Photosynthesis and Singlet Fission – #WATOC2017 PO1-296
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).
Jablonski’s Diagram for SF
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.
FMO Complex. Trimer (left), monomer (center), pigments (right)
BChQRU chlorosome. 600 Bchl-d molecules
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!
A New Graduate Student – Medicinal #CompChem on HIV-1
Last Friday Durbis Castillo-Pazos became officially a graduated chemist. His presentation about the in silico development of entry inhibtors for the HIV-1 virus was very clear and straightforward and he performed admirably during the examination for which he graduated with Honors from the Mexico State Autonomous University (UAEMex).
Ever since he came to my lab, Durbis was adamant to work in some project related to medicinal chemistry which is not really my thing. Still, we were able to take an old project related to the in silico design of entry inhibitors for the protein GP120 which is responsible for the entry of the HIV-1 virus into human T-cells; the very first stage of the HIV infection which leads to AIDS. The lengths to which Durbis took the project were remarkable but none of that would have been possible without the leadership of my good friend and colleague Dr. Antonio Romo from the Queretaro Autonomous University (UAQ) who not only guided us through the nuances of the field but also on the intricacies of working with Schrödinger Inc.’s software, MAESTRO.
Suffice it to say that based on a piperazine core and four thousand fragments with commercial pharmaceutical applications, Durbis built a 16.3 million compounds library from which a few candidates are selected as potential drug molecules after several docking steps at various precision levels and molecular dynamics simulations plus QSAR related analysis. More about the details in future posts.
The knowledge gained by Durbis allowed him to become a member of the QSAR team led by Dr. Karina Martinez, at the Institute of Chemistry UNAM which provides QSAR studies to various companies who want to submit this kind of studies for regulatory purposes. Durbis has his sight abroad and has been admitted to a prestigious graduate program where he will keep seeking to improve his knowledge on medicinal chemistry. I’m sure he will succeed in anything on which he sets his mind.
Collaborations in Inorganic Chemistry
I began my path in computational chemistry while I still was an undergraduate student, working on my thesis under professor Cea at unam, synthesizing main group complexes with sulfur containing ligands. Quite a mouthful, I know. Therefore my first calculations dealt with obtaining Bond indexed for bidentate ligands bonded to tin, antimony and even arsenic; yes! I worked with arsenic once! Happily, I keep a tight bond (pun intended) with inorganic chemists and the recent two papers published with the group of Prof. Mónica Moya are proof of that.
In the first paper, cyclic metallaborates were formed with Ga and Al but when a cycle of a given size formed with one it didn’t with the other (fig 1), so I calculated the relative energies of both analogues while compensating for the change in the number of electrons with the following equation:
Fig 1
Under the same conditions 6-membered rings were formed with Ga but not with Al and 8-membered rings were obtained for Al but not for Ga. Differences in their covalent radii alone couldn’t account for this fact.
ΔE = E(MnBxOy) – nEM + nEM’ – E(M’nBxOy) Eq 1
A seamless substitution would imply ΔE = 0 when changing from M to M’
Hipothetical compounds optimized at the B3LYP/6-31G(d,p) level of theory
The calculated ΔE were: ΔE(3/3′) = -81.38 kcal/mol; ΔE(4/4′) = 40.61 kcal/mol; ΔE(5/5′) = 70.98 kcal/mol
In all, the increased stability and higher covalent character of the Ga-O-Ga unit compared to that of the Al analogue favors the formation of different sized rings.
Additionally, a free energy change analysis was performed to assess the relative stability between compounds. Changes in free energy can be obtained easily from the thermochemistry section in the FREQ calculation from Gaussian.
This paper is published in Inorganic Chemistry under the following citation: Erandi Bernabé-Pablo, Vojtech Jancik, Diego Martínez-Otero, Joaquín Barroso-Flores, and Mónica Moya-Cabrera* “Molecular Group 13 Metallaborates Derived from M−O−M Cleavage Promoted by BH3” Inorg. Chem. 2017, 56, 7890−7899
The second paper deals with heavier atoms and the bonds the formed around Yttrium complexes with triazoles, for which we calculated a more detailed distribution of the electronic density and concluded that the coordination of Cp to Y involves a high component of ionic character.
This paper is published in Ana Cristina García-Álvarez, Erandi Bernabé-Pablo, Joaquín Barroso-Flores, Vojtech Jancik, Diego Martínez-Otero, T. Jesús Morales-Juárez, Mónica Moya-Cabrera* “Multinuclear rare-earth metal complexes supported by chalcogen-based 1,2,3-triazole” Polyhedron 135 (2017) 10-16
We keep working on other projects and I hope we keep on doing so for the foreseeable future because those main group metals have been in my blood all this century. Thanks and a big shoutout to Dr. Monica Moya for keeping me in her highly productive and competitive team of researchers; here is to many more years of joint work.
All you wanted to know about Hybrid Orbitals…
… but were afraid to ask
or
How I learned to stop worrying and not caring that much about hybridization.
The math behind orbital hybridization is fairly simple as I’ll try to show below, but first let me give my praise once again to the formidable Linus Pauling, whose creation of this model built a bridge between quantum mechanics and chemistry; I often say Pauling was the first Quantum Chemist (Gilbert N. Lewis’ fans, please settle down). Hybrid orbitals are therefore a way to create a basis that better suits the geometry formed by the bonds around a given atom and not the result of a process in which atomic orbitals transform themselves for better sterical fitting, or like I’ve said before, the C atom in CH4 is sp3 hybridized because CH4 is tetrahedral and not the other way around. Jack Simmons put it better in his book:
Taken from “Quantum Mechanics in Chemistry” by Jack Simmons
The atomic orbitals we all know and love are the set of solutions to the Schrödinger equation for the Hydrogen atom and more generally they are solutions to the hydrogen-like atoms for which the value of Z in the potential term of the Hamiltonian changes according to each element’s atomic number.
Since the Hamiltonian, and any other quantum mechanical operator for that matter, is a Hermitian operator, any given linear combination of wave functions that are solutions to it, will also be an acceptable solution. Therefore, since the 2s and 2p valence orbitals of Carbon do not point towards the edges of a tetrahedron they don’t offer a suitable basis for explaining the geometry of methane; even more so these atomic orbitals are not degenerate and there is no reason to assume all C-H bonds in methane aren’t equal. However we can come up with a linear combination of them that might and at the same time will be a solution to the Schrödinger equation of the hydrogen-like atom.
Ok, so we need four degenerate orbitals which we’ll name ζi and formulate them as linear combinations of the C atom valence orbitals:
ζ1= a12s + b12px + c12py + d12pz
ζ2= a22s + b22px + c22py + d22pz
ζ3= a32s + b32px + c32py + d32pz
ζ4= a42s + b42px + c42py + d42pz
to comply with equivalency lets set a1 = a2 = a3 = a4 and normalize them:
a12 + a22 + a32 + a42 = 1 ∴ ai = 1/√4
Lets take ζ1 to be directed along the z axis so b1 = c1 = 0
ζ1 = 1/√4(2s) + d12pz
since ζ1 must be normalized the sum of the squares of the coefficients is equal to 1:
1/4 + d12 = 1;
d1 = √3/2
Therefore the first hybrid orbital looks like:
ζ1 = 1/√4(2s) +√3/2(2pz)
We now set the second hybrid orbital on the xz plane, therefore c2 = 0
ζ2 = 1/√4(2s) + b22px + d22pz
since these hybrid orbitals must comply with all the conditions of atomic orbitals they should also be orthonormal:
〈ζ1|ζ2〉 = δ1,2 = 0
1/4 + d2√3/2 = 0
d2 = –1/2√3
our second hybrid orbital is almost complete, we are only missing the value of b2:
ζ2 = 1/√4(2s) +b22px +-1/2√3(2pz)
again we make use of the normalization condition:
1/4 + b22 + 1/12 = 1; b2 = √2/√3
Finally, our second hybrid orbital takes the following form:
ζ2 = 1/√4(2s) +√2/√3(2px) –1/√12(2pz)
The procedure to obtain the remaining two hybrid orbitals is the same but I’d like to stop here and analyze the relative direction ζ1 and ζ2 take from each other. To that end, we take the angular part of the hydrogen-like atomic orbitals involved in the linear combinations we just found. Let us remember the canonical form of atomic orbitals and explicitly show the spherical harmonic functions to which the 2s, 2px, and 2pz atomic orbitals correspond:
ψ2s = (1/4π)½R(r)
ψ2px = (3/4π)½sinθcosφR(r)
ψ2pz = (3/4π)½cosθR(r)
we substitute these in ζ2 and factorize R(r) and 1/√(4π)
ζ2 = (R(r)/√(4π))[1/√4 + √2 sinθcosφ –√3/√12cosθ]
We differentiate ζ2 respect to θ, and set it to zero to find the maximum value of θ respect to the z axis we get the angle between the first to hybrid orbitals ζ1 and ζ2 (remember that ζ1 is projected entirely over the z axis)
dζ2/dθ = (R(r)/√(4π))[√2 cosθ –√3/√12sinθ] = 0
sinθ/cosθ = tanθ = -√8
θ = -70.53°,
but since θ is measured from the z axis towards the xy plane this result is equivalent to the complementary angle 180.0° – 70.53° = 109.47° which is exactly the angle between the C-H bonds in methane we all know! and we didn’t need to invoke the unpairing of electrons in full orbitals, their promotion of any electron into empty orbitals nor the ‘reorganization‘ of said orbitals into new ones. Orbital hybridization is nothing but a mathematical tool to find a set of orbitals which comply with the experimental observation and that is the important thing here!
To summarize, you can take any number of orbitals and build any linear combination you want, in order to comply with the observed geometry. Furthermore, no matter what hybridization scheme you follow, you still take the entire orbital, you cannot take half of it because they are basis functions. That is why you should never believe that any atom exhibits something like an sp2.5 hybridization just because their bond angles lie between 109 and 120°. Take a vector v = xi+yj+zk, even if you specify it to be v = 1/2i that means x = 1/2, not that you took half of the unit vector i, and it doesn’t mean you took nothing of j and k but rather than y = z = 0.
This was a very lengthy post so please let me know if you read it all the way through by commenting, liking, or sharing. Thanks for reading.
No, seriously, why can’t orbitals be observed?
The concept of electronic orbital has become such a useful and engraved tool in understanding chemical structure and reactivity that it has almost become one of those things whose original meaning has been lost and replaced for a utilitarian concept, one which is not bad in itself but that may lead to some wrong conclusions when certain fundamental facts are overlooked.
Last week a wrote -what I thought was- a humorous post on this topic because a couple of weeks ago a viewpoint in JPC-A was published by Pham and Gordon on the possibility of observing molecular orbitals through microscopy methods, which elicited a ‘seriously? again?‘ reaction from me, since I distinctly remember the Nature article by Zuo from the year 2000 when I just had entered graduate school. The article is titled “direct observation of d-orbital holes.” We discussed this paper in class and the discussion it prompted was very interesting at various levels: for starters, the allegedly observed d-orbital was strikingly similar to a dz2, which we had learned in class (thanks, prof. Carlos Amador!) that is actually a linear combination of d(z2-x2) and d(z2-y2) orbitals, a mathematical -lets say- trick to conform to spectroscopic observations.
Pham and Gordon are pretty clear in their first paragraph: “The wave function amplitude Ψ*Ψ is interpreted as the probability density. All observable atomic or molecular properties are determined by the probability and a corresponding quantum mechanical operator, not by the wave function itself. Wave functions, even exact wave functions, are not observables.” There is even another problem, about which I wrote a post long time ago: orbitals are non-unique, this means that I could get a set of orbitals by solving the Schrödinger equation for any given molecule and then perform a unit transformation on them (such as renormalizing them, re-orthonormalizing them to get a localized version, or even hybridizing them) and the electronic density derived from them would be the same! In quantum mechanical terms this means that the probability density associated with the wave function internal product, Ψ*Ψ, is not changed upon unit transformations; why then would a specific version be “observed” under a microscope? As Pham and Gordon state more eloquently it has to do with the Density of States (DOS) rather than with the orbitals. Furthermore, an orbital, or more precisely a spinorbital, is conveniently (in math terms) separated into a radial, an angular and a spin component R(r)Ylm(θ,φ)σ(α,β) with the angular part given by the spherical harmonic functions Ylm(θ,φ), which in turn -when plotted in spherical coordinates- create the famous lobes we all chemists know and love. Zuo’s observation claim was based on the resemblance of the observed density to the angular part of an atomic orbital. Another thing, orbitals have phases, no experimental observation claims to have resolved those.
Now, I may be entering a dangerous comparison but, can you observe a 2? If you say you just did, well, that “2” is just a symbol used to represent a quantity: two, the cardinality of a set containing two elements. You might as well depict such quantity as “II” or “⋅⋅” but still cannot observe “a two”. (If any mathematician is reading this, please, be gentle.) I know a number and a function are different, sorry if I’m just rambling here and overextending a metaphor.
Pretending to having observed an orbital through direct experimental methods is to neglect the Born interpretation of the wave function, Heisenberg’s uncertainty principle and even Schrödinger’s cat! (I know, I know, Schrödinger came up with this gedankenexperiment in order to refute the Copenhagen interpretation of quantum mechanics, but it seems like after all the cat is still not out of the box!)
So, the take home message from the viewpoint in JPC is that molecular properties are defined by the expected values of a given wave function for a specific quantum mechanical operator of the property under investigation and not from the wave function itself. Wave functions are not observables and although some imaging techniques seem to accomplish a formidable task the physical impossibility hints to a misinterpretation of facts.
I think I’ll write more about this in a future post but for now, my take home message is to keep in mind that orbitals are wave functions and therefore are not more observable (as in imaging) than a partition function is in statistical mechanics.
The Gossip Approach to Scientific Writing
Communication of scientific findings is an essential skill for any scientist, yet it’s one of those things some students are reluctant to do partially because of the infamous blank page scare. Once they are confronted to writing their thesis or papers they make some common mistakes like for instance not thinking who their audience is or not adhering to the main points. One of the the highest form of communication, believe it or not, is gossip, because gossip goes straight to the point, is juicy (i.e. interesting) and seldom needs contextualization i.e. you deliver it just to the right audience (that’s why gossiping about friends to your relatives is almost never fun) and you do it at the right time (that’s the difference between gossips and anecdotes). Therefore, I tell my students to write as if they were gossiping; treat your research in a good narrative way, because a poor narrative can make your results be overlooked.
I’ve read too many theses in which conclusions are about how well the methods work, and unless your thesis has to do with developing a new method, that is a terrible mistake. Methods work well, that is why they are established methods.
Take the following example for a piece of gossip: Say you are in a committed monogamous relationship and you have the feeling your significant other is cheating on you. This is your hypothesis. This hypothesis is supported by their strange behavior, that would be the evidence supporting your hypothesis; but be careful because there could also be anecdotal evidence which isn’t significant to your own as in the spouse of a friend had this behavior when cheating ergo mine is cheating too. The use of anecdotal evidence to support a hypothesis should be avoided like the plague. Then, you need an experimental setup to prove, or even better disprove, your hypothesis. To that end you could hack into your better half’s email, have them followed either by yourself or a third party, confronting their friends, snooping their phone, just basically about anything that might give you some information. This is the core of your research: your data. But data is meaningless without a conclusion, some people think data should speak for itself and let each reader come up with their own conclusions so they don’t get biased by your own vision and while there is some truth to that, your data makes sense in a context that you helped develop so providing your own conclusions is needed or we aren’t scientists but stamp collectors.
This is when most students make a terrible mistake because here is where gossip skills come in handy: When asked by friends (peers) what was it that you found out, most students will try to convince them that they knew the best algorithms for hacking a phone or that they were super conspicuous when following their partners or even how important was the new method for installing a third party app on their phones to have a text message sent every time their phone when outside a certain area, and yeah, by the way, I found them in bed together. Ultimately their question is left unanswered and the true conclusion lies buried in a lengthy boring description of the work performed; remember, you performed all that work to reach an ultimate goal not just for the sake of performing it.
Writers say that every sentence in a book should either move the story forward or show character; in the same way, every section of your scientific written piece should help make the point of your research, keep the why and the what distinct from the how, and don’t be afraid about treating your research as the best piece of gossip you’ve had in years because if you are a science student it is.
Some Comp.Chem. Tweeps
Out of some +1000 twitter accounts I follow about a quarter are related computational chemistry. The following public list isn’t comprehensive and prone to errors and contains researchers, programmers, students, journals, products and companies who gravitate around the use of in silico methods for the understanding and design of chemical and biochemical compounds.
A Twitter List by joaquinbarroso //platform.twitter.com/widgets.js
Dealing with Spin Contamination
Most organic chemistry deals with closed shell calculations, but every once in a while you want to calculate carbenes, free radicals or radical transition states coming from a homolytic bond break, which means your structure is now open shell.
Closed shell systems are characterized by having doubly occupied molecular orbitals, that is to say the calculation is ‘restricted’: Two electrons with opposite spin occupy the same orbital. In open shell systems, unrestricted calculations have a complete set of orbitals for the electrons with alpha spin and another set for those with beta spin. Spin contamination arises from the fact that wavefunctions obtained from unrestricted calculations are no longer eigenfunctions of the total spin operator <S^2>. In other words, one obtains an artificial mixture of spin states; up until now we’re dealing only with single reference methods. With each step of the SCF procedure the value of <S^2> is calculated and compared to s(s+1) where s is half the number of unpaired electrons (0.75 for a radical and 2.0 for triplets, and so on); if a large deviation between these two numbers is found, the then calculation stops.
Gaussian includes an annihilation step during SCF to reduce the amount of spin contamination but it’s not 100% reliable. Spin contaminated wavefunctions aren’t reliable and lead to errors in geometries, energies and population analyses.
One solution to overcome spin contamination is using Restricted Open Shell calculations (ROHF, ROMP2, etc.) for which singly occupied orbitals is used for the unpaired electrons and doubly occupied ones for the rest. These calculations are far more expensive than the unrestricted ones and energies for the unpaired electrons (the interesting ones) are unreliable, specially spin polarization is lost since dynamical correlation is hardly accounted for. The IOP(5/14=2) in Gaussian uses the annihilated wavefunction for the population analysis if acceptable but since Mulliken’s method is not reliable either I don’t advice it anyway.
The case of DFT is different since rho.alpha and rho.beta can be separated (similarly to the case of unrestricted ab initio calculations), but the fact that both densities are built of Kohn-Sham orbitals and not true canonical orbitals, compensates the contamination somehow. That is not to say that it never shows up in DFT calculations but it is usually less severe, of course for the case of hybrid functional the more HF exchange is included the more important spin contamination may become.
So, in short, for spin contaminated wavefunctions you want to change from restricted to unrestricted and if that doesn’t work then move to Restricted Open Shell; if using DFT you can use the same scheme and also try changing from hybrid to pure orbitals at the cost of CPU time. There is a last option which is using spin projection methods but I’ll discuss that in a following post.
Stability of Unnatural DNA – @PCCP #CompChem
As is the case of proteins, the functioning of DNA is highly dependent on its 3D structure and not just only on its sequence but the difference is that protein tertiary structure has an enormous variety whereas DNA is (almost) always a double helix with little variations. The canonical base pairs AT, CG stabilize the famous double helix but the same cannot be guaranteed when non-canonical -unnatural- base pairs (UBPs) are introduced.
Figure 1
When I first took a look at Romesberg’s UBPS, d5SICS and dNaM (throughout the study referred to as X and Y see Fig.1) it was evident that they could not form hydrogen bonds, in the end they’re substituted naphtalenes with no discernible ways of creating a synton like their natural counterparts. That’s when I called Dr. Rodrigo Galindo at Utah University who is one of the developers of the AMBER code and who is very knowledgeable on matters of DNA structure and dynamics; he immediately got on board and soon enough we were launching molecular dynamics simulations and quantum mechanical calculations. That was more than two years ago.
Our latest paper in Phys.Chem.Chem.Phys. deals with the dynamical and structural stability of a DNA strand in which Romesberg’s UBPs are introduced sequentially one pair at a time into Dickerson’s dodecamer (a palindromic sequence) from the Protein Data Bank. Therein d5SICS-dNaM pair were inserted right in the middle forming a trisdecamer; as expected, +10 microseconds molecular dynamics simulations exhibited the same stability as the control dodecamer (Fig.2 left). We didn’t need to go far enough into the substitutions to get the double helix to go awry within a couple of microseconds: Three non-consecutive inclusions of UBPs were enough to get a less regular structure (Fig. 2 right); with five, a globular structure was obtained for which is not possible to get a proper average of the most populated structures.
X and Y don’t form hydrogen bonds so the pairing is pretty much forced by the scaffold of the rest of the DNA’s double helix. There are some controversies as to how X and Y fit together, whether they overlap or just wedge between each other and according to our results, the pairing suggests that a C1-C1′ distance of 11 Å is most stable consistent with the wedging conformation. Still much work is needed to understand the pairing between X and Y and even more so to get a pair of useful UBPs. More papers on this topic in the near future.
