities high, and the non-adibatic coupling vector high in magnitude. When electronic states are well-separated and the temperature is low, it is a good approximation to say that the system evolves adiabatically for long periods of time followed by nonadiabatic switches at conical intersections that are restricted to small regions in phase space.

Surface hopping stochastic dynamics [45] makes this approximate picture literal, by propagating the nuclei along single excited state potentials, and switching them randomly to other state potentials with a rate that is a function of $ \dot{\mathbf{R}}_{nuc} \cdot \mathbf{d}_{jk}$. Tully's popular minimum switching algorithm [28] executes this switching in an efficient way that preserves fluxes and minimizes the abruptness of switching from one state to another. The excited state potentials can be parameterized from quantum calculations or experiment; or DFT, Car-Parinello, or TDDFT methods can be used to compute them on the fly.

next up previous contents
Next: Bibliography Up: The electron force field, Previous: Appendix A: Wave packet   Contents
Julius 2008-04-29