Montag, 10. Juli 2017

ADC/SS-PCM at glance? Part one.

Dear colleagues,
in the framework of a recent project published in this article, I for the first time had the chance to systematically study the performance of ADC/SS-PCM for a group of click-chemistry generated molecules with varying charge-transfer character in their emitting excited state.
The common motif of these molecule is that they consist of an electron-pushing triphenylamin group (right side of the molecule in the figure below) and an electron-pulling group (left side), whose strength can be modulated by oxidizing the contained sulphur atom. In the lowest excited singlet state of all of these systems, an electron is excited from the electron-pushing triphenylamine to the electron pulling part of the molecule. While the excitation-hole turned out to be essentially identical in all of these systems and all solvents, the location and structure of the excited electron differed significantly depending on the molecule and environment.
The figure below shows the structures of the molecules as well as the location of the excited electron computed at the ADC2/SS-PCM/SV level of theory for the non-polar solvent cyclohexane on the left and for the polar solvent acetonitrile in the right.

Since the discussion of the performance of the methodology was (as usual) quite brief in the article, I will share some more methodological insights here and in the following posts.
The central observation, which also made it into the paper, was that the solvent effects are systematically overestimated for this group of molecules, leading to too low emission energies in polar solvents (here acetonitrile, ACN). In the non-polar solvent cyclohexane (CHX) the error seems to cancel out with the systematic overestimation of the emission energies by the quantum-chemical method (ADC2 with a small, non-polarized split-valence basis), leading to a very reasonable agreement. 

Something that we only mentioned but did not show in the article was that the spin-opposite scaled variant of ADC(2), SOS-ADC(2) [short: SOS(2)] improves on this systematic error. After the article was published I took a closer look at this and recomputed all of the numbers also with SOS(2). Turns out this method pretty much nails the emission energies in ACN, with those in CHX getting a little worse:
Fluorescence energies of a series of molecules calculated with ADC(2) and SOS-ADC(2)/SS-PCM/SV using TD-uPBE/6-31+G(d) geometries. Geometry optimizations were conducted in the gas phase, i.e., without any solvent model.
While these trends are most certainly interesting to know for future investigations (use SOS-ADC/SV for polar solvents and ADC/SV for non-polar ones), the question that remains is if this systematic overestimation of the solvent-stabilization is due to A) the quantum-chemical methodology, B) the solvent-model, C) the nature of these molecules or D) a combination of all three. 
Speaking for A) is that I have observed before how ADC(2) yields too low energies for charge-transfer states compared to locally excited states. Since in this case the picture was very consistent, including the fact that the problem is corrected by ADC(3), I'm pretty confident that A) contributes at least to some extent. The question that really bothers me, however, is if B) is also the case, because so far, I am pretty sure that the self-consistent treatment of solvent polarization is the one and only physically correct way to do it. You can find my arguments for that in the respective article (same as above).
To further look into the issue, let us eliminate the inherent error of the quantum-chemical methodology as far as possible by looking at the CHX to ACN shifts in the emission energy. Let us furthermore include the emission energies from the first solvent-field iteration in the comparison, i.e. the ones calculated with the solvent field obtained for the excited-state computed in the relaxed ground-state solvent field (ptSS-calculation).

Seeing this plot for the first time literally shocked me. The agreement of the one-shot approach with the experimental data is so apparent and convincing that I immediately starting looking for the fundamental flaw in the self-consistent approach. The latter apparently overestimated the shifts by about a factor of two. Did I say about? For ADC(2), its pretty much exactly a factor of two:

Can this be a coincidence? Exactly a factor of two? The lines are pretty much on top of each other. Yet, I'm pretty sure (as sure as it gets for a scientist) that this is indeed a coincidence, for a number of convincing reasons, which I will elaborate in the next post. And the best thing will be, that we can still learn from this coincidence!

So long,

Mittwoch, 5. Juli 2017

Boron-substituted aromatics - part two: thermochemical calculations

In my last post I described how I calculated vibrationally resolved UV/vis absorption spectra for a couple of boronaromatics and their acetonitrile (ACN) adducts and how this helped to resolve the mystery of their odd solvatochromism (solvent dependent absorption spectra). The one question that remained open was: Do DBI and DBA exclusively form 1:1 monoadducts with ACN or also the 1:2 diadducts? Since the latter are predicted to have essentially no absorption in the relevant spectral range, i.e., between 300 and 700 nm, I could neither confirm nor exclude their formation.

To answer this question, I started to conduct what was planned as a "brief" thermochemical analysis of formation of the respective mono- and diadducts. However, since the first results were inconclusive the I ended up employing a hierarchy of methods of increasing sophistication (and computational demand), which only eventually (at the very highest level of theory) cumulated in a good enough agreement with the experimental observations. I think the convergence of the results with respect to level of theory and employed basis-sets is quite instructive (without having much experience in quantum-thermochemistry), but at the same time too long and theoretical to be included in the article, and hence provides a nice topic for this blog. Lets get to it!

The initial plan was to just collect and investigate the results from the calculations I had already conducted to model the vibrationally resolved spectra. These include all optimizations and normal-mode analyses for naked DBI, DBA and DBP as well as their monoadducts and trans-diadducts, such that solely the respective calculations for ACN were missing. Having completed the latter and putting together all the numbers (computed at the same level of theory as the vibronic spectra, i.e., B3LYP-D3BJ/SVP) afforded the following picture:
Thermochemical data for the formation of the ACN mono- and diadduct of DBI, DBA and DBP (only trans-diadduct of DBI is shown) calculated at the B3LYP-D3BJ/SVP level of theory (using the COSMO solvation model to mimic the influence of the solvent ACN here and in all of the following results). The leftmost value termed "electronic" is computed from the electronic energies only. "ZPE" additionally includes the energy changes due to zero-point vibrations, which make the formation of additional B-N lewis-bonds more expensive compared to electronic energies only (since some energy goes into the zero-point vibrations of these new bonds). "RT" adds thermal corrections for room-temperature on top of that, which is the energy is is stored in the low-frequency (torsional and collective) modes that are thermally excited already at room temperature. ZPE and RT corrections are computed from the result of the normal-mode analyses of the molecules.
The good news is that at least the ordering suits the experimental obervsations with the DBI monoadduct attaining the lowest energy, followed by DBA and eventually DBP. The bad news is that all values are about 20 kJ/mol too low to explain the experiment. The negative values of the free energies of formation ("+RT" values) for all mono-adducts suggest a quantitative formation for all of them, even if only traces of ACN are present. Remembering the experimental observations (no changes in the spectrum of DBP even in pure ACN, weak influence for DBA at high ACN concentrations, major changes only for DBI), the values are apparently systematically too low.
Considering that thermochemical calculations are (in contrast to the excited-state calculations) quite sensitive to basis-set size and that I have employed the small SVP basis, this should not really be a surprise. The magnitude of the effect, however, did surprise me. The underlying problem is that small incomplete basis sets are susceptible to the so-called basis-set superposition error (BSSE). In a nutshell, BSSE is the result of an artificially constrained electronic wavefunction. Consequently, any calculation for a larger system (e.g. adducts, dimers) attain energies that are systematically lowered compared to respective calculation for the respective subs systems (their isolated constituents, mononers), since in the supermolecular calculation the wavefunction of a fragment A can also use the basis-functions on fragment B and vice versa.
Although there are corrections for the BSSE, the straightforward approach is typically to use a larger (augmented) basis set, which is what I did. To save time in the calculations with the larger (def2-TZVP) basis set, I skipped the optimization of the the geometries and used the ones from before, which is a quite common thing to do since geometries are not as sensitive as energies to basis-set size. To save even more time I used another common trick: I did not redo the normal-mode analyses for the ZPE and RT corrections but used the ones obtained with SVP, since they are also much less basis-set dependent. Such a combined approach is abbreviated B3LYP-D3BJ/def2-TZVP//SVP (behind  the double-slash follows the level used for the geometries/corrections).
Electronic energies obtained at the B3LYP-D3BJ/def2-TZVP//SVP level of theory.
With the larger basis set, the results are already in much better agreement with the experimental observations. With the main difference being a systematic shift to lower energies of formation by about 25 kJ/mol, also the energy of the DBP adduct has moved up slightly relative to the DBI and DBA adducts. Although the values do not yet fully agree with the experiment, we can at this point most certainly answer the initial question about the formation of diadduct: Judging from these results, their formation can most certainly be excluded. The values for the DBA and DBP diaddcuts (not shown) are very similar to those of DBI with all free energies of formation well above +30 kJ/mole.

However, I wasn't quite satisfied with the agreement and more importantly just curious how ab-initio methods would compare to the DFT results. So I eventually conducted additional SCS-MP2 and CEPA/1 (a coupled-cluster variant) calculations. To also improve on the structures, I re-optimized them at the SCS-MP2/SVP level of theory and later refined them at the SCS-MP2/def2-TZVP level of theory (this was JUST possible for the DBI diadduct). For the final energies at the SCS-MP2 and CEPA/1 levels of theory, I even conducted complete basis-set (CBS) extrapolations using the def2-SVP and def2-TZVP sets. This technique estimates the energy that would be obtained with a hypothetical, complete set of basis functions from the differences between the energies of two limited sets.  The free energies of formation for the monoadducts of DBI DBA and DBP are summarized in this final figure:
Summary of the free energies of formation of the monoadducts at (from left to right) increasing levels of theory.
SCS-MP2 with the small SVP basis yields an even stronger overbinding than B3LYP, most certainly for the very same reasons (mainly BSSE). This is not too surprising since correlated methods are said to be in general more prone to basis-set incompleteness errors than DFT.
The systematic overbinding is improved but not eliminated at the mixed approach with the larger basis (def2-TZVP//SVP), and becomes slightly worse at the fully consistent SCS-MP2/def2-TZVP level of theory. Note that the difference between the mixed SVP//def2-TZVP and fully consistent def2-TZVP approaches is quite small, whereas the mixed approach is MUCH cheaper.
The CBS extrapolation, which I would have expected to correct the systematic errors, actually worsens the agreement with the experiment. All energies of formation are again systematically reduced, such that even the DBP monoadduct is suggested to be stable at this level. I had heard that MP2 tends to overbind organic molecules, but this is a larger error than I expected. In particular since I have used the spin-component scaled variant, which should be less prone to these problems.
Only at the CEPA/1/CBS level of theory do the calculated free energies of formation ultimately agree with the observed behaviour: DBI showing quantitative monoadduct formation is weakly bound, DBA showing weak but significant monoadduct formation at high MeCN concentrations is energy-neutral, and DBP, which does not show any signs of adduct formation, is strongly endothermic. To make the coupled-cluster calculations possible for these already quite large systems, I employed the very handy domain-localized pair natural orbital (DLPNO) approximation implemented in Orca, and still burned a lot of computer time.

At the time, I found it quite instructive (and ultimately satisfying) to see how these numbers eventually converge to agree with the experimentally observed behaviour at the highest level of theory, and how all of the methods show the issues I had heard about, but never observed before. I hope you has a similar experience and can take something home. If you have any questions please leave a comment.

The next post will be about the investigation of the mechanism underlying the different fluorescence quantum yields of DBI, DBA and DBP.

So long!

Montag, 19. Juni 2017

Q-Chem 5.0 with ADC/SS-PCM has been released!

While I'm still in the process of writing the second post about boron-subtituted aromatics, let me quickly mention that Q-Chem 5.0 has recently been released. 

Why is this special, you may ask? It is, because it contains the ADC/SS-PCM approach that I have developed during my PhD and that has been discussed numerous times in this blog. With this approach, you can accurately model your favourite excited states and transitions from (emission) and to (absorption) them in solution at the ADC level of theory at up to third order in perturbation theory, i.e., ADC(3), a very accurate benchmark method.
You can also employ e.g. the very efficient riSOS-ADC(2) approach and investigate quite large molecules (say: materials) with up to 500 basis functions. Since ADC is, in contrast to TD-DFT, an ab-initio method, it does provide complete and physically sound description of the electrostatics of the systems as well as its interaction with the solvent/dielectric environment. We compared the ADC2/SS-PCM and ADC3/SS-PCM approaches already in the publication presenting the method.

I'm currently in the final steps of publishing another article about a project with my friend and colleague Felix (find his blog in the menu), which turned out to be a nice showcase for the model. Turns out I just made this table-of-contents graphic, which I want to share with you:

It shows the excitation-hole (in blue) and excited electron (in red) of the lowest excited singlet state (S1) of an already quite large push-push system that we studied in the article. In the top left corner, it shows the excited electron computed with the SS-PCM simulating the non-polar solvent cyclohexane, and in the bottom right corner with parameters for the polar solvent acetonitrile. From this quite intuitive visualization it becomes clear how the electron-hole or in other word charge separation (the distance between the blue and red blops) increases as it is stabilized by the polar solvent and, more importantly, how this affects the properties of the system. I'll post a link to the article as soon as it appears online. (*Link)

A brief description of the theory behind the ADC/SS-PCM model and its capabilities can be found in the Q-Chem 5.0 online manual, which you can find here.

Please note that very unfortunately, due to a last minute change of some defaults of the PCM solvent model that was not communicated very well, the description of ADC/SS-PCM in the in the manual is incomplete concerning one detail: For all calculations with the model the line "ChargeSeparation Marcus" has to be included in the $pcm block of the input file. This bug will be fixed with the next release (5.0.1) in July.

So long,

Dienstag, 6. Juni 2017

A story about boron-substituted aromatics - part one

Another project I have been working on last year was concerned with the quantum-chemical investigation of a bunch of boron-containing aromatics that have been synthesized and studied in the group of Prof. Wagner in Frankfurt. This project is interesting for a number of reasons, the most story-worthy of which is that Prof. Wagners lecture "Allgemeine und Anorganische Chemie" marked the very beginning of my academic career in October 2005. This cooperation was a nice opportunity to "turn the spit" and explain aspects of photochemistry to the Professor who introduced me to chemistry as a scientific discipline some 10 years ago. But also from a scientific point of view this project is exciting. These boron compounds do in general have a number of quite unique photophysical properties (some e.g. exhibit temperature-dependent delayed fluorescence with a curios solvent-dependency), and the project turned out to provide the opportunities to try out a couple of methods I had wanted to try out for a while. The paper itself can be found here. In the following I want to shed some light onto aspects that didn't get too much attention in the brevity of a scientific article.

Two of the four investigated molecules can be derived from anthracene and pentacene by replacing the middle carbons with boron atoms and are named accordingly: Dibora-anthracene (DBA) and dibora-pentacene (DBP). Diborinine (DBI) consists of two antiaromatic biphenylene subunits bridged via the dibora-benzylic motif, while iso-DBP is an asymetric variant of DBP, in which the boron-containing ring is shifted to one side.

The initial motivation for the project was to find an explanation for the surprisingly different fluorescence quantum yields of these closely related molecules. While isoDBP and DBP exhibit high quantum yields of 90% and 50%, respectively, DBA is hardly fluorescent (2%) and DBI completely dark. But let us postpone this question to one of the upcoming posts about this project and start  off with another question that popped up while I was inspecting the absorption spectra of these compounds in various solvents: DBI and to some extent also DBA exhibit a huge influence of some solvents onto the shape of the absorption spectra. Specifically, the main peaks of the absorption spectrum of DBI in non-polar cyclohexane (CHX) are blue-shifted and almost vanish in acetonitrile (ACN), while this is not the case in other polar solvents. We concluded that there has to exist an specific interaction between the lewis-basic solvent ACN and the lewis-acidic boron-atoms in DBI, or in other words that ACN behaves as a non-innocent solvent.

As a first step to confirm this hypothesis, I computed the structures and electronic energies of one mono- and two di-adducts (cis and trans) with a cheap and fast (exploratory) methodology (B3LYP-D3BJ/SVP, today I would use PBEh-3c). The resulting structures are shown together with the structure of bare DBI in the following picture:

Calculated ground-state minimum structures of bare DBI (a), the ACN-monoadduct (b), the trans-diadduct (c) and the cis-diadduct (d). Already from these cheap, approximate calculations it was evident that only the mono- and trans-diadduct are energetically feasible.
For the energetically feasible adducts of DBI and DBA, I subsequently calculated vibrationally resolved absorption spectra, which can be directly compared to the experimental spectra. For this purpose, I used the same methodology, namely TD-B3LYP with a small SVP basis, since it is fast and due to a surprisingly stable inherent error compensation typically also quite accurate (but also not reliable).

The calculation of vibrationally resolved spectra is one of the things I wanted to try out for a while now and hence, I'll briefly explain the hows and whys in the following. To make this type of calculation possible for such a large molecule, one has to employ a drastic-sounding approximation with a very long name: Independent-mode-shifted-harmonic-oscillator (IMDHO)-approximation. This is necessary to reduce the untraceable problem of knowing the shape of all potential-energy surfaces of all excited-states of interest around the initial (ground-state) geometry to something actually doable. Using IMDHO, the problem at hand is reduced to knowing the vibrational frequencies (normal modes) of the ground-state and the gradients along them in each of the excited states of interest. The assumptions required to get there are:
  1. All modes can be approximated as a harmonic oscillator. (Common and not too bad for the ground state, crude for excited states)
  2. The vibrational frequencies in the excited states are so similar to those of the ground state than we can treat them as being identical (Very crude)
From these assumptions, it follows that the only difference between the ground- and excited-state PES is the position of the respective minima, which is why its called "shifted"HO. Since we are in the harmonic approximation, all we need to know this shift w.r.t. to ground-state minimum are the gradients along these normal modes in the excited states of interest. Hence, to obtain the vibrationally resolved absorption spectrum within the IMDHO approximation, one first computes the ground state vibrational frequencies (normal-modes), and subsequently the gradient along these normal-modes for each of the excited-states. Having these quantities, we (actually the handy tool ORCA_ASA does that for us) can build the approximate PES and compute the coupling between the electronic and vibrational excitations, yielding the spectra, which are shown together with the absorption spectra recorded in various mixtures of CHX and ACN in the following picture:

Comparison of measured (top) and calculated (bottom) absorption spectra of DBI and DBA and their respective ACN mono- and diadducts. Calculations of the vibrationally resolved spectra were carried out at the (TD)B3LYP/SVP level of theory employing the so-called independent-mode-shifted-harmonic-oscillator (IMDHO)-approximation. Considering the crude IMDHO approximation and the poor level of electronic structure theory, the agreement between experiment and calculation is surprisingly good.

Due to the very reasonable agreement between experiment and calculation, in particular for DBI, we can doubtlessly identify the ACN-monoadduct in the spetra recorded in ACN. However, it is difficult to say this with certainty since the trans-diadduct is essentially dark (no absorption) and would not show up in the spectra. Thus, to further investigate the issue I conducted a thorough thermochemical analysis of the formation of the ACN adducts for DBA, DBI and DBP, which I will present and discuss in the upcoming post.

So long!

Freitag, 26. Mai 2017

Reporting Back

Fellow scientists,

it has been quiet here for way over a year now, and I feel I'm owing an explanation: After I was awarded a Feodor-Lynen fellowship early in 2016 (thank you very much for this opportunity, Alexander-von-Humboldt Foundation!), I spent much of the year preparing to move to the other side of the planet to become a Postdoc in Peter Schwerdtfegers research group. On top of that my daughter was born in February, which did the rest to keep me from posting here. But now that we are settled in and out of the worst baby-issues I want to keep this blog alive. For this purpose, I'll first write a couple of posts about articles I've published since my last contribution and eventually about my current projects in the Schwerdtfeger group, namely the development of a polarizable forcefield for Methyl-ammonium lead iodide and the generation of pseudopotentials for super-heavy elements.

Let me begin this series with a project that I have already mentioned in my last three posts: The work on self-consistent, state-specific PCM equilibrium solvation. This self-consistent variant of the SS-PCM approach constitutes a theoretical framework to treat "long-lived" excited states in solution and completes the developments started during my PhD. It allows to equilibrate the excited-state wavefunction with its self-induced polarization, which is the way to go whenerver an excited-states lives for a couple of picoseconds ("long-lived"). Since this is the case for most of the experimentally accessible properties and/or processes originating from excited states, this is a quite important capability. Combined with the perturbation-theoretical variant of the state-specific approach that I've implemented during my PhD, these models enable a calculation and investigation of virtually all photochemical processes in solution, like e.g. ground- and excited state absorption, emission as well as photochemical reactivity. The interface works with all orders and variants of the Algebraic-Diagrammatic Construction (ADC) method developed in Andreas Dreuws group. The article on the topic eventually appeared in PCCP in early 2017. It is available in Q-Chem from version 4.4.2, but documented in the manual starting only from version 5 (release date 1st of June), where you can also find a brief introduction into the theory. It follows a brief summary of the article:

We demonstrated that a general, state-specific PCM in  combination with an ADC(2) or ADC(3) description of the solute’s electronic structure provides excellent energies of solvent-relaxed states and vertical transitions in solution. Since we limited our approach to the state-specific picture, where the solvent effect enters the quantum-chemical calculation only via one-electron charge-density Coulomb integrals, the underlying ADC  equations are unmodified and the model can be used in combination with any flavor of ADC. Moreover, due to this clear separation between quantum-chemical part of the calculation and the solvent model, the results are presumably of general validity for both, the SS-PCM approach as well as the excited-state method.
To validate ADC/SS-PCM, a set of symmetric, ionized dimers was employed, whose lowest energy CT states are formally identical to the broken-symmetry ground state. Computing the latter using the well-established MP/PTE approach and comparing the results to the CT state computed using ADC/SS-PCM, the  deviation between the two methods was found to be 0.02 eV over a wide range of dielectric constants. This holds even for the challenging nitromethane case where electron correlation effects are large.
Ultimately, ADC/SS-PCM was employed to investigate solvent-relaxed potential energy surfaces of 4-(N,N)-dimethylamino-benzonitrile. The agreement with experimental fluorescence data is excellent for the LE state under all circumstances, in particular with ADC(2). For the CT state, however, it was demonstrated that an intra-molecular twisting coordinate has to be considered in detail to achieve a similar agreement. In general, the agreement of ADC(2)/SS-PCM is consistently better than for ADC(3) for fluorescence energies. For the relative energies of the LE and CT states, however, only ADC(3) yields results that are consistent with the experimental observation of dual fluorescence in polar solvents but not in non-polar ones. This was traced back to an underestimation of the energy of the CT state compared to the LE state at second order of perturbation theory.

In my next post, I'll write about an article that originated from a cooperation with the experimental group of Prof. Wagner in Frankfurt, who basically introduced me to chemistry at university level.

So long!

Freitag, 8. Januar 2016

A summary of recent publications and their connection to equilibrium solvation - happy 2016!

Let me begin 2016 with a summary of a couple of papers that have been accepted for publication in recent weeks before I use one of them to lead over to the ongoing work I reported in the last two posts.

A few weeks ago, my wife Steffi and I celebrated our first "Mewes & Mewes et al" paper making it into PCCP. It offers a fresh perspective onto the electronic structure of excitons in poly(para-phenylene vinylene) or short PPV, which is something like the guinea pig of the organic electronics community. We studied PPV using highly accurate ab-initio quantum-chemical methods (ADC of up to third order) in combination with the very handy wavefunction analysis routines developed by Felix and Steffi.

Furthermore, two other papers to which I contributed were published recently:
  • Firstly, there is this work in PNAS on signatures and control of strong-field dynamics in a complex (i.e. molecular) system. My contribution to this work were quantum-chemical insights, which eventually helped Kristina and her co-workers to set up a theoretical model to simulate the exposure and response of molecules to strong laser pulses.
  • Secondly, we finally managed to publish the second paper on our  non-equilibrium PCM in Q-Chem. This second article is concerned with unexpected discrepancies between two alternative schemes of separating the fast and slow components of the polarization (I explained this in one of my recent posts). Zhi-Qiang found that the so-called Pekar and Marcus partitioning of the polarization, which should formally lead to identical results, yield significantly different results. In the article, he traces this back to the discretization of the PCM equations for the actual, numerical implementation. Eventually, he demonstrates that SSVPE (one of the three common flavours of PCMs besides the conductor-like approximation (C-PCM) and the integral-equation formalism (IEF-PCM)) exhibit the largest deviations (they are still rather small and in the ballpark of 0.01~eV)
Interestingly, this last finding directly relates to my current work on equilibrium solvation models. I fortuitously found very similar results for C-PCM, IEF-PCM and SSVPE while doing a little sanity test of my solvent-field equilibration code. For this purpose, I combined the perturbative non-equilibrium formalism with the new, self-consistent equilibrium solvation functionality. My expectation was that the non-equilibrium corrections for any state should become zero during the solvent-field equilibration for the respective state. Why? Because the non-equilibrium correction constitutes a perturbative estimate of how the relaxation of the fast component of the polarization with respect to a certain excited state would reduce its energy. But obviously, the energy should not be changed if the solvent field is already fully equilibrated for the respective state.
Surprisingly, the non-equilibrium terms do become zero only with C-PCM and IEF-PCM, but not with SSVPE. The deviations I find are in the same ballpark as the differences between the Pekar and Marcus schemes. Currently, I'm in the process of writing up these results and hence, expect more updates the in near future.

So long!

Mittwoch, 2. Dezember 2015

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!