## IACS Research Day Faculty Talks: Matthew G. Reuter

>>So, good afternoon, everyone. Thanks for sticking around past lunch in research

day. We have nothing to motivate you anymore, there’s

no more food. Anyway, so my name is Nat [assumed spelling]

Reuter. I’m here in ICS. I’m also in the department of applied math

and statistics, but don’t let that fool you. I’m a physical chemist by training. And so, what I’m going to tell you a little

bit today, are some results ongoing in my research group about reconciling experiment

and computation in electron transport studies through molecules. And so, we can think about, sort of, just

broadly, since I’m a physical chemist working in an applied math department in a computation

institute, what exactly do I do? I respond to emails, I merge PDF documents,

and I invert big matrices. Okay. So basically research in my group, sort of,

goes in two directions depending on, sort of, what the starting point was. Where in some cases we use nanoscience in

a general sense. So, looking at really small things where quantum

mechanics becomes really important. And that usually can help motivate us with

some questions. Maybe they’re more chemistry-related, maybe

they were physics-related. But anyway, what we ended up doing, is sort

of, instead of saying, well, how can we develop some nice models for this type of stuff? And how can we then use these models, whether

they’re phenomenological, whether there may be a little bit more sophisticated, depends

on the question we’re asking. And how can we dig into the, now, sort of

the computational math of what’s going on there? Can we find some interesting questions to

ask on the mathematical side? Can we find what the physicist might call

is an exotic case? Can we find what the mathematicians would

call a pathological case. And somehow relate the two back together. And so sometimes nanoscience makes us look

and say, well, what computational math can we use? Turns out there’s some really wonderful things

in the dusty corners of linear algebra. If you’ve never heard of anti-eigenvalue analysis,

or pseudospectral, there’s some really cool shit out there, pardon my French. But anyway, sometimes we can use the computational

math, and come back and find interesting questions to ask in nanoscience. And so the two, sort of, particular areas

that we’ve been focusing on in recent years. Number one, going back to some of my PhD work

is an electron transport. So how do you electrons go through some sort

of quantum mechanical system. Again, inverting big matrices. Some of the stuff that my two students Chris

and Jonathan talked about this morning in the lightning talks on nanomaterials, sort

of, start to look at the same thing in this idea of how do we use modern linear algebra

tools to better understand and characterize material properties. And we’ve been spending a lot of time looking

at complex fan structure in the last couple of years. At the end of the day, where this all sort

of fits together, in sort of chemistry, physics, applied math, computer, computational science

is we’re trying to develop efficient and accurate algorithms. And in a lot of cases, this reaches out to

using some very modern linear algebra tools, whether developed for computation for pure

math, or in some cases because of some interesting physical and chemistry questions. Okay, that’s sort of the overview of the group. So the title said, we’re going to sort of

compare and try to reconcile experiment and computation. And so, when [inaudible] the system that we’re

looking at, we’ve got some sort of molecule. I apologize, I didn’t put a picture on the

slide. There’s a molecule, there’s big electrodes,

and we’re going to put some sort of bias across, and say, how much electric current flows through

the molecule. I will ruin the punchline for you here and

say that experimented computation do not agree. So most of you, you’re thinking we want to

do high-level computation and get accurate results. You’re hoping for, you know, quantitative

accuracy. Maybe you get two decimal places, right, or

three decimal places, right? In the electron transport community, if we

could get within three orders of magnitude, we’d have a beer, and go home, and say that’s

great. Okay, so there’s not a lot of agreement. And there’s a whole bunch of reasons in the

community that suggested for why this this agreement is so poor. They’re all plausible. We don’t need to get into them here. The big issue, or two of the big issues that

we [inaudible] that experiment actually measures one thing. The conductance, the actual property that

we’re interested in. Computation in certain limits can calculate

conductance, but in most limits is actually calculating something called the transmission. So basically, it’s saying, what’s the transmission

probability that electron will tunnel, or get from the one electrode through the molecule

over to the other electrode. And it turns out, those two things, they’re

not the same thing. There’s a reason one is called the conductance

and one is called the transmission. Okay, so, they can be related through various

theories. The simplest one is Landauer Boudicca Theory

where you get the conductance is equal to this constant of proportionality called the

quantum of conductance times the transmission evaluated at the Fermi energy. Okay, so they’re nice and they’re proportional,

that’s fine and good. Couple issues. So, it turns out, calculating the transmission

is hard. We don’t actually know what the Fermi energy

is, but we know what the charge of the electron is. We know what a Planck’s constant is. So hey, we know the constant of proportionality. Go, team. Okay, and so in this regard, this sort of

complicates doing a comparison because you’re sort of comparing, you know, apples and orangutans. All right. So one of the things and advances that we’ve

been doing in recent years, is trying to develop new ways using some of this math. And using some of these computational tools,

new ways to better compare computation and experiment, recognizing that one calculates

transmission, one measures conductance. And so, basically the way experiments are

done is everything statistical. We don’t need to go into the details of how

the experiments are done, but they’re extremely uncontrollable. They’re irreproducible. This sounds like good science, right? Okay. So the way they get around this is they start

doing the experiment, I don’t know, a thousand, ten thousand times. And they basically compile this to whole bunch

of Statistics. It turns out the statistics are reproducible. And so, what is it happens we get this sort

of thing called a conductance histogram. So we’ve got our conductance that’s measured

and essentially how many times we measured it. We get different peaks in it, and we can attribute

those peaks to say, well, there’s the molecular conductance. Okay, now a couple things to point out. Full width, half mass of this peak is about

three quarters of an order of magnitude. Okay, that’s the molecular conductance. Yeah, there’s some error bars here. Okay. So the way that these have traditionally been

read as you would pick on a table. There’s the mode of the P. We won’t call it

the mode, will call it the average. And we’ll say that’s our single molecule conductance. And just sort of throw out all of this other

data. Okay, what we’ve been doing instead of saying

well, can we somehow use all of these statistics. Number one, can we understand why these Peaks

have the shapes that they do. The answer is yes, so I have some references

on the left slide. But can we actually use the statistics to

actually predict from experiment what that transmission would look like with error bars. And the answer is, again, yes, we can. And so we did a couple of just simple test

systems here using the simulated data. And actually now here’s a real computation

that was done from some of my colleagues at Lawrence Berkeley National Lab and Molecular

Foundry, comparing against experiments done on the same molecule from one of the groups

in just down the road at Columbia. And what we’re able to find, [inaudible] in

the red error bar, this would be the conductance that the computation, or excuse me, that experiment

predicted with one standard deviation. And there’s the computation that was predicting. This is using the best computational methods

that have been developed for doing these types of studies, and when you go and look at their

paper that was published, probably three years ago on this, they’re very sad that there wasn’t,

you know they only worked right on top of each other. But the issue is the computation is within

two standard deviations, of what the extent we predict, so they’re sort of crying and

saying we weren’t really on the money. And I’m like, I don’t think you can say you

weren’t on the money. You’re within two standard deviations. Now you’re right at two standard deviations. But you’re right there. But what we’ve ended up doing is finding better

ways to do the comparisons, that we can actually try to put both the conductance and the transmission

on equal footing. Okay, so that’s sort of one area. Another one that we’re doing sort of coming

back towards the computation is, how do we do better computations? It’s possible there’s something wrong in our

computations that might contribute to these three, four, five order of magnitude difference

that we sometimes get between the computation and the experiment. And so, it turns out if you start digging

into the literature, there are some well-known numerical artifacts. In these transport calculations, they know

for instance that you typically don’t get basic set convergence. So you throw more computational resources

at your problem. And it basically turns into a random number

generator. Your number doesn’t converge to serve, keeps

flopping around like a fish. That’s usually not regarded as a good thing. There’s another called ghost transmission

a numerical artifact where you basically say take the molecule out of the system, but just

leave like a three nanometer gap. And look at this tumbling transmission probability,

which Physics 1 student could figure out that the probability of the electron getting through

that Gap is zero. It’s not, well, it’s like 10 to the minus

120, but we can all call that zero. Okay. And so, you can then go and actually employ

the common state of the art techniques on these systems. And actually find that the computation produces

a transmission probability of 1 and 10. That’s close to a 10 to the minus 120. Right. Okay. So what we’ve been doing in recent years,

this is very much ongoing work with Pannu (assumed spelling) who’s here in the audience,

is trying to develop and Implement different solutions for how do we, number 1, figure

out what are the causes of some of these problems.That’s been, well, Robert and I have hypothesized

on this a couple of years ago. It’s implementing, it’s been a bit of a bit

more fun. But working on the implementation now and

to demonstrate that not only can we understand where these numerical artifacts are coming

from, but that, if you’ll let me use the verb in a scientifically appropriate context, we

can exorcise ghost transmission. Okay. And so what we plotted here, some preliminary

results from a few years ago– we’re basically in the black line. We’ve got, sort of, what their traditional

codes are saying you would get to the transmissions. All of the jumpiness doesn’t matter. In the gray line is where we sort of use our

fix [assumed spelling]. And what you, sort of, see — sorry, other

way around, black lines the fix — is that, in general, when you put in this fix, things

sort of shift down a little bit. And we can start to account for why things

are a little bit higher than what they would want, what they would necessarily should be. So, anyway, I will end there, and just put

up the slide. Thank Pannu for his ongoing work, Steven Shudey

[assumed spelling] was a absolutely phenomenal undergrad here about two or three years ago,

who helped do the [inaudible] histogram [assumed spelling] work. And my colleagues and collaborators, Torsten

from University of Copenhagen, and Robert here. So I’d be happy to take, maybe, one quick

question if you have any. Thanks for your attention. [ Applause from the Audience ]>>[Inaudible] I ask a question. Don’t you think that the most important aspect

for the difference is that you observe the [inaudible] theory and experiment is the detailed

knowledge of the attachment of the molecule to the [inaudible].>>So that is certainly very much– so the

question was, for other people because you didn’t hear it — is the actual geometry of

the molecule [inaudible] support. Absolutely, it is. And so, the issue though, is if that were

the only factor, I would argue that half the time, experiment would be lower than theory,

and half the time it would be higher. To me, the fact that computation is always,

always two, three, four, five, seven, eight orders of magnitude bigger tells me, I think,

there’s something else that’s a little bit more systemic and pernicious going on. I mean, I can’t discount that fact. It’s certainly a plausible explanation, but

I don’t think it’s the only one. Right. Thank you.

## Leave a Reply