Coffee

Dear Diary,

I finally accepted Kieron's offer of coffee today.  Justin kept saying it was a mean, wholesome cup of joe, so I had to have some at some point.  He wasn't far off.

Kieron actually hesitated before acquiescing.  Apparently, caffeine is addictive. I should be careful, but I drank it slow and avoided a caffeine fit.  The group's drink of choice is definitively coffee and I'm on the tea end of drinks.  It was even a possibly prohibiting factor from joining the group, but I swore I'd come around and have a cup before graduating!

Little does Kieron know (and how could he?  I miscounted when I told him how many cups of coffee I'd ever had!), but I've had coffee from the coffee store of all coffee stores:  Juan Valdez in Colombia, no less!  (Recommended)

Afterward, I washed my cup out and placed it back on the desk...and felt alive for several hours afterward! (Kieron's Coffee: Also Recommended)

Oops! Should have known that (#328/420,870)

Dear Diary,

Yikes.  Li and I have been running some binding curves of H2.  Thing is, they don't look like the binding curves of H2 that either John or Lucas published in their papers.  The dips of our curves are located around zero separation.

But the dips of those published figures are around 2-something!  A conundrum!

Lucas set me straight, like always.  By, as he says, going back to basics.  The regular DFT calculations don't include nuclear-nuclear repulsion.  So, when you calculate the electronic density, the atoms are fixed.  If you add in the nuclear-nuclear repulsion term, then you get the dip at a higher number.  This is where the molecule was bound if we did a calculation where the nuclei were allowed to move around.

These turn out to be accurate (depending on what flavor of DFT you use), but that's exactly what we're trying to fix!  Trying to calculate these properties can help with a lot things like climate models and other things!  Yay!

Inspiration for DFT

Dear Diary,

Yesterday, I heard a meaningful and inspirational speech from Kieron.

Honorary graduate student Fan asked Kieron what he thought of some tools used in DFT.  Kieron mused on the usefulness versus utility of reformulations of the same problem.  Quite wisely, Kieron noticed that some alternative formulations of quantum mechanics have a harder time bringing new insight to the table (he didn't mention any names, but I'm looking at you, Bohmian mechanics!).

I asked him what DFT brings to the community.  Kieron indicated that there is a structure that is of yet unknown that the community would benefit greatly from by reaching it.  My answer was the Hohenburg-Kohn theorems which I can't easily think of a good analogy with a quantum field theory treatment.  That you can describe everything except the potential is one functional.  That's quite an insight.  I bet Kieron is right and more things are around the corner.

Hopefully, I can get some of these things in front of me out of the way and get closer to finding out that unknown thing!  And get working on that paper that I'm stalled on!

A visiting Professor: Weitao Yang! (Part 2)

Dear Diary,

Weitao Yang has left to go back to snow-ridden Duke University.  He and I had a chance to sit down and discuss physics.

It was the first time that I've ever been around someone well known who I could actually have a conversation with.  In comparison, Kieron and I have had about two pages worth of dialogue over the last year for him (which is perhaps a slight exaggeration...maybe three pages--and that's mainly because I haven't had any real results yet).

Anyway, we discussed gaps and my previous research!  He was interested in double excitations (which he graciously spoke to us at the joint Group meeting between Professor Furche's group and our own).  He was really nice!  And really smart!

Wow!  That was great!

We ride together, we derive together, bad boys for life

Dear Diary,

I had an individual meeting that got very hectic!  At first, I came in at the end of Li's meeting and sat down to wait my turn.  Turns out, they were just getting to my and Li's collaboration.  Li explained some details and just about as I was to being, Justin arrived!

Justin didn't have much, so Kieron and he postponed, giving me more time (not always good!).  So, I went to the board and began discussing a last figure that Lucas and I were making.  Then, who should call but Lucas!

Kieron and Lucas discussed some items over Skype when who should call but John!  He said he missed my voice!  So, Kieron put his tablet and computer facing each other so that John and Lucas could chat for a bit.  Then Kieron wanted Li back up to be there when Kevin arrived.  Kevin didn't arrive, but when Kieron called the office to get Li, an old voicemail from Lucas answered!

Now, I'm used to having chaotic meetings.  They happened with my old advisor all of the time.  Kieron is busy a lot of the time and I'm able to find other things to fill my schedule with if I can't get all of my answers in the individual meeting window.  So, I've turned my attention to matters I can get done and stay productive (including a project that Kieron doesn't even know about yet!).

A visiting professor: Weitao Yang!

Dear Diary,

This week, one Professor Weitao Yang will be visiting the campus and consequently us.

We're supposed to have a boat load of questions for him, so I've been studying up on the work that I've done that is related to what he's done.

He's coming tomorrow. Better get back to it.

What is dreamed and math that should never be (part 3)

From Lucas!

 

"You can also diagonalize large matrices without explicit diagonalization, and it's also related to the inverse.  The key is to use the Green's function G(z) = (z S - H)^{-1}, defined for the generalized eigenvalue problem H psi = S psi E.  Taking a contour integral C in the complex plane which encloses the eigenvalues you want, you can get the density matrix rho = -1/(2 pi i) INT[z on C] G(z), which is essentially psi psi^T.  Go complex analysis and residue theorems!

There's a neat method which avoids the full inverse and approximates the contour integral nicely, and it's super parallelizable.  I think Zhenfei or Zenghui gave it at a previous group talk.  Or maybe John.  You can find it at <a href="http://arxiv.org/abs/0901.2665">arxiv:0901.2665</a>, but see later work by Polizzi."

What is dreamed and math that should never be (part 2)

Dear Diary,

It just so happens my two best friends from high school have math Ph.D.s.  One responded with this:

"I started thinking about what you wrote, and it was pretty interesting, so I couldn't resist...

Your strategy for finding eigenvalues works, but you have to be careful with the intuition that 2 equations in 2 unknowns will give you a solution. Let's look at the 2x2 case:

xy = d (the determinant)
x + y = t (the trace)

This gives y = t - x so x(t-x) = d, and this is a quadratic in x: you should expect 2 solutions, and you won't know which one is right. But here's a tricky observation: the two equations are symmetric, meaning that if you swap the variables x and y, you get the same equations. So if (x = a, y = b) is a solution, so is (x = b, y = a) - the two solutions you get are actually the same pair of eigenvalues.

Now you'll have to forgive me for taking a mathematical tangent...... The fact that you get 2 solutions here instead of 1 is an example of Bezout's theorem, which says roughly that if you have two curves in the plane, given by polynomial equations of degrees d_1 and d_2 (the degree is the highest number of variables appearing in a single monomial, so xy has degree 2), you should expect them to intersect (d_1)(d_2) times. This generalizes to higher dimensions: if you have n hypersurfaces in n-space (a hypersurface is the solution set to polynomial equation in n variables), you should expect them to intersect a number of times equal to the product of their degrees.

Now this might not happen: they might not intersect at all (this corresponds to their intersection being "at infinity" - think of parallel lines), or they might intersect fewer times because some intersection points have "higher multiplicity" (think of a parabola that just brushes the x-axis at the origin - this intersection point has "multiplicity 2"). But there's a precise way of asserting that "these situations almost never happen".

This explains something special about linear algebra: a system of n linear equations in n unknowns will always have a unique solution if it is consistent (i.e. the solution isn't "at infinity") - all the equations have degree 1, so Bezout's theorem tells us to multiply a bunch of 1s together.

Now let's look at the 3x3 case. You have 3 equations:

xyz = d (the determinant) - degree 3

x + y + z = t (the trace) - degree 1

1/x + 1/y + 1/z = s (the trace of the inverse) - not a polynomial

The last equation isn't a polynomial, but we can replace it with yz + xz + xy = sd (which has degree 2) by multiplying by xyz = d without losing any information, as long as d is nonzero.

This gives 3 equations in 3 variables of degrees 1, 2, and 3. Bezout's theorem says that for almost all d, t, and s, there will be 6 solutions to this system of equations. So your strategy cuts down the number of possible eigenvalue triples to 6... but now, miraculously, we can apply the same observation as in the 2x2 case: your three equations are symmetric under all permutations of the three variables, and there are 6 of these. So the 6 solutions we get are really all the same solution, just with their names permuted.

To check this, I selected 3 random values of d, t, and s, and asked Wolfram Alpha to solve the equations: http://www.wolframalpha.com/input/?i=solve+xyz+%3D+1%2C+x+%2B+y+%2B+z+%3D+2%2C+1%2Fx+%2B+1%2Fy+%2B+1%2Fz+%3D+1%2F2

Yep, 6 solutions! And (up to some very small error that Alpha gets from solving numerically) they're all the same set of 3 eigenvalues. Phew, math works.

What if d = 0? Then we have a problem, because we don't have 3 equations: the matrix has no inverse, so we can't take the trace of its inverse."

I hadn't thought if the determinant was zero.  You're right!  What an insight! See, Diary, math is cool!

What is dreamed and math that should never be

Dear Diary,

I've been catching up on sleep over the weekend.  I didn't get out of bed all of Saturday!

But my mind will not be put down so easily.  Getting away from computer screens while the power is out has thrown me into full mathematical mode.  Maybe this is some Freudian-level way of my subconscious craving more pen and paper work to satisfy some latent math craving.  I even dreamed about a math problem!

Now, I've done that before.  Most notably, I was staying up quite late to get a take home final for classical mechanics.  I was concentrating so hard as I shut my eyes, that I woke up with the answer!  I did this one other time more recently with a quantum fields homework.  If only I could wake up with my thesis or that paper Kieron keeps asking me for!

I've also had dreams that posed math problems.  For example, a trio of lighthouses whose beacons are spinning at different rates...what area is illuminated by the triangles bounded between the lighthouses and where they must meet if they're in an equilateral triangle?  It took a few minutes after waking up to get that one.

However, this puzzle was more interesting!  I was talking to someone who had one of those "evil" goatees and he wondered if I could find the eigenvalues of a matrix without doing the whole diagonalization procedure (i.e. avoid finding the characteristic polynomial).  Challenged accepted!

I immediately leapt for the 2x2 case and actually solved it while I was still asleep.  Taking the trace (sum of the diagonals) and the determinant (there's only one!) gives two coupled equations that  can be solved for the eigenvalues (remember, the trace and determinant don't change under a similarity transform, so they're the same even when it's a diagonal matrix with the eigenvalues in the diagonal).

But evil goatee man wanted the 3x3 case!

This I actually had to wake up to check my answer.  Honest, I was thinking about the inverse matrix in the dream, but numbers and writing don't work the same in a dream. So, it gets hard fast to legibly read anything (or even flip on a light switch!).

Since we now have three values to solve for, we need one more equation.  The trace of the inverse matrix looks like the sum of the reciprocal eigenvalues, so voi'la!  We win!

Awake me:  1, Asleep me: 0!

Now, you could quibble with me since my choice solution for solving for the inverse of a matrix is to find the characteristic polynomial and use the Cayley-Hamilton equation to obtain it (see this post), but there are other ways to find the inverse matrix...like seeing what row operations need to be performed to change it to the identity.  I'm ok with this solution, and I guess I'll see how I was graded tonight.

The great university blackout!

Dear Diary,

The university will be switching some power station generator something-something this weekend and turning off the power to do it.  We've been asked to conserve electricity* presumably by filling the powerless hours with card games lit up by hurricane lamp or going outside (eek!).

Since the computers will be affected, I decided to send an email around to everyone:

"Hi everyone.

Just a reminder before the weekend to turn your computers to avoid consequences from the power outages."

But as Justin points out:

"Why?  I'm pretty sure their spatial orientation has nothing to do with electrical charge."

Well, yes.  It's true that currents are not affected by direction unless something (like a magnetic field) are present.  I suppose this is the part where I try to explain that the Earth's magnetic field is always present (for now!), but it's futile because rotating my computer will not prevent it from turning off.

I think I'll be using the time to have a hard look at myself in the darkened liquid crystal display and see my true self looking back--Fahrenheit 451 style...or go outside and exercise or something.  It's sunny.

*But charge is always conserved!