excited-state solvent-field equilibration in pratice

Sketch of a solvent-field equilibration for
the cationic ethene dimer.

In my last post, I introduced the cationic ethene dimer shown on the right to illustrate and evaluate a solvent model for excited states. Despite its neat size and simplicity,  quantum-chemical calculations on this system were a bit tricky to converge onto the desired states. This is due to its open-shell (doublet) electronic structure exhibiting multiple, energetically almost degenerate SCF solutions. While these problems can be circumvented for the gas-phase calculation with one little trick, the actual solvent-field equilibration requires for more drastic measures, some of which I want to present in this post. Here are the three main problems and how I approached them:

1. The SCF converges on an undesired solution: Without a PCM as well as for low dielectric constants and large basis set, the initial SCF converges onto the symmetrically charged (both ethenes + ½) solution, instead of the asymmetrically charged one I want to investigate.
   
   To convince the SCF algorithm to converge onto the asymmetrically charged solution, I use sequential jobs. In my favorite quantum-chemistry program, this can be triggered by adding a line “@@@” after the first input and appending the second one. The advantage of this is that results stored on disk, like e.g. MOs, state-densities and PCM surface-charges are available in the following step. I want to use the first job to create a set of asymmetric orbitals, which I employ as initial guess in the second one. For this purpose, I shorten the C-H bonds of one of the molecules by 0.1 A to introduce an asymmetry that triggers the SCF to converge onto the asymmetrical solution. Using the resulting orbitals as starting point for the SCF, I can give the second job the undistorted geometry (otherwise it would by default use the one from the first job). Starting from the asymmetrical guess orbitals, the second SCF also converges onto the desired solution, even for the now symmetric geometry. This trick was necessary for all gas-phase calculations, as well as for the calculations with large (cc-pVQZ) or augmented (aug-cc-pVDZ) basis sets, even if a PCM with high epsilon is used.
   
   
2. The energetic ordering of the excited states switches: While the charge-inverse excited state is one of the higher lying states in the ground-state equilibrated solvent field (2.-4., depending on epsilon and the basis), it becomes the lowest one as soon as its solvent field is relaxed.
   
   Changes in the energetic ordering of the excited-states during excited state geometry optimizations are very common, if not inherent to the problem and I guess every theoretical photochemist knows the case. Using symmetry in the calculation can help, but does not necessarily solve the problem. Surprisingly few quantum-chemistry codes feature a decent root homing algorithm to identify excited-states via their overlap with the results from the previous step to guide the optimization. For the solvent-field optimizations of this predictable toy system the solution is rather trivial: Since I have not yet implemented a proper iteration loop, but use a script to generate sequential input files with a fixed number of iteration steps, I can just adapt the state_to_opt variable separately in each of the iteration steps. Nevertheless, I wanted to point out how practical root homing would be as a common feature in quantum-chemistry codes.
   
   
3. The SCF "follows" the solvent field: The first SCF calculation in the solvent-field of the excited state converges onto a solution resembling the former excited-state wavefunction, since the latter is much lower in energy in the new, excited-state relaxed solvent field.
   
   This one was the trickiest one to tackle. Although I do per default use the orbitals of the previous step as a guess for the solvent-field iterations all of the available SCF algorithms slowly crawl to the solution of the energetically much lower state. Obviously, the excited-state solvent field is just too attractive for the positive charge. In my favorite quantum-chemistry program, however, there is a very nice feature which I want to introduce in a little more detail in the following. The maximum overlap method (MOM) is basically a kind of inter-step root-homing for the SCF, which was developed by Andrew Gilbert and Peter Gill for the calculation of excited-states wavefunctions. For this purpose, converged HF orbitals from a previous calculation are rotated in such a way that they resemble the excited state, e.g. via  "excitation" of an electron from the HOMO to the LUMO. One drawback of the method is that for this to succeed, the overlap between the ground- and respective excited state wavefunction has to be rather small. Hence, the approach does in general only work for dark excited states. If you are interested any further: I discussed the advantages and drawbacks of this method for the description of excited states in combination with coupled-cluster theory somewhat more extensively in a work on nitrobenzene. For the problem at hand, i.e. forcing the SCF to converge onto the local minimum provided in the guess orbitals, MOM works like a charm and I consider making its use the default for my equilibrium solvation approach.

After all, the results I obtained with these tricks are promising: If the solvent-fields are fully relaxed for the respective, correlated MP2 or ADC2 density, the energy difference between the ground- and charge-inverse excited state stay within 0.01 eV of the gas-phase value for the whole range of epsilon:

Energy difference between the solvent equilibrated ground (MP2/cc-pVTZ) and charge-inverse excited (ADC2/cc-pVTZ) states (left y-axis) as a function of the dielectric constant (x-axis). For clarity, I have subtracted the gas-phase value of 0.34 eV from the results. For relation, the total solute-solvent interaction energy for one of the states is given with respect to the right y-axis.

Since this post has already become quite long and the day quite old I will continue with a more detailed survey of the results as well as a description of the differences between the PTE, PTD and PTED schemes in my next post.

So long!
Jan