Since March 14, 2020, President Trump has given semi-regular press updates on the global COVID-19 outbreak. While this pandemic is impacting nearly every aspect of everyday life, it is important to understand how our leaders perceive and communicate the next steps as we tackle these unprecedented issues.
The New York Times (NYT) recently ran a fascinating article that painstakingly analyzed over 260,000 words Trump has publicly spoken about the novel coronavirus. It took three journalists at the NYT to review all these statements by hand, and they found that the President’s comments were particularly self-congratulatory, exaggerated, and startlingly lacking in empathy when compared to how often he praises himself and his administration.
I wondered, though, if a more data-driven approach to Trump’s remarks would yield any new insights. In particular, using data and natural language processing (NLP) to understand:
What topics Trump thinks are particularly important?
What are some of the keywords for each day’s briefing?
Trump’s attitude (sentiment) during his speeches?
What underlying factors in the news or world correlate with his message?
Given that all his remarks are publicly recorded on the White House webpage, I knew we had what we needed to answer these questions.
Findings
What does Trump talk about?
I wanted to get an idea of what topics Trump talks about during his briefings. To answer this, I used non-negative matrix factorization (NMF) to cluster keywords around certain topics. The number of topics and the interpretation of the topic is up to the scientist. There’s no right answer as this is an unsupervised learning problem, but I found that the topics made the most sense if you limit them to six.
The most important keywords are listed and my intepretation of the topic. For example, the first topic is clearly talking about new treatments, cures, and vaccines for COVID-19. We should expect this to be a topic duing his briefings! Other ones make sense to me as well, like economic recovery. I was a little surprised (OK maybe not too surprised) how often he remarked on his media coverage. And the COVID-19 Task Force topic generally seems to bring up how well his administration is handling the pandemic. This is consistent with what the NYT article found.
You can see for yourself below!
Click for examples of Trump remarks by topic
Topic 1: Vaccine and treatment
April 13: We have things happening that are unbelievable. I saw a presentation today that I can’t talk about yet, but it’s incredible. Plus, I think they’re doing — Tony — I think they’re doing very well in the vaccines. They’re working hard on the vaccines and I think you’ll have an answer for vaccines. I believe that there’s some great things coming out with respect to that. Now you need a testing period, but you’re going to have some great things.
Topic 2: Global impact
April 19: I had a G7 call and their economies are in tatters. They’re shattered, the G7 countries. You have Japan and Germany and France, and the different countries. Italy — look at what happened to Italy. Look at what happened to these countries. Look at what happened to Spain. Look what happened to Spain, how — how incredible. It’s just been shattered. And so many other countries are shattered.
Topic 3: Media coverage
March 29: I mean, even the media is much more fair. I wouldn’t say all of it, but that’s okay. They should be fair because they should want this to end. This is — this is about death.
Topic 4: Farms, trade, and tariffs
April 22: We don’t want to do — you know, the border has been turned off a number of times over the years. And you know what happened? Our farmers all went out of business. They were out of business. They couldn’t farm. We’re taking care of our farmers. Nobody ever took care of farmers like I take care of farmers.
Topic 5: COVID Task Force
April 16: I would now like to ask Vice President Mike Pence and Dr. Birx to further explain the new guidelines. I want to thank Dr. Birx. I want to thank Dr. Fauci. And I want to thank, really especially, a man who has devoted 24 hours a day to his task force and done such an incredible job — our great Vice President, Mike Pence.
Topic 6: Economic recovery
April 2: And we’ve learned a lot. We’ve learned about borders. We’ve learned about reliance on other countries. We’ve learned so much — so much that I think we really have a chance to be bigger and better and stronger. And I think it’s going to come back very quickly, but first we have to defeat this enemy.
In general, when Trump speaks his statements fall into one of these topics 18% of the time. That’s pretty good coverage for our topics given that we limit ourselves to only 6 topics. That said, relatively speaking, what does he talk most about?
Topic
% Relative
% Total
Vaccine & treatment
16.6
2.9
Global impact
21.4
3.7
Media coverage
10.2
1.8
Farms, trade, & tariffs
24.7
4.3
COVID Task Force
14.1
2.5
Economic recovery
12.9
2.3
When Trump’s statements fall into one of these six topics, Trump talks about farmers, trade, and tariffs nearly 25% of the time (4.3% overall). If we lump this group in with the economic recovery topic, it seems Trump talks about the economy well over a third of the time, nearly double that of the actual health impact (vaccines and treatment). Another quarter of the time is spent commenting on the media and how his administration is handling the crisis. These are rough numbers, but the pattern emerges that Trump’s preferred topics are the economy and his image (combining both press coverage and his administration’s COVID Task Force).
Keywords for each day’s briefing
It’s interesting to see what Trump thinks is important each day. One way to address this is to pull out the top keywords for each day’s briefing. The technique we use is term-frequency inverse-document-frequency (TF-IDF) and treat each day’s briefing as a “document”. It’s a cheap and effective way to pull out those unique keywords that distinguish each day’s briefing.
Here’s an example of the important keywords found on March 14:
Date
Keyword 1
Keyword 2
Keyword 3
Keyword 4
Keyword 5
March 14
fed
proactive
mic
closure
refinance
So here you can see that “mic” (or “microphone”) was a keyword of interest. Interestingly, Trump went on a bit of a rant during his talk because he was accused of touching the microphone:
Somebody said yesterday I touched the microphone. I was touching it because we have different height people and I’m trying to make it easy for them because they’re going to have to touch, because they wouldn’t be able to reach the mic; they wouldn’t be able to speak in the mic. So I’ll move the mic down. And they said, “Oh, he touched the microphone.” Well, if I don’t touch it, they’re going to have to touch it. Somebody is going to have to, so I might as well be the one to do it.
Amusing that the model identified this (somewhat) bizarre tangent.
You can see for yourself below. Near the beginning of the briefings, things seem much more focused on health-care and government action (“proactive”, “self swab”, “buyback”). But as the economy progressively gets worse (see March 23!) then we see more of an economic focus (“exchange”, “cure is worse than the problem”). The economic shutdown is no small deal!
Click to show important keywords by day
Date
Keyword 1
Keyword 2
Keyword 3
Keyword 4
Keyword 5
March 14
fed
proactive
mic
closure
refinance
March 15
predict
Federal Reserve
Google
relax
store
March 16
postponing
symptom
Friday night
wash
Rochelle
March 17
payroll tax
Boeing
telehealth
tourism
screened
March 18
self swab
calm
invincible
target
passenger
March 19
Syria
chloroquine
Tice
mother
chaos
March 21
student loan
penalty
inaccurately
buyback
religious leader
March 22
veteran
VA
Federal Medical Station
preexisting
rich
March 23
Boeing
cure is worse than the problem
watching closely
exchange
learning
March 24
timeline
independence
Larry
large section
established
March 25
Steve
Olympics
South Korea
specification
significance
March 26
steel
Brady
Tom
South Korea
96
March 27
appreciative
Peter
smartest
Boeing
handle
March 30
June 1st
rating
waive
300,000
22 million
March 31
scarf
guest
ride
impeached
oil
April 01
cash
violence
train
Logan
Facebook
April 02
gesture
cut
flat
Iran
bump
April 03
voter id
81
voluntary
mail
oil
April 04
whistleblower
transcript
informer
commissioner
180
April 05
182
sue
rushing
oil
sign
April 06
letter
London
intensive
nuclear
replacement
April 08
delegate
Bernie
memo
Easter
voting
April 09
oil
mental
Scalia
storage
OPEC
April 10
pastor
Denver
interruption
Mexico
World Trade Organization
April 13
January
committee
clip
Jan 17th
Wuhan
April 15
judge
approved
Sonny
session
nominee
April 16
count
wide
arena
devastated
capacity
April 19
swab
aroused
ran
pressure
hotel
April 20
defending
squeeze
sustainable
60,000
rally
April 22
corona
parlor
misquoted
ember
flu
April 23
sun
heat
laboratory
surface
Brian Kemp
April 24
Stephen
approving
declined
payment
Tim
April 27
pharmacy
damage
Attorney General
General Flynn
quarter
You can also see the importance of oil in April. It wasn’t a huge issue early on, but you see it become more important starting around April 3, peaking around April 9. That day, if you recall, was when OPEC made huge cuts in oil production. So naturally it was a topic in Trump’s briefing that day!
How positive/negative is Trump during his speeches?
We can investigate how positive or negative Trump is speaking during his briefings. To do this, we look at the compound sentiment for each briefing using the Valence Aware Dictionary and sEntiment Reasoner (VADER). Although it was trained on social media, it works well for our purposes. It ranges from -1 (negative) to +1 (positive).
So given the statement: “I hate you!”, VADER assigns it a -1. But given the statement “I love you!” VADER assigns it a +1. For statements like “I love pizza, but hate the color red”, VADER gives it a compound sentiment of zero. For Trump, we take the average compound sentiment for a given day.
You can see that, on the whole, Trump is pretty positive-to-neutral. Politically, I think this would be better to lean positive during the briefings. Of course, you don’t want to be too positive either during these tough times!
On average, his sentiment lies around 0.4, with a standard deviation of 0.1. So leaning positive, but by no means elated, which is probably the right tone to hit.
It’s interesting to note the spike in Trump’s positivity on April 24. Was he really more positive than usual? The model suggests so — and he was!
POLITICO noticed then same thing, and ran an article about it. After getting rebuked the day before for claiming sunlight and ingesting Lysol as a cure for COVID-19, Trump took a more conventional approach. In this particular briefing, he didn’t bash any political opponents, didn’t talk about any unproven cures, or argue with reporters. He only predicted quick economic turnaround. The article states:
For a ritual that in recent days has displayed some the trappings of a political rally, the most unusual thing about the briefing was how conventional it was … There was no touting of unproven remedies, no swipes at political foes; only prediction of a swift economic rebound, praise for governors who are rolling out plans to reopen their states and updates on his administration’s public health and economic response to the crisis that has killed more than 50,000 Americans. (POLITICO 04/24/2020)
Fascinating to see the change in attitude reflected in the VADER analysis!
What might be influencing Trump’s attitude?
Perhaps more interesting is to try and find patterns in Trump’s sentiment. Initially, I noticed the big spike in sentiment on March 24, as well as a pretty big drop in sentiment on April 1. On March 24, there was a big stock market rally, and on April 1, the market suffered a few days of losses. I wondered: could it be that Trump’s mood is correlated with stock market performance?
To investigate, I plotted Trump’s sentiment versus the percent change in the closing value of the Dow Jones Industrial Average. On first look, you can see that the briefing sentiment shows some correlation with changes in the DJIA. If the market is doing bad, Trump is generally more negative, and if the market is doing well, Trump is more positive. It’s tough to say which direction the correlation goes (correlation is not causation!) but given that Trump usually holds his briefings later in the day, I’m inclined to think the market bears more influence on his mood. But the issues are complicated and there are many more variables at play, so take it with a grain of salt.
We can actually make the correlation between stock market and Trump’s mood a bit tighter. Below we have a regression plot, and find that the Pearson correlation is around +0.45. Note that we don’t include the cases where Trump gives a briefing on a weekend or holiday! (Which shouldn’t be correlated with the markets.) We also see that the R^{2} is 0.2 with a p-value of 0.03. So the Dow daily variation can account for 20% of Trump’s sentiment and is statistically significant (p-value < 0.05). This is good enough to say that there is likely a moderate positive correlation between Trump’s sentiment and the market performance. Makes sense, as he prides himself on his business acumen and being able to keep the American economy strong. (No comment if that corresponds to reality.)
Details on Data Acquisition
The raw data was scraped from https://www.whitehouse.gov/remarks/ using requests and BeautifulSoup4. For now, all we explore Trump’s statements during the Coronavirus Task Force press meetings (which as of today, May 6, may be ending).
The president’s name is listed before he speaks, so once the page is downloaded locally, it is relatively easy to extract just Trump’s statements. We treat a “statement” as any of Trump’s spoken words separated by a newline. This gave a variety of statements, usually around 3 to 5 sentences long. If any statement was less than 16 words, we ignored it (these usually are statements like “Thank you John, next question”, etc.) as these are not meaningful. If a statement was too long (longer than 140 words) we split them into smaller chunks. This foresight allows us to use the BERT transformer in future work, which in the default model cannot handle sentences with more than 512 word embeddings.
Once the raw text was extracted to a pandas DataFrame, we applied standard text-cleaning routines to make all lowercase, remove stopwords, and add bigrams and trigrams (so mayor, de, blasio is not three separate words, but instead is treated as mayor_de_blasio, which is much more informative!).
Most scientists are aware of the importance and significance of neural networks. Yet for many, neural networks remain mysterious and enigmatic.
Here, I want to show that neural networks are simply generalizations of something we scientists are perhaps more comfortable with: linear regression.
Regression models are models that find relationships among data. Linear regression models fit linear functions to data, like the equations we learned in algebra. If you have one variable, it’s single-variate linear regression; if you have more than one variable, it’s multi-variate linear regression.
One example of a linear regression model would be Charles’ Law which says that temperature and volume of a gas are proportional. By collecting temperatures and volumes of a gas, we can derive the proportionality constant between the two. For a multi-variable case, one example might be finding the relationship between the price of a house and its square footage and how many bedrooms it has.
It turns out we can write linear regression in terms of a graph like the one below. In this example we have two types of input data (features), and . The function to be fit takes each of these data points, multiplies it by a weight or and adds them together. The is the “bias”, which is analogous to the in a linear equation .
To make a model, we have to solve for the weights and the bias . This can be done by many types of optimization algorithms (say, steepest descent or by pseudoinverse). Don’t worry about how the weights are obtained, just realize that that, one, we can obtain them and, two, once we have the weights and bias then we now have a functional relationship among our data.
So for example, if is housing price, and is square footage and is number of bedrooms, then we can predict the housing price given any house area and bedroom number via . So far so good.
Now we take things a bit further. The next step in our regression “evolution” is to pass the whole above linear regression model to a function. In the example below, we pass it to a sigmoid or logistic function . This is what logistic regression looks like as a graph:
This is known as a “generalized linear model.” For our purposes, the exact function you pass the linear model into doesn’t matter so much. The important part is that we can make a generalized linear model by passing the linear model into a function. If you pass it into a sigmoid function, like we did above, you get a logistic regression model which can be useful for classification problems.
Like if I tell you the color and size of an animal, the logistic model can predict whether the animal is either a flamingo or an elephant.
So to recap, we started with a linear model, then by adding a function to the output, we got a generalized linear model. What’s the next step?
Glad you asked.
The next step with to take our linear model, pass it into a function (generalized linear model), then take that output and use it as an input for another linear model. If you do this, as depicted in the graph below, you will have obtained what is a called a shallow neural network.
At this point, we have obtained a nonlinear regression model. That’s really all neural networks are: models for doing nonlinear regression.
At the end of the day all these models are doing the same thing, namely, finding relationships among data. Nonlinear regression allows you to find more complex relationships among the data. This doesn’t mean nonlinear regression is better, rather it is just a more flexible model. For many real-world problems, simpler linear regression is better! As long as it is suitable to handle your problem, it will be faster and easier to interpret.
So to recap, we went from linear regression, to logistic regression (a generalized linear model), to shallow neural network (nonlinear regression). The final step to get to deep neural networks is to pass the output of the shallow network into another function and make a linear model based off that output. This is depicted below:
To reiterate, all these deep networks do is take data, make a linear model, transform with a function, then make another linear model with that output, transform that with a function, then make another linear model…and so on and so forth.
Data linear model function linear model function linear model…
It’s a series of nested functions that give you even more flexibility to handle strange nonlinear relationships among your data. And if that’s what your problem requires, that’s what it’s there for!
Obviously, modern developments of neural networks are far more complicated. I’m simplifying a lot of the details, but the general idea holds: neural networks are models to do nonlinear regression, and they are built up from linear models. They take one of the workhorses of scientific analysis, linear regression, and generalize it to handle complex, nonlinear relationships among data.
But as long as you keep in mind that it can be broken down to the pattern of passing the output of linear regression to a function, and then passing that output to another linear model, you can get pretty far.
I have a new review on (and titled) real-time time-dependent electronic structure theory out now, which has been one of my active research areas over the past five years or so. Like other electronic structure methods, real-time time-dependent (RT-TD) methods seeks to understand molecules through a quantum treatment of the motions of their electrons. What sets RT-TD theories and methods apart is that they explore how electrons evolve in time, especially as they respond to things like lasers, magnetic fields, X-rays, and so on. Real time methods look at molecular responses to these things explicitly in time, and gives an intuitive and dynamic view of how molecules behave under all sorts of experimental conditions. From this, we can predict and explain how certain molecules make better candidates for solar energy conversion, molecular electronics, or nanosensors, etc.
The truth is that there is a wealth of information that can be obtained by real-time time-dependent methods. Because of this, RT-TD methods are often criticized as inefficient and overkill for many predictive applications. In some cases this is true, for example when computing absorption spectra of small organic molecules (linear response methods are usually a better choice). However, to dismiss all RT-TD methods is a mistake, as the methods comprise a general technique for studying extremely complex phenomena. RT-TD methods allow for complete control over the number and strength of interacting external perturbations (multiple intense lasers, for example) in the study of molecular responses. In the review, we address many of these unique applications, ranging from non-equilibrium solvent dynamics to dynamic hyperpolarizability to the emerging real-time quantum electrodynamics (QED), where even the photon field is quantized.
The article was extremely fun to write, and I hope you find something useful and interesting in it. RT-TD comprises a broad field with extremely talented scientists working on its development. You can find the article here.
The goal of all real-time electronic dynamics methods is to solve the
time-dependent Schrödinger equation (TDSE)
where is the time-dependent Hamiltonian and is the
time-dependent wave function. The goal of the Magnus expansion is to
find a general solution for the time-dependent wave function in the case
where is time-dependent, and, more crucially, when does not
commute with itself at different times, e.g. when
. In the following we will follow
closely the notation of Blanes, et al..
First, for simplicity we redefine and
introduce a scalar as a bookkeeping device, so that
At the heart of the Magnus expansion is the idea of solving the TDSE by
using the quantum propagator that connects wave functions at
different times, e.g. Furthermore, the
Magnus expansion assumes that can be represented as an
exponential, This
yields the modified TDSE
Now, for scalar and , the above has a simple solution,
namely
However, if and are matrices this is not necessarily true. In
other words, for a given matrix the following expression does not
necessarily hold:
because the matrix and its derivatives do not necessarily commute.
Instead, Magnus proved that in general
satisfies
where are the Bernoulli numbers. This equation may be solved by
integration, and iterative substitution of . While it may
appear that we are worse off than when we started, collecting like
powers of (and setting ) allows us to obtain a
power-series expansion for ,
This is the Magnus expansion, and here we have given up to the
third-order terms. We have also made the notational simplification that
. This is the basis for nearly all
numerical methods to integrate the many-body TDSE in molecular physics.
Each subsequent order in the Magnus expansion is a correction that
accounts for the proper time-ordering of the Hamiltonian.
The Magnus expansion immediately suggests a route to
many numerical integrators. The simplest would be to approximate the
first term by
leading to a forward-Euler-like time integrator of
which we can re-write as
where subscript gives the node of the time-step stencil. This gives
a first-order method with error . A more accurate
second-order method can be constructed by approximating the first term
in the Magnus expansion by the midpoint rule, leading to an
time integrator
Modifying the stencil to eliminate the need to evaluate the Hamiltonian
at fractional time steps (e.g. change time step to ) leads
to the modified midpoint unitary transformation (MMUT) method
which is a leapfrog-type unitary integrator. Note that the midpoint
method assumes is linear over its time interval, and the
higher order terms (containing the commutators) in this approximation go to zero. There are many other types of integrators
based off the Magnus expansion that can be found in the
literature. The key point for all of these
integrators is that they are symplectic, meaning they preserve
phase-space relationships. This has the practical effect of conserving
energy (within some error bound) in long-time dynamics, whereas
non-symplectic methods such as Runge-Kutta will experience energetic
“drift” over long times. A final note: in each of these schemes it is
necessary to evaluate the exponential of the Hamiltonian. In real-time
methods, this requires computing a matrix exponential. This is not a
trivial task, and, aside from the construction of the Hamiltonian
itself, is often the most expensive step in the numerical solution of
the TDSE. However, many elegant solutions to the construction of the
matrix exponential can be found in the literature.
References
Blanes, S., Casas, F., Oteo, J.A. and Ros, J., 2010. A pedagogical approach to the Magnus expansion. European Journal of Physics, 31(4), p.907.
Magnus, W., 1954. On the exponential solution of differential equations for a linear operator. Communications on pure and applied mathematics, 7(4), pp.649-673.
In quantum chemistry, we are often interested in evaluating integrals
over Gaussian basis functions. Here I am going to take much of the
theory for granted and talk about how one might actually implement the
integrals for quantum chemistry by using recursion relationships with
Hermite Gaussians. Our goal is to have clean, readable code. I’m writing
this for the people are comfortable with the mathematics behind the
Gaussian integrals, but want to see a readable computer implementation.
I’ll talk a bit about some computational considerations at the end, but
my goal is to convert equations to code. In fact, I’ve tried to
structure the equations and the code in such a way that the two look
very similar.
For a very good overview of integral evaluation, please see:
Helgaker, Trygve, and Peter R. Taylor. “Gaussian basis sets and
molecular integrals.” Modern Electronic Structure (1995).
I will try and follow the notation used in the above reference.
Mathematical preliminaries
Let’s start with some of the basics. First, we have our 3D Gaussian
functions
with orbital exponent , electronic coordinates ,
origin , and
also, are the angular quantum numbers (e.g. is -type,
is type, etc.) Cartesian Gaussians are separable in 3D along
so that
with the 1D Gaussian
So far, so good. Let’s consider the overlap integral of two 1D Gaussians, and
where we used the Gaussian product theorem so that
When using Hermite Gaussians, we can express as
where
are expansion coefficients (to be determined recursively) and
is the Hermite Gaussian overlap of two Gaussians and
. It has a simple expression that kills the sum via the Kronecker
delta . It can be shown that the expansion coefficients can
be defined using the following recursive definitions
The first equation gives us a way to reduce the index and the second
gives us a way to reduce index so that we can get to the third
equation, which is our base case. The last equation tells us what to do
if we go out of index bounds.
Overlap integrals
The first thing we need to do is implement a function E which returns
our expansion coefficients . Aside from angular momentum
and from the Gaussian functions, we also need the distance between
Gaussians and the orbital exponent coefficients and
as inputs.
This is simple enough! So for a 1D overlap between two Gaussians we
would just need to evaluate and multiply it by
. Overlap integrals in 3D are just a product of
the 1D overlaps. We could imagine a 3D overlap function like
so
Note that we are using the NumPy package in order to take advantage of
the definitions of and the fractional power to the . The
above two functions overlap and E are enough to get us the overlap
between two Gaussian functions (primitives), but most basis functions
are contracted, meaning they are the sum of multiple Gaussian
primitives. It is not too difficult to account for this, and we can
finally wrap up our evaluation of overlap integrals with a function
S(a,b) which returns the overlap integral between two contracted
Gaussian functions.
Basically, this is just a sum over primitive overlaps, weighted by
normalization and coefficient. A word is in order for the arguments,
however. In order to keep the number of arguments we have to pass into
our functions, we have created BasisFunction objects that contain all
the relevant data for the basis function, including exponents,
normalization, etc. A BasisFunction class looks like
So, for example if we had a STO-3G Hydrogen 1s at origin (1.0, 2.0,
3.0), we could create a basis function object for it like so
Where we used the EMSL STO-3G definition
So doing S(a,a) = 1.0, since the overlap of a basis function with
itself (appropriately normalized) is one.
Kinetic energy integrals
Having finished the overlap integrals, we move on to the kinetic
integrals. The kinetic energy integrals can be written in terms of
overlap integrals
where
For a 3D primitive, we can form a kinetic function analogous to
overlap,
and for contracted Gaussians we make our final function T(a,b)
Nuclear attraction integrals
The last one-body integral I want to consider here is the nuclear
attraction integrals. These differ from the overlap and kinetic energy
integrals in that the nuclear attraction operator is Coulombic,
meaning we cannot easily factor the integral into Cartesian components
.
To evaluate these integrals, we need to set up an auxiliary Hermite
Coulomb integral that handles the
Coulomb interaction between a Gaussian charge distribution centered at
and a nuclei centered at . The Hermite Coulomb
integral, like its counterpart , is defined recursively:
where is the Boys function
which is a special case of the Kummer confluent hypergeometric function,
which is convenient for us, since SciPy has an implementation of as
a part of scipy.special. So for R we can code up the recursion like
so
and we can define our boys(n,T) function as
There are other definitions of the Boys function of course, in case you
do not want to use the SciPy built-in. Note that R requires
knowledge of the composite center from two Gaussians
centered at and . We can determine
using the Gaussian product center rule
which is very simply coded up as
Now that we have a the Coulomb auxiliary Hermite integrals
, we can form the nuclear attraction integrals with respect
to a given nucleus centered at , , via the
expression
And, just like all the other routines, we can wrap it up to treat
contracted Gaussians like so:
Important: Note that this is the nuclear repulsion integral
contribution from an atom centered at . To get the full
nuclear attraction contribution, you must sum over all the nuclei, as
well as scale each term by the appropriate nuclear charge!
Two electron repulsion integrals
We are done with the necessary one-body integrals (for a basic
Hartree-Fock energy code, at least) and are ready to move on to the
two-body terms: the electron-electron repulsion integrals. Thankfully,
much of the work has been done for us on account of the nuclear
attraction one-body integrals.
In terms of Hermite integrals, to evaluate the two electron repulsion
terms, we must evaluate the summation
which looks terrible and it is. However, recalling that
letting (that is, the
Gaussian exponents on and , and and ), we can write the
equation in a similar form to the nuclear attraction integrals
And, for completeness’ sake, we wrap the above to handle contracted
Gaussians
And there you have it! All the integrals necessary for a Hartree-Fock
SCF code.
Computational efficiency considerations
Our goal here has been to eliminate some of the confusion when it comes
to connecting mathematics to actual computer code. So the code that we
have shown is hopefully clear and looks nearly identical to the
mathematical equations they are supposed to represent. This is one
reason we chose Python as the code of choice to implement the
integrals. It emphasizes readability.
If you use the code as is, you’ll find that you can only really handle
small systems. To that end, I’ll give a few ideas on how to improve the
integral code to actually be usable.
First, I would recommend not using pure Python. The problem is that we
have some pretty deep loops in the code, and nested for loops will
kill your speed if you insist on sticking with Python. Now, you can
code in another language, but I would suggest rewriting some of the
lower level routines with Cython (http://www.cython.org). Cython
statically compiles your code to eliminate many of the Python calls
that slow your loops down. In my experience, you will get several orders
of magnitude speed up. This brings me to my second point. One of the
problems with Python is that you have the global interpreter lock
(GIL) which basically means you cannot run things in parallel. Many of
the integral evaluation routines would do well if you could split the
work up over multiple CPUs. If you rewrite some of the low level
routines such that they do not make Python calls anymore, you can turn
off the GIL and wrap the code in OpenMP directives, or even use
Cython’s cython.parallel module. This will take some thought, but
can definitely be done. Furthermore, removing the explicit recursion in the
E and R functions and making them iterative would go a long way to speed
up the code.
A couple other thoughts: be sure to exploit the permutational symmetry
of the integrals. The two electron repulsion integrals, for example, can
be done in of the time just by exploiting these symmetries, which
are unrelated to point group. Also, you can exploit many of the integral
screening routines, since many of the two electron integrals are
effectively zero. There are a lot of tricks out there in the literature,
go check them out!