Category Archives: TD-DFT
Electronic excitations are calculated vertically according to the Frank—Condon principle, this means that the geometry does not change upon the excitation and we merely calculate the energy required to reach the next electronic state. But for some instances, say calculating not only the absorption spectra but also the emission, it is important to know what the geometry minimum of this final state looks like, or if it even exists at all (Figure 1). Optimizing the geometry of a given excited state requires the prior calculation of the vertical excitations whether via a multireference method, quantum Monte Carlo, or the Time Dependent Density Functional Theory, TD-DFT, which due to its lower computational cost is the most widespread method.
Most single-reference treatments, ab initio or density based, yield good agreement with experiments for lower states, but not so much for the higher excitations or process that involve the excitation of two electrons. Of course, an appropriate selection of the method ensures the accuracy of the obtained results, and the more states are considered, the better their description although it becomes more computationally demanding in turn.
In Gaussian 09 and 16, the argument to the ROOT keyword selects a given excited state to be optimized. In the following example, five excited states are calculated and the optimization is requested upon the second excited state. If no ROOT is specified, then the optimization would be carried out by default on the first excited state (Where L.O.T. stands for Level of Theory).
#p opt TD=(nstates=5,root=2) L.O.T.
Gaussian16 includes now the calculation of analytic second derivatives which allows for the calculation of vibrational frequencies for IR and Raman spectra, as well as transition state optimization and IRC calculations in excited states opening thus an entire avenue for the computation of photochemistry.
If you already computed the excited states and just want to optimize one of them from a previous calculation, you can read the previous results with the following input :
#p opt TD=(Read,Root=N) L.O.T. Density=Current Guess=Read Geom=AllCheck
Common problems. The following error message is commonly observed in excited state calculations whether in TD-DFT, CIS or other methods:
No map to state XX, you need to solve for more vectors in order to follow this state.
This message usually means you need to increase the number of excited states to be calculated for a proper description of the one you’re interested in. Increase the number N for nstates=N in the route section at higher computational cost. A rule of thumb is to request at least 2 more states than the state of interest. This message can also reflect the fact that during the optimization the energy ordering changes between states, and can also mean that the ground state wave function is unstable, i.e., the energy of the excited state becomes lower than that of the ground state, in this case a single determinant approach is unviable and CAS should be used if the size of the molecule allows it. Excited state optimizations are tricky this way, in some cases the optimization may cross from one PES to another making it hard to know if the resulting geometry corresponds to the state of interest or another. Gaussian recommends changing the step size of the optimization from the default 0.3 Bohr radius to 0.1, but obviously this will make the calculation take longer.
If the minimum on the excited state potential energy surface (PES) doesn’t exist, then the excited state is not bound; take for example the first excited state of the H2 molecule which doesn’t show a minimum, and therefore the optimized geometry would correspond to both H atoms moving away from each other indefinitely (Figure 2). Nevertheless, a failed optimization doesn’t necessarily means the minimum does not exist and further analysis is required, for instance, checking the gradient is converging to zero while the forces do not.
I have written about extracting information from excited state calculations but an important consideration when analyzing the results is the proper use of the keyword density.
This keyword let’s Gaussian know which density is to be used in calculating some results. An important property to be calculated when dealing with excited states is the change in dipole moment between the ground state and any given state. The Transition Dipole Moment is an important quantity that allows us to predict whether any given electronic transition will be allowed or not. A change in the dipole moment (i.e. non-zero) of a molecule during an electronic transition helps us characterize said transition.
Say you perform a TD-DFT calculation without the density keyword, the default will provide results on the lowest excited state from all the requested states, which may or may not be the state of interest to the transition of interest; you may be interested in the dipole moment of all your excited states.
Three separate calculations would be required to calculate the change of dipole moment upon an electronic transition:
1) A regular DFT for the ground state as a reference
2) TD-DFT, to calculate the electronic transitions; request as many states as you need/want, analyze it and from there you can see which transition is the most important.
3) Request the density of the Nth state of interest to be recovered from the checkpoint file with the following route section:
# TD(Read,Root=N) LOT Density=Current Guess=Read Geom=AllCheck
replace N for the Nth state which caught your eye in step number 2) and LOT for the Level of Theory you’ve been using in the previous steps. That should give you the dipole moment for the structure of the Nth excited state and you can compare it with the one in the ground state calculated in 1). Again, if density=current is not used, only properties of N=1 will be printed.
This is a guest post by our very own Gustavo “Gus” Mondragón whose work centers around the study of excited states chemistry of photosynthetic pigments.
When you’re calculating excited states (no matter the method you’re using, TD-DFT, CI-S(D), EOM-CCS(D)) the analysis of the orbital contributions to electronic transitions poses a challenge. In this post, I’m gonna guide you through the CI-singles excited states calculation and the analysis of the electronic transitions.
I’ll use adenine molecule for this post. After doing the corresponding geometry optimization by the method of your choice, you can do the excited states calculation. For this, I’ll use two methods: CI-Singles and TD-DFT.
The route section for the CI-Singles calculation looks as follows:
#p CIS(NStates=10,singlets)/6-31G(d,p) geom=check guess=read scrf=(cpcm,solvent=water)
adenine excited states with CI-Singles method
I use the same geometry from the optimization step, and I request only for 10 singlet excited states. The CPCP implicit solvation model (solvent=water) is requested. If you want to do TD-DFT, the route section should look as follows:
#p FUNCTIONAL/6-31G(d,p) TD(NStates=10,singlets) geom=check guess=read scrf=(cpcm,solvent=water)
adenine excited states with CI-Singles method
Where FUNCTIONAL is the DFT exchange-correlation functional of your choice. Here I strictly not recommend using B3LYP, but CAM-B3LYP is a noble choice to start.
Both calculations give to us the excited states information: excitation energy, oscillator strength (as f value), excitation wavelength and multiplicity:
Excitation energies and oscillator strengths:
Excited State 1: Singlet-A 6.3258 eV 196.00 nm f=0.4830 <S**2>=0.000
11 -> 39 -0.00130
11 -> 42 -0.00129
11 -> 43 0.00104
11 -> 44 -0.00256
11 -> 48 0.00129
11 -> 49 0.00307
11 -> 52 -0.00181
11 -> 53 0.00100
11 -> 57 -0.00167
11 -> 59 0.00152
11 -> 65 0.00177
The data below corresponds to all the electron transitions involved in this excited state. I have to cut all the electron transitions because there are a lot of them for all excited states. If you have done excited states calculations before, you realize that the HOMO-LUMO transition is always an important one, but not the only one to be considered. Here is when we calculate the Natural Transition Orbitals (NTO), by these orbitals we can analyze the electron transitions.
For the example, I’ll show you first the HOMO-LUMO transition in the first excited state of adenine. It appears in the long list as follows:
35 -> 36 0.65024
The 0.65024 value corresponds to the transition amplitude, but it doesn’t mean anything for excited state analysis. We must calculate the NTOs of an excited state from a new Gaussian input file, requesting from the checkpoint file we used to calculate excited states. The file looks as follows:
#p SP geom=allcheck guess=(read,only) density=(Check,Transition=1) pop=(minimal,NTO,SaveNTO)
I want to say some important things right here for this last file. See that no level of theory is needed, all the calculation data is requested from the checkpoint file “adenine.chk”, and saved into the new checkpoint file “adNTO1.chk”, we must use the previous calculated density and specify the transition of interest, it means the excited state we want to analyze. As we don’t need to specify charge, multiplicity or even the comment line, this file finishes really fast.
After doing this last calculation, we use the new checkpoint file “adNTO1.chk” and we format it:
formchk -3 adNTO1.chk adNTO1.fchk
If we open this formatted checkpoint file with GaussView, chemcraft or the visualizer you want, we will see something interesting by watching he MOs diagram, as follows:
We can realize that frontier orbitals shows the same value of 0.88135, which means the real transition contribution to the first excited state. As these orbitals are contributing the most, we can plot them by using the cubegen routine:
cubegen 0 mo=homo adNTO1.fchk adHOMO.cub 0 h
This last command line is for plotting the equivalent as the HOMO orbital. If we want to plot he LUMO, just change the “homo” keyword for “lumo”, it doesn’t matter if it is written with capital letters or not.
You must realize that the Natural Transition Orbitals are quite different from Molecular Orbitals. For visual comparisson, I’ve printed also the molecular orbitals, given from the optimization and from excited states calculations, without calculating NTOs:
These are the molecular frontier orbitals, plotted with Chimera with 0.02 as the isovalue for both phase spaces:
The frontier NTOs look qualitatively the same, but that’s not necessarily always the case:
If we analyze these NTOs on a hole-electron model, the HOMO refers to the hole space and the LUMO refers to the electron space.
Maybe both orbitals look the same, but both frontier orbitals are quite different between them, and these last orbitals are the ones implied on first excited state of adenine. The electron transition will be reported as follows:
If I can do a graphic summary for this topic, it will be the next one:
NTOs analysis is useful no matter if you calculate excited states by using CIS(D), EOM-CCS(D), TD-DFT, CASSCF, or any of the excited states method of your election. These NTOs are useful for population analysis in excited states, but these calculations require another software, MultiWFN is an open-source code that allows you to do this analysis, and another one is called TheoDORE, which we’ll cover in a later post.
The canonical molecular orbital depiction of an electronic transition is often a messy business in terms of a ‘chemical‘ interpretation of ‘which electrons‘ go from ‘which occupied orbitals‘ to ‘which virtual orbitals‘.
Natural Transition Orbitals provide a more intuitive picture of the orbitals, whether mixed or not, involved in any hole-particle excitation. This transformation is particularly useful when working with the excited states of molecules with extensively delocalized chromophores or multiple chromophoric sites. The elegance of the NTO method relies on its simplicity: separate unitary transformations are performed on the occupied and on the virtual set of orbitals in order to get a localized picture of the transition density matrix.
 R. L. Martin, J. Chem. Phys., 2003, DOI:10.1063/1.1558471.
After running a TD-DFT calculation with the keyword TD(Nstates=n) (where n = number of states to be requested) we need to take that result and launch a new calculation for the NTOs but lets take it one step at a time. As an example here’s phenylalanine which was already optimized to a minimum at the B3LYP/6-31G(d,p) level of theory. If we take that geometry and launch a new calculation with the TD(Nstates=40) in the route section we obtain the UV-Vis spectra and the output looks like this (only the first three states are shown):
Excitation energies and oscillator strengths: Excited State 1: Singlet-A 5.3875 eV 230.13 nm f=0.0015 <S**2>=0.000 42 -> 46 0.17123 42 -> 47 0.12277 43 -> 46 -0.40383 44 -> 45 0.50838 44 -> 47 0.11008 This state for optimization and/or second-order correction. Total Energy, E(TD-HF/TD-KS) = -554.614073682 Copying the excited state density for this state as the 1-particle RhoCI density. Excited State 2: Singlet-A 5.5137 eV 224.86 nm f=0.0138 <S**2>=0.000 41 -> 45 -0.20800 41 -> 47 0.24015 42 -> 45 0.32656 42 -> 46 0.10906 42 -> 47 -0.24401 43 -> 45 0.20598 43 -> 47 -0.14839 44 -> 45 -0.15344 44 -> 47 0.34182 Excited State 3: Singlet-A 5.9254 eV 209.24 nm f=0.0042 <S**2>=0.000 41 -> 45 0.11844 41 -> 47 -0.12539 42 -> 45 -0.10401 42 -> 47 0.16068 43 -> 45 -0.27532 43 -> 46 -0.11640 43 -> 47 0.16780 44 -> 45 -0.18555 44 -> 46 -0.29184 44 -> 47 0.43124
The oscillator strength is listed on each Excited State as “f” and it is a measure of the probability of that excitation to occur. If we look at the third one for this phenylalanine we see f=0.0042, a very low probability, but aside from that the following list shows what orbital transitions compose that excitation and with what energy, so the first line indicates a transition from orbital 41 (HOMO-3) to orbital 45 (LUMO); there are 10 such transitions composing that excitation, visualizing them all with canonical orbitals is not an intuitive picture, so lets try the NTO approach, we’re going to take excitation #10 for phenylalanine as an example just because it has a higher oscillation strength:
%chk=Excited State 10: Singlet-A 7.1048 eV 174.51 nm f=0.3651 <S**2>=0.000 41 -> 45 0.35347 41 -> 47 0.34685 42 -> 45 0.10215 42 -> 46 0.17248 42 -> 47 0.13523 43 -> 45 -0.26596 43 -> 47 -0.22995 44 -> 46 0.23277
Each set of NTOs for each transition must be calculated separately. First, copy you filename.chk file from the TD-DFT result to a new one and name it after the Nth state of interest as shown below (state 10 in this case). NOTE: In the route section, replace N with the number of the excitation of interest according to the results in filename.log. Run separately for each transition your interested in:
#chk=state10.chk #p B3LYP/6-31G(d,p) Geom=AllCheck Guess=(Read,Only) Density=(Check,Transition=N) Pop=(Minimal,NTO,SaveNTO) 0 1 --blank line--
By requesting SaveNTO, the canonical orbitals in the state10.chk file are replaced with the NTOs for the 10th excitation, this makes it easier to plot since most visualizers just plot whatever set of orbitals they read in the chk file but if they find the canonical MOs then one would need to do some re-processing of them. This is much more straightforward.
Now we format our chk files into fchk with the formchk utility:
formchk -3 filename.chk filename.fchk
formchk -3 state10.chk state10.fchk
If we open filename.fchk (the file where the original TD-DFT calculation is located) with GaussView we can plot all orbitals involved in excited state number ten, those would be seven orbitals from 41 (HOMO-3) to 47 (LUMO+2) as shown in figure 1.
If we now open state10.fchk we see that the numbers at the side of the orbitals are not their energy but their occupation number particular to this state of interest, so we only need to plot those with highest occupations, in our example those are orbitals 44 and 45 (HOMO and LUMO) which have occupations = 0.81186; you may include 43 and 46 (HOMO-1 and LUMO+1, respectively) for a much more complete description (occupations = 0.18223) but we’re still dealing with 4 orbitals instead of 7.
The NTO transition 44 -> 45 is far easier to conceptualize than all the 10 combinations given in the canonical basis from the direct TD-DFT calculation. TD-DFT provides us with the correct transitions, NTOs just paint us a picture more readily available to the chemist mindset.
NOTE: for G09 revC and above, the %OldChk option is available, I haven’t personally tried it but using it to specify where the excitations are located and then write the NTOs of interest into a new chk file in the following way, thus eliminating the need of copying the original chk file for each state:
NTOs are based on the Natural Hybrid orbitals vision by Löwdin and others, and it is said to be so straightforward that it has been re-discovered from time to time. Be that as it may, the NTO visualization provides a much clearer vision of the excitations occurring during a TD calculation.
Thanks for reading, stay home and stay safe during these harsh days everyone. Please share, rate and comment this and other posts.
Photosynthesis, the basis of life on Earth, is based on the capacity a living organism has of capturing solar energy and transform it into chemical energy through the synthesis of macromolecules like carbohydrates. Despite the fact that most of the molecular processes present in most photosynthetic organisms (plants, algae and even some bacteria) are well described, the mechanism of energy transference from the light harvesting molecules to the reaction centers are not entirely known. Therefore, in our lab we have set ourselves to study the possibility of some excitonic transference mechanisms between pigments (chlorophyll and its corresponding derivatives). It is widely known that the photophysical properties of chlorophylls and their derivatives stem from the electronic structure of the porphyrin and it is modulated by the presence of Mg but its not this ion the one that undergoes the main electronic transitions; also, we know that Mg almost never lies in the same plane as the porphyrin macrocycle because it bears a fifth coordination whether to another pigment or to a protein that keeps it in place (Figure 1).
During our calculations of the electronic structure of the pigments (Bacteriochlorophyll-a, BChl-a) present in the Fenna-Matthews-Olson complex of sulfur dependent bacteria we found that the Mg²⁺ ion at the center of one of these pigments could in fact create an intermolecular interaction with the C=C double bond in the phytol fragment which lied beneath the porphyrin ring.
This would be the first time that a dihapto coordination is suggested to occur in any chlorophyll and that on itself is interesting enough but we took it further and calculated the photophysical implications of having this fifth intramolecular dihapto coordination as opposed to a protein or none for that matter. Figure 3 shows that the calculated UV-Vis spectra (calculated with Time Dependent DFT at the CAM-B3LYP functional and the cc-pVDZ, 6-31G(d,p) and 6-31+G(d,p) basis sets). A red shift is observed for the planar configuration, respect to the five coordinated species (regardless of whether it is to histidine or to the C=C double bond in the phytyl moiety).
Before calculating the UV-Vis spectra, we had to unambiguously define the presence of this observed interaction. To that end we calculated to a first approximation the C-Mg Wiberg bond indexes at the CAM-B3LYP/cc-pVDZ level of theory. Both values were C(1)-Mg 0.022 and C(2)-Mg 0.032, which are indicative of weak interactions; but to take it even further we performed a non-covalent interactions analysis (NCI) under the Atoms in Molecules formalism, calculated at the M062X density which yielded the presence of the expected critical points for the η²Mg-(C=C) interaction. As a control calculation we performed the same calculation for Magnoscene just to unambiguously assign these kind of interactions (Fig 4, bottom).
This research is now available at the International Journal of Quantum Chemistry. A big shoutout and kudos to Gustavo “Gus” Mondragón for his work in this project during his masters; many more things come to him and our group in this and other research ventures.
If you work in the field of photovoltaics or polyacene photochemistry, then you are probably aware of the Singlet Fission (SF) phenomenon. SF can be broadly described as the process where an excited singlet state decays to a couple of degenerate coupled triplet states (via a multiexcitonic state) with roughly half the energy of the original singlet state, which in principle could be centered in two neighboring molecules; this generates two holes with a single photon, i.e. twice the current albeit at half the voltage (Fig 1).
It could also be viewed as the inverse process to triplet-triplet annihilation. An important requirement for SF is that the two triplets to which the singlet decays must be coupled in a 1(TT) state, otherwise the process is spin-forbidden. Unfortunately (from a computational perspective) this also means that the 3(TT) and 5(TT) states are present and should be taken into account, and when it comes to chlorophyll derivatives the task quickly scales.
SF has been observed in polyacenes but so far the only photosynthetic pigments that have proven to exhibit SF are some carotene derivatives; so what about chlorophyll derivatives? For a -very- long time now, we have explored the possibility of finding a naturally-occurring, chlorophyll-based, photosynthetic system in which SF could be possible.
But first things first; The methodology: It was soon enough clear, from María Eugenia Sandoval’s MSc thesis, that TD-DFT wasn’t going to be enough to capture the whole description of the coupled states which give rise to SF. It was then that we started our collaboration with SF expert, Prof. David Casanova from the Basque Country University at Donostia, who suggested the use of Restricted Active Space – Spin Flip in order to account properly for the spin change during decay of the singlet excited state. A set of optimized bacteriochlorophyll-a molecules (BChl-a) were oriented ad-hoc so their Qy transition dipole moments were either parallel or perpendicular; the rate to which SF could be in principle present yielded that both molecules should be in a parallel Qy dipole moments configuration. When translated to a naturally-occurring system we sought in two systems: The Fenna-Matthews-Olson complex (FMO) containing 7 BChl-a molecules and a chlorosome from a mutant photosynthetic bacteria made up of 600 Bchl-d molecules (Fig 2). The FMO complex is a trimeric pigment-protein complex which lies between the antennae complex and the reaction center in green sulfur dependent photosynthetic bacteria such as P. aestuarii or C. tepidium, serving thus as a molecular wire in which is known that the excitonic transfer occurs with quantum coherence, i.e. virtually no energy loss which led us to believe SF could be an operating mechanism. So far it seems it is not present. However, for a crystallographic BChl-d dimer present in the chlorosome it could actually occur even when in competition with fluorescence.
I will keep on blogging more -numerical and computational- details about these results and hopefully about its publication but for now I will wrap this post by giving credit where credit is due: This whole project has been tackled by our former lab member María Eugenia “Maru” Sandoval and Gustavo Mondragón. Finally, after much struggle, we are presenting our results at WATOC 2017 next week on Monday 28th at poster session 01 (PO1-296), so please stop by to say hi and comment on our work so we can improve it and bring it home!
With pleasure I announce that last week our very own Gustavo “Gus” Mondragón became the fifth undergraduate student from my lab to defend his BSc thesis and it has to be said that he did it admirably so.
Gus has been working with us for about a year now and during this time he not only worked on his thesis calculating excited states for bacteriochlorophyl pigments but also helped us finishing some series of calculations on calix[n]arene complexes of Arsenic (V) acids, which granted him the possibility to apear as a co-author of the manuscript recently published in JIPH. Back in that study he calculated the interaction energies between a family of calix macrocycles and arsenic acid derivatives in order to develop a suitable extracting agent.
For his BSc thesis, Gus reproduced the UV-Vis absorption spectra of bacteriochlorophyll-a pigments found in the Fenna-Matthews-Olson complex of photosynthetic purple bacteria using Time Dependent Density Functional Theory (TD-DFT) with various levels of theory, with PBEPBE yielding the best results among the tried set. These calculations were performed at the crystallographic conformation and at the optimized structure, also, in vacuo results were compared to those in implicit solvent (SMD, MeOH). He will now move towards his masters where he will further continue our research on photosynthesis.
Thank you, Gustavo, for your hard work and your sense of humor. Congratulations on this step and may many more successes come your way.
It is with great pride that I’d like to announce that for the first time we have a Masters Student graduated from this Comp.Chem. lab: María Eugenia “Maru” Sandoval-Salinas has finished her graduate studies and just last Friday defended her thesis admirably earning not only the degree of Masters of Science in Chemistry but doing so with the highest honors given by the National Autonomous University of Mexico.
Maru’s thesis is for many reasons a landmark in this lab not only because it is the first graduate thesis published from our lab but also the first document on our work about the study of Photosynthesis, a long sought after endeavor now closer to publication. It must also be said that Maru came to this lab when she was an undergraduate student five years ago when I just recently joined UNAM as a researcher fresh out of a postdoc stay. After getting her B.Sc. degree and publishing an article in JCTC (DOI: 10.1021/ct4004178) she now is about to publish more papers that I’m sure will be as highly ranked as the previous one. Thus, Maru was a pioneer in our lab giving it a vote of confidence when we had little to nothing to show for; thanks to her hard work and confidence, along with that of the students who have followed her, we managed to succeed as a consolidated research group in the field of computational chemistry.
More specifically, her thesis centered around finding a mechanism for the excitonic transference between pigments (bacteriochlorophyl-a, BChl-a) in the Fenna-Matthews-Olson (FMO) complex, a protein trimer with seven BChl-a molecules in each monomer, located between the antenna complex and the reaction center in green sulfur bacteria. Among the possible mechanisms explored were Förster’s theory, a modification to Marcus’ theory and finally we explored the possibility of Singlet Fission occurring between adjacent molecules with the help of Dr. David Casanova from the Basque Country University where Maru took a short research stay last autumn. Since nature doesn’t conform to any specific mechanism -specially in a complex arrangement such as the FMO- then it could be possible that a combination of the above might also occur but lets just wait for the papers to be published to discuss it. Calculations were performed through the TD-DFT and the C-DFT formalisms using G09 and Q-Chem; comparing experimental data in CH3OH (SMD implicit calculations with the SVWN5 functional) were undertaken previously for selection of the level of theory.
Now, after two original theses written and successfully defended, an article published in JCTC and more in process, at least five posters, a couple of oral presentations and countless hours at her desk, Maru will go pursuit a PhD abroad where I’m sure she will exceed anyone’s expectations with her work, drive, dedication and scientific curiosity. Thank you, Maru, for all your hard work and trust when this lab needed it the most, we wish you the best for you earn it. You will surely be missed.
Last week at the congress of the Mexican Society of Chemistry I presented some of our results in the study of photosynthesis. Below I embeded the talk. Unfortunately for the wider audience of this blog, the talk is in Spanish (if anyone out there is willing to make subtitles for it I’ll hire you on the spot!)
The slides are also in Spanish although they should be easier to follow for non-Spanish speakers and they are uploaded in SlideShare at this link.
A big thank you to Maria Eugenia “Maru” Sandoval for all the hard work and time invested in this project!
Thanks for clicking!