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WE20 Oceanography Historical Perspective

 

Maple is a trademark of Waterloo Maple Inc.

 

Adapted from Introductory Oceanography 10th ed. by Harold V. Thurman and Alan P. Trujillo

 

1.  How did the view of the ocean by early Mediterranean cultures influence the naming of planet Earth?

 

Early cultures envisioned the world as composed of large landmasses surrounded by marginal bodies of water.

 

2.  What is the difference between an ocean and a sea?  Which ones are the seven seas?

 

A sea is

   •  Smaller and shallower than an ocean (this is why the Arctic Ocean might be more appropriately considered a sea)

   •  Composed of salt water (many "seas" such as the Caspian Sea in Asia, are actually large fresh water lakes)

   •  Somewhat enclosed by land (but some seas, such as the Sargasso Sea in the Atlantic Ocean, are defined by strong ocean currents rather than land.

 

The seven seas:  the North Pacifiic, the South Pacific, the North Atlantic, the South Atlantic, the Indian, the Arctic, and the Southern or Antarctic.

 

3.  Describe the development of navigation techniques that have enabled sailors to navigate the open ocean far from land.

 

Initially, navigators in the Northern Hemisphere measured the angle between the horizon and the North Star (Polaris), which is directly above the North Pole.  Latitude could be determined by noting the angular difference between the horizon and the North Star.  In the Southern Hemisphere, the angle between the horizon and the Southern Cross was used because the Southern Cross is directly overhead at the South Pole.

 

There was no method of determining longitude until one based on time was developed at the end of the 18th century.  Pendulum-driven clocks in use in the early 1700s would not work for long on a rocking ship at sea.  In 1728 a cabinetmaker in Lincolnshire, England, named John Harrison, began working on a new type of time peace.  The "chronometer" was driven by a helical balance spring that remained horizontal and independent of ship motion.

 

As Earth turns on its rotational axis it moves through 2Pi radian every 24 hours (one complete rotation).  Earth, therefore, rotates through (1/12)*Pi radian of longitude per hour.  As a result, a navigator need only know the time at the prime or Greenwich at the exact same time the sun is at its highest point (local noon) at the ship's location.

 

Suppose a ship sets sail west across the Atlantic Ocean from Europe, checking its longitude each day when the sun is at its highest point (solar noon) at the ship's location.  On one day the chronometer reads 16:18 hours.  What is the location of the ship?

 

The ship is 4 hours and 18 minutes behind (west of) Greenwich time.  To determine the longitude we must convert time into longitude.  

 

(1/12)*Pi radian of longitude per hour

(1/12)*Pi*(1/60) = (1/720)*Piradian of longitude per minute

(1/720)*Pi*(1/60) = (1/43200)*Piradian of longitude per second

 

4*((1/12)*Pi)+18*((1/720)*Pi) = 43*Pi*(1/120)

 

180*(43*Pi*(1/120))/Pi = 7740*(1/120) and 7740*(1/120) = 129/2 and 129/2 = 64.5degree west of the Greenwich meridian.

 

In Maple:

dg := proc (hr, mn, sc) local r, d, d_dec; r := (1/12)*hr*Pi+(1/720)*mn*Pi+(1/4320)*sc*Pi; d := 180*r/Pi; d_dec := evalf[5](d); print(d, d_dec) end proc

dg(4, 18, 0)

129/2, 64.500

(1)

4.  Construct a time line that includes the major events of human history that have resulted in greater understanding of our planet in general and the oceans in particular.

 

 

 

 

 

 

5.  Using a diagram, illustrate the method used by Eratosthenes to calculate the circumference of Earth.

 

Eratoshenes (276–192 BCE) had heard that at noon on the longest day of the year the Sun shone directly into the waters of a deep,vertical well in Syene (Aswan), which was 800 kilometers (500 miles) to the south.  At the same time in Alexandria, he noticed a vertical pole cast a slight shadow when the Sun was at its apex in the sky over Alexandria.  He accurately measured the shadow the pole cast, which was 7.2 degree.

 

Convert 7.2 ° to radian  alpha^R = ((1/180)*Pi*`#msup(mi("α",fontstyle = "normal"),mo("∘"))`*Pi)*`#msup(mn("7.2"),mo("∘"))` and ((1/180)*Pi*`#msup(mi("α",fontstyle = "normal"),mo("∘"))`*Pi)*`#msup(mn("7.2"),mo("∘"))` = (1/25)*Pi

7.2*(1/180)

0.4000000000e-1

(2)

convert(0.4000000000e-1, 'rational', 'exact')

1/25

(3)

(2*Pi*800)*km/((1/25)*Pi) = 40000*km

(2*Pi*800)/((1/25)*Pi)

40000

(4)

 The equatorial circumference of Earth is about 24,901 miles (40,075 km). However, from pole-to-pole — the meridional circumference — Earth is only 24,860 miles (40,008 km) around. This shape, caused by the flattening at the poles, is called an oblate spheroid.  Since Eratoshenes was calculating the meridional circumference, his calculation was only 8 km different from the accepted modern measurement.

 

The following calculations are for the diagram.

  

NULLNULL`implies`(40000/(2*Pi) = 20000*alpha/Pi and 20000*alpha/Pi = (1/25)*Pi*sin(alpha) and (1/25)*Pi*sin(alpha) = y/c, c*sin(alpha) = y)

`≈`(20000*sin((1/25)*Pi)/Pi, 798*kg)NULLNULL

20000*sin((1/25)*Pi)/Pi

20000*sin((1/25)*Pi)/Pi

(5)

evalf[10](20000*sin((1/25)*Pi)/Pi)

797.8961462

(6)

`implies`(tan(alpha) = y/x implies tan(alpha)/y = 1/x, y/tan(alpha) = 20000*x*sin((1/25)*Pi)/(Pi*tan((1/25)*Pi)) and `≈`(20000*x*sin((1/25)*Pi)/(Pi*tan((1/25)*Pi)), 6316*km))

20000*sin((1/25)*Pi)/(Pi*tan((1/25)*Pi))

20000*sin((1/25)*Pi)/(Pi*tan((1/25)*Pi))

(7)

evalf[10](20000*sin((1/25)*Pi)/(Pi*tan((1/25)*Pi)))

6315.998350

(8)

NULL

 

with(plottools); with(plots)

p1 := circle([0, 0], 20000/Pi)

p2 := line([0, 0], [6316, 798])

p3 := line([0, 0], [6316, 0])

p4 := pointplot([[6316, 0], [6316, 798]], color = yellow, symbol = solidcircle, symbolsize = 25, transparency = .3)

p5 := textplot({[4400, -600, "Syene (Aswan)"], [4400, 1300, "Alexandria"]}, font = [Perpetua, bold, 18])

display(p1, p2, p3, p4, p5, scaling = constrained, size = [350, 350], axes = none)

 

 

6.  While the Arabs dominated the Mediterranean region during the Middle Ages, what were the most significant ocean-related events taking place in Europe?

 

Between 831 and 1071, the Emirate of Sicily was one of the major centers of Islamic culture in the Mediterranean. After its conquest by the Christian Normans, the island developed its own distinct culture with the fusion of Latin and Byzantine influences. Palermo remained a leading artistic and commercial center of the Mediterranean well into the Middle Ages.

 

Europe was reviving, however, as more organized and centralized states began to form in the later Middle Ages after the Renaissance of the 12th century. Motivated by religion and dreams of conquest, the kings of Europe launched a number of Crusades to try to roll back Muslim power and retake the holy land. The Crusades were unsuccessful in this goal, but they were far more effective in weakening the already tottering Byzantine Empire that began to lose increasing amounts of territory to the Seljuk Turks and later to the Ottoman Turks. They also rearranged the balance of power in the Muslim world as Egypt once again emerged as a major power in the eastern Mediterranean.

 

7.  Describe the important events in oceanography that occurred during the Age of Discovery in Europe.

 

One of the most famous voyages during the Age of Discovery began in 1768 when the HMS Endeavour left Portsmouth, England, under the command of Captain James Cook. Over 10 years Cook led three world-encircling expeditions and mapped many countries, including Australia, New Zealand and the Hawaiian Islands. He was an expert seaman, navigator and scientist who made keen observations wherever he went. He was also one of the first ship captains to recognize that a lack of Vitamin C in sailors’ diets (due mostly to a lack of fresh fruit) caused scurvy, a serious disease that killed many sailors in those times. Cook always sailed with lots of pickled cabbage, which he insisted that the sailors eat. Scurvy was never a problem on his ships because the cabbage contained lots of Vitamin C.

 

8.  List some of the major achievements of Captain James Cook.

.com

Three important voyages commanded by Captain Cook:  

1771–1768  Endeavor observes the transit of Venus in Tahiti; charts New Zealand; charts the eastern coastline of Australia and continues on to New Guinea and Java.

 

1772–1776 Resolution crosses the Anartic circle for the first time in history.  On returning to England Cook writes about preventing scurvy on long voyages.

 

1776–1779 Resolution and Discovery visit New Zealand, Tonga, Tahiti, and Christmas Island; Hawaiian islands discovered; expedition crosses the Pacific eastwards to make landfall off the coast of Oregon.  Cook sails into the Bering Straits but foiled by ice he returns to Hawaii where he is killed by natives on 14 Feb 1779.

 

9.  Describe, in general, the voyages of  HMS Challenger and HMS Beagle.

 

The Beagle sailed from Plymouth Sound on 27 December 1831 under the command of Captain Robert FitzRoy. While the expedition was originally planned to last two years, it lasted almost five—the Beagle did not return until 2 October 1836. Darwin spent most of this time exploring on land (three years and three months on land; 18 months at sea). The book he wrote is a vivid travel memoir as well as a detailed scientific field journal covering biology, geology, and anthropology that demonstrates Darwin's keen powers of observation, written at a time when Western Europeans were exploring and charting the whole world.

 

Darwin's notes made during the voyage include comments hinting at his changing views on the inflexibility of species. On his return to England, he wrote the book based on these notes, at a time when he was first developing his theories of evolution through common descent and natural selection. The book includes some suggestions of his ideas, particularly in the second edition of 1845 (Thurman and Trujillo, p. 23).

 

On December 21, 1872 the HMS. Challenger sailed from Portsmouth, England, for an epic voyage which would last almost three and a half years. It  was the first expedition organized and funded for a specific scientific purpose: to examine the deep-sea floor and answer questions about the ocean environment.  Although the mission was scientific, another purpose was to collect information that could be used in the laying of communication cables along the sea floor.

 

The expedition covered 69,000 miles (about 130,000 km) and gathered data on currents, water chemistry, temperature, bottom deposits and marine life at 362 oceanographic stations. More than 4700 new species of marine animals were discovered during the course of the voyage, many of which were found on the seafloor – an environment that scientists originally believed to be too inhospitable to support life (https://paleonerdish.wordpress.com/2013/07/01/the-challenger-expedition-and-the-beginning-of-oceanography/).

 

10.  Describe the voyages of the Fram and how it helped prove there was no continent beneath the Arctic ice pack.

 

The Fram was the first ship specially built (in Norway) for polar research. She was used on three important expeditions: with Fridtjof Nansen on a drift over the Arctic Ocean 1893-96, with Otto Sverdrup to the arctic archipelago west of Greenland - now the Nunavut region of Canada - 1898-1902, and with Roald Amundsen to Antarctica for his South Pole expedition 1910-12. The Fram is now housed and exhibited in the Fram Museum at Bygdøynes, Oslo

(https://frammuseum.no/polar_history/vessels/the_polar_ship_fram/).

 

11.  Why did Benjamin Franklin want to know about the surface current pattern in the North Atlantic Ocean?

 

As Deputy Postmaster General of the American colonies, Franklin promoted using the Gulf Stream to speed up delivery of mail from America to Europe, as well as to improve other commercial shipping (https://divediscover.whoi.edu/history-of-oceanography/benjamin-franklin-discovering-the-gulf-stream/).

 

12.  What was Alexander Agassiz's major contribution to an increased knowledge of the oceans?

 

Agassiz is widely acknowledged as the driving force that brought oceanography recognition as a science.  His training as a mining engineer led him to develop ingenious oceanographic sampling devices—the prototypes for many devices in use on research vessels today—that improved the quantitative value of biological sampling (Thurman and Trujillo, p. 27).

 

13.  What important oceanographic inventions and data came out of Word Wars I and II?

The need to detect submarines led the navies of the world to greatly expand their studies of the sea. This led to the founding of oceanography departments at state universities, including Oregon State, Texas A&M University, University of Miami, and University of Rhode Island,

and the founding of national ocean laboratories such as the various Institutes of Oceanographic Science (https://earthweb.ess.washington.edu/booker/ESS514/stewart/ stewart_ocean_book.pdf).

 

14.  List features of the oceans that can be studied remotely by use of satellites.

 

• phytoplankton concentrations

• heat storage and aerosol formation and other ocean influences on climate processes

• cycles of carbon, sulfur, and nitrogen concentrations

(https://www.nrcan.gc.ca/earth-sciences/geomatics/satellite-imagery-air-photos/satellite-imagery-products/educational-resources/9359).

• Bathymetry/Seafloor (Bathymetry is the measurement of depth of water in oceans seas, or lakes)

• Topography

• Costal Process

• Marine Geophysics

• Ocean Acoustics

• Ocean Chemistry

• Ocean Circulation

• Ocean Heat Budget

• Ocean Optics

• Ocean Pressure

• Ocean Temperature

• Ocean Waves

• Ocean Winds

• Salinity/Density

• Sea ice

• Sea Surface Topography

(https://earthdata.nasa.gov/learn/discipline/ocean)

 

15.  Discuss what problems the human body can experience as a result of diving underwater.

 

Nitrogen narcosis (also referred to as inert gas narcosis, raptures of the deep, and the Martini effect) :  While scuba diving does sometimes involve breathing air that’s mixed with nitrogen, this condition is caused when the diver goes deeper into the water, where the partial pressure of nitrogen increases and more nitrogen ends up getting absorbed into the bloodstream. The higher the nitrogen concentration in the bloodstream, the slower the nervous system will be, and the more likely the diver will experience intoxicating effects that can seriously impair their judgment underwater—possibly enough to lead them into dangerous situations (https://www.leisurepro.com/blog/scuba-guides/dealing-nitrogen-narcosis-2/).

 

Barotrauma: Trauma caused by rapid or extreme changes in air pressure, especially affecting enclosed cavities within the body such as the middle ear (otic barotrauma), the sinuses (sinus barotrauma), and the lungs (pulmonary barotrauma) (https://www.medicinenet.com/script/main/art.asp?articlekey=31722).

 

Decompression illness is caused by intravascular or extravascular bubbles that are formed as a result of reduction in environmental pressure (decompression). The term covers both arterial gas embolism, in which alveolar gas or venous gas emboli (via cardiac shunts or via pulmonary vessels) are introduced into the arterial circulation, and decompression sickness, which is caused by in-situ bubble formation from dissolved inert gas. Both syndromes can occur in divers, compressed air workers, aviators, and astronauts, but arterial gas embolism also arises from iatrogenic causes unrelated to decompression. Risk of decompression illness is affected by immersion, exercise, and heat or cold. Manifestations range from itching and minor pain to neurological symptoms, cardiac collapse, and death. First-aid treatment is 100% oxygen and definitive treatment is recompression to increased pressure, breathing 100% oxygen. Adjunctive treatment, including fluid administration and prophylaxis against venous thromboembolism in paralysed patients, is also recommended. Treatment is, in most cases, effective although residual deficits can remain in serious cases, even after several recompressions (https://www.thelancet.com/journals/lancet/article/PIIS0140-6736(10)61085-9/fulltext).


 

Download WE20_Oceanography_Historical_Persepceive.mw

I experienced a significant obstacle while trying to generate independent random samples with Statistics:-Sample on different nodes of a Grid multi-processing environment. After many hours of trial-and-error, I discovered an astonishing workaround, and I achieved excellent time and memory performance. Since this seems like a generally useful computation, I thought that it was worthy of a Post.

This Post is also worth reading to learn how to use Grid when you need to initialize a substantial environment on each node before using Grid:-Map or Grid:-Seq.

All remaining details are in the following worksheet.
 

How to use Statistics:-Sample in the `Grid` environment

Author: Carl Love <carl.j.love@gmail.com> 1 August 2019

 

I experienced a significant obstacle while trying to generate indenpendent random samples with Statistics:-Sample on the nodes of a multi-processor Grid (on a single computer). After several hours of trial-and-error, I discovered that two things are necessary to do this:

1. 

The random number generator needs to be seeded differently in each node. (The reason for this is easy to understand.)

2. 

The random variables generated by Statistics:-RandomVariable need to have different names in each node. This one is mind-boggling to me. Afterall, each node has its own kernel and so its own memory It's as if the names of random variables are stored in a disk file which all kernels access. And also the generator has been seeded differently in each node.

 

Once these things were done, the time and memory performance of the computation were excellent.

restart
:

Digits:= 15
:

#Specify the size of the computation:
(n1,n2,n3):= (100, 100, 1000):
# n1 = size of each random sample;
# n2 = number of samples in a batch;
# n3 = number of batches.

#
#Procedure to initialize needed globals on each node:
Init:= proc(n::posint)
local node:= Grid:-MyNode();
   #This is wrapped in parse so that it'll act globally. Otherwise, an environment
   #variable would be reset when this procedure ends.
   parse("Digits:= 15;", 'statement');

   randomize(randomize()+node); #Initialize independent RNG for this node.
   #If repeatability of results is desired, remove the inner randomize().

   (:-X,:-Y):= Array(1..n, 'datatype'= 'hfloat') $ 2;

   #Perhaps due to some oversight in the design of Statistics, it seems necessary that
   #r.v.s in different nodes **need different names** in order to be independent:
   N||node:= Statistics:-RandomVariable('Normal'(0,1));
   :-TRS:= (X::rtable)-> Statistics:-Sample(N||node, X);
   #To verify that different names are needed, change N||node to N in both lines.
   #Doing so, each node will generate identical samples!

   #Perform some computation. For the pedagogical purpose of this worksheet, all that
   #matters is that it's some numeric computation on some Arrays of random Samples.
   :-GG:= (X::Array, Y::Array)->
      evalhf(
         proc(X::Array, Y::Array, n::posint)
         local s, k, S:= 0, p:= 2*Pi;
            for k to n do
               s:= sin(p*X[k]);  
               S:= S + X[k]^2*cos(p*Y[k])/sqrt(2-sin(s)) + Y[k]^2*s
            od
         end proc
         (X, Y, n)
      )      
   ;
   #Perform a batch of the above computations, and somehow numerically consolidate the
   #results. Once again, pedagogically it doesn't matter how they're consolidated.  
   :-TRX1:= (n::posint)-> add(GG(TRS(X), TRS(Y)), 1..n);
   
   #It doesn't matter much what's returned. Returning `node` lets us verify that we're
   #actually running this on a grid.
   return node
end proc
:

The procedure Init above uses the :- syntax to set variables globally for each node. The variables set are X, Y, N||node, TRS, GG, and TRX1. Names constructed by concatenation, such as N||node, are always global, so :- isn't needed for those.

#
#Time the initialization:
st:= time[real]():
   #Send Init to each node, but don't run it yet:
   Grid:-Set(Init)
   ;
   #Run Init on each node:
   Nodes:= Grid:-Run(Init, [n1], 'wait');
time__init_Grid:= time[real]() - st;

Array(%id = 18446745861500764518)

1.109

The only purpose of array Nodes is that it lets us count the nodes, and it lets us verify that Grid:-MyNode() returned a different value on each node.

num_nodes:= numelems(Nodes);

8

#Time the actual execution:
st:= time[real]():
   R1:= [Grid:-Seq['tasksize'= iquo(n3, num_nodes)](TRX1(k), k= [n2 $ n3])]:
time__run_Grid:= time[real]() - st

4.440

#Just for comparison, run it sequentially:
st:= time[real]():
   Init(n1):
time__init_noGrid:= time[real]() - st;

st:= time[real]():
   R2:= [seq(TRX1(k), k= [n2 $ n3])]:
time__run_noGrid:= time[real]() - st;

0.16e-1

24.483

R1 and R2 will be different because different random numbers were used, but they should have similar histograms.

plots:-display(
   Statistics:-Histogram~(
      <R1 | R2>, #side-by-side plots
      'title'=~ <<"With Grid\n"> | <"Without Grid\n">>,
      'gridlines'= false
   )
);

(Plot output deleted because MaplePrimes cannot handle side-by-side plots!)

They look similar enough to me!

 

Let's try to quantify the benefit of using Grid:

speedup_factor:= time__run_noGrid / time__run_Grid;

5.36319824753560

Express that as a fraction of the theoretical maximum speedup:

efficiency:= speedup_factor / num_nodes;

.670399780941950

I think that that's really good!

 

The memory usage of this code is insignificant, which can be verified from an external memory monitor such as Winodws Task Manager. It's just a little bit more than that needed to start a kernel on each node. It's also possible to measure the memory usage programmatically. Doing so for a Grid:-Seq computation is a little bit beyond the scope of this worksheet.

 


 

Download GridRandSample.mw

Here are the histograms:

In this post, the Numbrix Puzzle is solved by the branch and bound method (see the details of this puzzle in  https://www.mapleprimes.com/posts/210643-Solving-A-Numbrix-Puzzle-With-Logic). The main difference from the solution using the  Logic  package is that here we get not one but all possible solutions. In the case of a unique solution, the  NumbrixPuzzle procedure is faster than the  Numbrix  one (for convenience, I inserted the code for Numbrix procedure into the worksheet below). In the case of many solutions, the  Numbrix  procedure is usually faster (see all the examples below).

 

restart;

NumbrixPuzzle:=proc(A::Matrix)
local A1, L, N, S, MS, OneStepLeft, OneStepRight, F1, F2, m, L1, p, q, a, b, T, k, s1, s, H, n, L2, i, j, i1, j1, R;
uses ListTools;
S:=upperbound(A); N:=nops(op(A)[3]); MS:=`*`(S);
A1:=convert(A, listlist);
for i from 1 to S[1] do
for j from 1 to S[2] do
for i1 from i to S[1] do
for j1 from 1 to S[2] do
if A1[i,j]<>0 and A1[i1,j1]<>0 and abs(A1[i,j]-A1[i1,j1])<abs(i-i1)+abs(j-j1) then return `no solutions` fi;
od; od; od; od;
L:=sort(select(e->e<>0, Flatten(A1)));
L1:=[`if`(L[1]>1,seq(L[1]-k, k=0..L[1]-2),NULL)];
L2:=[seq(seq(`if`(L[i+1]-L[i]>1,L[i]+k,NULL),k=0..L[i+1]-L[i]-2), i=1..nops(L)-1), `if`(L[-1]<MS,seq(L[-1]+k,k=0..MS-L[-1]-1),NULL)];
  

OneStepLeft:=proc(A1::listlist)
local s, M, m, k, T;
uses ListTools;
s:=Search(a, Matrix(A1));   
M:=[[s[1]-1,s[2]],[s[1]+1,s[2]],[s[1],s[2]-1],[s[1],s[2]+1]];
T:=table(); k:=0;
for m in M do
if m[1]>=1 and m[1]<=S[1] and m[2]>=1 and m[2]<=S[2] and A1[op(m)]=0 then k:=k+1; T[k]:=subsop(m=a-1,A1);
fi;
od;
convert(T, list);
end proc;

 
OneStepRight:=proc(A1::listlist)
local s, M, m, k, T, s1;
uses ListTools;
s:=Search(a, Matrix(A1));  s1:=Search(a+2, Matrix(A1));  
M:=[[s[1]-1,s[2]],[s[1]+1,s[2]],[s[1],s[2]-1],[s[1],s[2]+1]];
T:=table(); k:=0;
for m in M do
if m[1]>=1 and m[1]<=S[1] and m[2]>=1 and m[2]<=S[2] and A1[op(m)]=0 and `if`(a+2 in L, `if`(is(abs(s1[1]-m[1])+abs(s1[2]-m[2])>1),false,true),true) then k:=k+1; T[k]:=subsop(m=a+1,A1);
fi;
od;
convert(T, list);   
end proc;

F1:=LM->ListTools:-FlattenOnce(map(OneStepLeft, LM));
F2:=LM->ListTools:-FlattenOnce(map(OneStepRight, LM));

T:=[A1];
for a in L1 do
T:=F1(T);
od;

for a in L2 do
T:=F2(T);
od;

R:=map(t->convert(t,Matrix), T);
if nops(R)=0 then return `no solutions` else R[] fi;

end proc:

Numbrix := proc( M :: ~Matrix, { inline :: truefalse := false } )

local S, adjacent, eq, i, initial, j, k, kk, m, n, one, single, sol, unique, val, var, x;

    (m,n) := upperbound(M);

    initial := &and(seq(seq(ifelse(M[i,j] = 0
                                   , NULL
                                   , x[i,j,M[i,j]]
                                  )
                            , i = 1..m)
                        , j = 1..n));

    adjacent := &and(seq(seq(seq(x[i,j,k] &implies &or(NULL
                                                       , ifelse(i>1, x[i-1, j, k+1], NULL)
                                                       , ifelse(i<m, x[i+1, j, k+1], NULL)
                                                       , ifelse(j>1, x[i, j-1, k+1], NULL)
                                                       , ifelse(j<n, x[i, j+1, k+1], NULL)
                                                      )
                                 , i = 1..m)
                             , j = 1..n)
                         , k = 1 .. m*n-1));

    one := &or(seq(seq(x[i,j,1], i=1..m), j=1..n));   


    single := &not(&or(seq(seq(seq(seq(x[i,j,k] &and x[i,j,kk], kk = k+1..m*n), k = 1..m*n-1)
                                , i = 1..m), j = 1..n)));

    sol := Logic:-Satisfy(&and(initial, adjacent, one, single));
    
    if sol = NULL then
        error "no solution";
    end if;
if inline then
        S := M;
     else
        S := Matrix(m,n);
    end if;

    for eq in sol do
        (var, val) := op(eq);
        if val then
            S[op(1..2, var)] := op(3,var);
        end if;
    end do;
    S;
end proc:

           Two simple examples

A:=<0,0,5; 0,0,0; 0,0,9>;
# The unique solution
NumbrixPuzzle(A);

A:=<0,0,5; 0,0,0; 0,8,0>;
# 4 solutions
NumbrixPuzzle(A);

Matrix(3, 3, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 5, (2, 1) = 0, (2, 2) = 0, (2, 3) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = 9})

 

Matrix(3, 3, {(1, 1) = 3, (1, 2) = 4, (1, 3) = 5, (2, 1) = 2, (2, 2) = 7, (2, 3) = 6, (3, 1) = 1, (3, 2) = 8, (3, 3) = 9})

 

Matrix(3, 3, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 5, (2, 1) = 0, (2, 2) = 0, (2, 3) = 0, (3, 1) = 0, (3, 2) = 8, (3, 3) = 0})

 

Matrix(%id = 18446746210121682686), Matrix(%id = 18446746210121682806), Matrix(%id = 18446746210121674750), Matrix(%id = 18446746210121674870)

(1)


Comparison with Numbrix procedure. The example is taken from
http://rosettacode.org/wiki/Solve_a_Numbrix_puzzle 

 A:=<0, 0, 0, 0, 0, 0, 0, 0, 0;
 0, 0, 46, 45, 0, 55, 74, 0, 0;
 0, 38, 0, 0, 43, 0, 0, 78, 0;
 0, 35, 0, 0, 0, 0, 0, 71, 0;
 0, 0, 33, 0, 0, 0, 59, 0, 0;
 0, 17, 0, 0, 0, 0, 0, 67, 0;
 0, 18, 0, 0, 11, 0, 0, 64, 0;
 0, 0, 24, 21, 0, 1, 2, 0, 0;
 0, 0, 0, 0, 0, 0, 0, 0, 0>;
CodeTools:-Usage(NumbrixPuzzle(A));
CodeTools:-Usage(Numbrix(A));

Matrix(9, 9, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 0, (1, 4) = 0, (1, 5) = 0, (1, 6) = 0, (1, 7) = 0, (1, 8) = 0, (1, 9) = 0, (2, 1) = 0, (2, 2) = 0, (2, 3) = 46, (2, 4) = 45, (2, 5) = 0, (2, 6) = 55, (2, 7) = 74, (2, 8) = 0, (2, 9) = 0, (3, 1) = 0, (3, 2) = 38, (3, 3) = 0, (3, 4) = 0, (3, 5) = 43, (3, 6) = 0, (3, 7) = 0, (3, 8) = 78, (3, 9) = 0, (4, 1) = 0, (4, 2) = 35, (4, 3) = 0, (4, 4) = 0, (4, 5) = 0, (4, 6) = 0, (4, 7) = 0, (4, 8) = 71, (4, 9) = 0, (5, 1) = 0, (5, 2) = 0, (5, 3) = 33, (5, 4) = 0, (5, 5) = 0, (5, 6) = 0, (5, 7) = 59, (5, 8) = 0, (5, 9) = 0, (6, 1) = 0, (6, 2) = 17, (6, 3) = 0, (6, 4) = 0, (6, 5) = 0, (6, 6) = 0, (6, 7) = 0, (6, 8) = 67, (6, 9) = 0, (7, 1) = 0, (7, 2) = 18, (7, 3) = 0, (7, 4) = 0, (7, 5) = 11, (7, 6) = 0, (7, 7) = 0, (7, 8) = 64, (7, 9) = 0, (8, 1) = 0, (8, 2) = 0, (8, 3) = 24, (8, 4) = 21, (8, 5) = 0, (8, 6) = 1, (8, 7) = 2, (8, 8) = 0, (8, 9) = 0, (9, 1) = 0, (9, 2) = 0, (9, 3) = 0, (9, 4) = 0, (9, 5) = 0, (9, 6) = 0, (9, 7) = 0, (9, 8) = 0, (9, 9) = 0})

 

memory used=7.85MiB, alloc change=-3.01MiB, cpu time=172.00ms, real time=212.00ms, gc time=93.75ms

 

Matrix(9, 9, {(1, 1) = 49, (1, 2) = 50, (1, 3) = 51, (1, 4) = 52, (1, 5) = 53, (1, 6) = 54, (1, 7) = 75, (1, 8) = 76, (1, 9) = 81, (2, 1) = 48, (2, 2) = 47, (2, 3) = 46, (2, 4) = 45, (2, 5) = 44, (2, 6) = 55, (2, 7) = 74, (2, 8) = 77, (2, 9) = 80, (3, 1) = 37, (3, 2) = 38, (3, 3) = 39, (3, 4) = 40, (3, 5) = 43, (3, 6) = 56, (3, 7) = 73, (3, 8) = 78, (3, 9) = 79, (4, 1) = 36, (4, 2) = 35, (4, 3) = 34, (4, 4) = 41, (4, 5) = 42, (4, 6) = 57, (4, 7) = 72, (4, 8) = 71, (4, 9) = 70, (5, 1) = 31, (5, 2) = 32, (5, 3) = 33, (5, 4) = 14, (5, 5) = 13, (5, 6) = 58, (5, 7) = 59, (5, 8) = 68, (5, 9) = 69, (6, 1) = 30, (6, 2) = 17, (6, 3) = 16, (6, 4) = 15, (6, 5) = 12, (6, 6) = 61, (6, 7) = 60, (6, 8) = 67, (6, 9) = 66, (7, 1) = 29, (7, 2) = 18, (7, 3) = 19, (7, 4) = 20, (7, 5) = 11, (7, 6) = 62, (7, 7) = 63, (7, 8) = 64, (7, 9) = 65, (8, 1) = 28, (8, 2) = 25, (8, 3) = 24, (8, 4) = 21, (8, 5) = 10, (8, 6) = 1, (8, 7) = 2, (8, 8) = 3, (8, 9) = 4, (9, 1) = 27, (9, 2) = 26, (9, 3) = 23, (9, 4) = 22, (9, 5) = 9, (9, 6) = 8, (9, 7) = 7, (9, 8) = 6, (9, 9) = 5})

 

memory used=1.21GiB, alloc change=307.02MiB, cpu time=37.00s, real time=31.88s, gc time=9.30s

 

Matrix(%id = 18446746210094669942)

(2)


In the example below, which has 104 solutions, the  Numbrix  procedure is faster.

C:=Matrix(5,{(1,1)=1,(5,5)=25});
CodeTools:-Usage(NumbrixPuzzle(C)):
nops([%]);
CodeTools:-Usage(Numbrix(C)):

Matrix(5, 5, {(1, 1) = 1, (1, 2) = 0, (1, 3) = 0, (1, 4) = 0, (1, 5) = 0, (2, 1) = 0, (2, 2) = 0, (2, 3) = 0, (2, 4) = 0, (2, 5) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = 0, (3, 4) = 0, (3, 5) = 0, (4, 1) = 0, (4, 2) = 0, (4, 3) = 0, (4, 4) = 0, (4, 5) = 0, (5, 1) = 0, (5, 2) = 0, (5, 3) = 0, (5, 4) = 0, (5, 5) = 25})

 

memory used=0.94GiB, alloc change=-22.96MiB, cpu time=12.72s, real time=11.42s, gc time=2.28s

 

104

 

memory used=34.74MiB, alloc change=0 bytes, cpu time=781.00ms, real time=783.00ms, gc time=0ns

 

 


 

Download NumbrixPuzzle.mw

Hare in the forest

The rocket flies

  

Быльнов_raketa_letit.mws

 

Plotting the function of a complex variable

Plotting_the_function_of_a_complex_variable.mws

 

Animated 3-D cascade of dolls

 

3d_matryoshkas_en.mws

 

With this application developed entirely in Maple using native syntax and embedded components for science and engineering students. Just replace your data and you're done.

Pearson_Coeficient.mw

Lenin Araujo Castillo

Ambassador of Maple

 

Foucault’s Pendulum Exploration Using MAPLE18

https://www.ias.ac.in/describe/article/reso/024/06/0653-0659

In this article, we develop the traditional differential equation for Foucault’s pendulum from physical situation and solve it from
standard form. The sublimation of boundary condition eliminates the constants and choice of the local parameters (latitude, pendulum specifications) offers an equation that can be used for a plot followed by animation using MAPLE. The fundamental conceptual components involved in preparing differential equation viz; (i) rotating coordinate system, (ii) rotation of the plane of oscillation and its dependence on the latitude, (iii) effective gravity with latitude, etc., are discussed in detail. The accurate calculations offer quantities up to the sixth decimal point which are used for plotting and animation. This study offers a hands-on experience. Present article offers a know-how to devise a Foucault’s pendulum just by plugging in the latitude of reader’s choice. Students can develop a miniature working model/project of the pendulum.

Exercises solved online with Maple exclusively in space. I attach the explanation links on my YouTube channel.

Part # 01

https://www.youtube.com/watch?v=8Aa2xzU8LwQ

Part # 02

https://www.youtube.com/watch?v=qyGT28CeSz4

Part # 03

https://www.youtube.com/watch?v=yf8rjSPbv5g

Part # 04

https://www.youtube.com/watch?v=FwHPW7ncZTg

Part # 05

https://www.youtube.com/watch?v=bm3frpukb0I

Link for download the file:

Vector_Exercises-Force_in_space.mw

Lenin AC

Ambassador of Maple

 

 

 

I just wanted to let everyone know that the Call for Papers and Extended Abstracts deadline for the Maple Conference has been extended to June 14.

The papers and extended abstracts presented at the 2019 Maple Conference will be published in the Communications in Computer and Information Science Series from Springer. We welcome topics that fall into the following broad categories:

  • Maple in Education
  • Algorithms and Software
  • Applications of Maple

You can learn more about the conference or submit your paper or abstract here: 

https://www.maplesoft.com/mapleconference/Papers-and-Presentations.aspx

Hope to hear from you soon!

It is a very good computational tool to perform modeling and simulation using our world as a reference. You can also teach math knowing how to choose the right icons.
I recommend this software to everyone who wants to simulate objects or multibodies. In any case, knowledge of physics and mathematics, especially vector mechanics, is necessary.
Very grateful to the Maplesoft company for sharing their projects through the MapleSim gallery.

From now on all projects will be with Maple and MapleSim.

Lenin AC

Ambassador of Maple

Submit your paper or extended abstract to the Maple Conference!

The papers and extended abstracts presented at the 2019 Maple Conference will be published in the Communications in Computer and Information Science Series from Springer. 

The deadline to submit is May 27, 2019. 

This conference is an amazing opportunity to contribute to the development of technology in academics. I hope that you, or your colleagues and associates, will consider making a contribution.

We welcome topics that fall into the following broad categories:

  • Maple in Education
  • Algorithms and Software
  • Applications of Maple

You can learn more about the conference or submit your paper or abstract here: 

https://www.maplesoft.com/mapleconference/Papers-and-Presentations.aspx

 

 

 

Maple 2019 has a new add-on package Maple Quantum Chemistry Toolbox from RDMChem for computing the energies and properties of molecules.  As a member of the team at RDMChem that developed the package, I would like to tell the story of its origins and provide a brief demonstration of the package.  

 

Thinking about Quantum Chemistry at Harvard

 

The story of the Maple Quantum Chemistry Toolbox begins with my graduate studies in Chemical Physics at Harvard University in the late 1990s.  Even in 1998 programs for computing the energies and properties of molecules were extremely complicated and nonintuitive.  Many of the existing programs had begun in the 1970s on computers whose programs would be recorded on punchcards.

Fig. 1: Used Punchcard by Pete Birkinshaw from Manchester, UK CC BY 2.0

 

Even today some of these programs have remnants of their early versions such as input files that must start on the second column to account for the margin of the now non-existent punchcards.  As a student, I made a bound copy of one of these manuals at a local Kinkos photocopy shop and later found myself in Harvard Yard, thinking that there must be a better way to present quantum chemistry computations.  The idea for a Maple-like package for quantum chemistry was born in that moment.

 

At the same time I was learning about something called the two-electron reduced density matrix (2-RDM).  The basic variable in quantum chemistry is the wave function which is the probability amplitude for finding each of the electrons in a molecule.  Because electrons are indistinguishable with pairwise interactions, the wave function contains much more information than is needed for computing the energies and electronic properties of molecules.  The energies and properties of any molecule with any number of electrons can be expressed as a function of a 2 electron matrix, the 2-RDM [1-3].  A quantum chemistry based on the 2-RDM, it was known, would have potentially significant advantages over wave function calculations in terms of accuracy and computational cost, especially for molecules far from the mean-field limit.  A 2-RDM approach to quantum chemistry became the focus of my Ph.D. thesis.

 

Representing Many Electrons with Only Two Electrons

 

The idea of using the 2-RDM in quantum chemistry can be attributed to four scientists: two physicists Kodi Husimi and Joseph Mayer, a chemist Per-Olov Lowdin, and a mathematician John Coleman [1-3].  In the early 1940s Husimi first published the idea in a Japanese physics journal, but in the midst of World War II the paper was not widely disseminated in the West.  In the summer of 1951 John Coleman, which attending a physics conference at Chalk River, realized that the ground-state energy of any atom or molecule could be expressed as functional of the 2-RDM, and similar ideas later occurred to Per-Olov Lowdin and Joseph Mayer who published their ideas in Physical Review in 1955.  It was soon recognized that computing the ground-state energy of an atom or molecule with the 2-RDM was potentially difficult because not every two-electron density matrix corresponds to an N-electron density matrix or wave function.  The search for the appropriate constraints on the 2-RDM, known as N-representability conditions, became known as the N-representability problem [1-3].  

 

Beginning in the late 1990s and early 2000s, Carmela Valdemoro and Diego Alcoba at the Consejo Superior de Investigaciones Científicas (Madrid, Spain), Hiroshi Nakatsuji, Koji Yasuda, and Maho Nakata at Kyoto University (Kyoto, Japan), Jerome Percus and Bastiaan Braams at the Courant Institute (New York, USA), John Coleman and Robert Erdahl at Queens University (Kingston, Canada), and my research group and I at The University of Chicago (Chicago, USA) began to make significant progress in the computation of the 2-RDM without computing the many-electron wave function [1-3].  Further contributions were made by Eric Cances and Claude Le Bris at CERMICS, Ecole Nationale des Ponts et Chaussées (Marne-la-Vallée, France), Paul Ayers at McMaster University (Hamilton, Canada), and Dimitri Van Neck at the University of Ghent (Ghent, Belgium) and their research groups.  By 2014 several powerful 2-RDM methods had emerged for the computation of molecules.  The Army Research Office (ARO) issued a proposal call for a company to develop a modern, built-from-scratch package for quantum chemistry that would contain two newly developed 2-RDM-based methods from our group: the parametric 2-RDM method [1] and the variational 2-RDM method with a fast algorithm for solving the semidefinite program [4,5,6].   The company RDMChem LLC was founded to work with the ARO to develop such a package built around RDMs, and hence, the name of the company RDMChem was selected as a hybrid of the RDM abbreviation for Reduced Density Matrices and the Chem colloquialism for Chemistry.  To achieve a really new design for an electronic structure package with access to numeric and symbolic computations as well as advanced visualizations, the team at RDMChem and I developed a partnership with Maplesoft to build something new that became the Maple Quantum Chemistry Package (or Toolbox), which was released with Maple 2019 on Pi Day.

 

Maple Quantum Chemistry Toolbox

 The Maple Quantum Chemistry Toolbox provides a powerful, parallel platform for quantum chemistry calculations that is directly integrated into the Maple 2019 environment.  It is optimized for both cutting-edge research as well as chemistry education.  The Toolbox can be used from the worksheet, document, or command-line interfaces.  Plus there is a Maplet interface for rapid exploration of molecules and their properties.  Figure 2 shows the Maplet interface being applied to compute the ground-state energy of 1,3-dibromobenzene by density functional theory (DFT) in a 6-31g basis set.           

Fig. 2: Maplet interface to the Quantum Chemistry Toolbox 2019, showing a density functional theory (DFT) calculation         

After entering a name into the text box labeled Name, the user can click on: (1) the button Web to import the geometry from an online database containing more than 96 million molecules,  (2) the button File to read the geometry from a standard XYZ file, or (3) the button Input to enter the geometry.  As soon the geometry is entered, the Maplet displays a 3D picture of the molecule in the window on the right of the options.  Dropdown menus allow the user to select the basis set, the electronic structure method, and a boolean for geometry optimization.  The user can click on the Compute button to perform the computation.  When the quantum computation completes, the total energy appears in the box labeled Total Energy.  The dropdown menu Analyze contains a list of data tables, plots, and animations that can be selected and then displayed by clicking the Analyze button.  The Maplet interface contains nearly all of the options available in the worksheet interface.   The Help Pages of the Toolbox include extensive curricula and lessons that can be used in undergraduate, graduate, and even high school chemistry courses.  Next we look at some sample calculations in the worksheet interface.     

 

Reproducing an Early 2-RDM Calculation

 

One of the earliest variational calculations of the 2-RDM was performed in 1975 by Garrod, Mihailović,  and  Rosina [1-3].  They minimized the electronic ground state of the 4-electron atom beryllium as a functional of only two electrons, the 2-RDM.  They imposed semidefinite constraints on the particle-particle (D), hole-hole (Q), and particle-hole (G) metric matrices.  They solved the resulting optimization problem of minimizing the energy as a linear function of the 2-RDM subject to the semidefinite constraints, known as a semidefinite program, by a cutting-plane algorithm.  Due to limitations of the cutting-plane algorithm and computers circa 1975, the calculation was a difficult one, likely taking a significant amount of computer time and memory.

 

With the Quantum Chemistry Toolbox we can use the command Variational2RDM to reproduce the calculation on a Windows laptop.  First, in a Maple 2019 worksheet we load the commands of the Add-on Quantum Chemistry Toolbox:

with(QuantumChemistry);

[AOLabels, ActiveSpaceCI, ActiveSpaceSCF, AtomicData, BondAngles, BondDistances, Charges, ChargesPlot, CorrelationEnergy, CoupledCluster, DensityFunctional, DensityPlot3D, Dipole, DipolePlot, Energy, FullCI, GeometryOptimization, HartreeFock, Interactive, Isotopes, MOCoefficients, MODiagram, MOEnergies, MOIntegrals, MOOccupations, MOOccupationsPlot, MOSymmetries, MP2, MolecularData, MolecularGeometry, NuclearEnergy, NuclearGradient, Parametric2RDM, PlotMolecule, Populations, RDM1, RDM2, ReadXYZ, SaveXYZ, SearchBasisSets, SearchFunctionals, SkeletalStructure, Thermodynamics, Variational2RDM, VibrationalModeAnimation, VibrationalModes, Video]

(1.1)

Then we define the atom (or molecule) using a Maple list of lists that we assign to the variable atom:

atom := [["Be",0,0,0]];

[["Be", 0, 0, 0]]

(1.2)

 

We can then perform the variational 2-RDM method with the Variational2RDM command to compute the ground-state energy and properties of beryllium in a minimal basis set like the one used by Rosina and his collaborators.  By default the method uses the D, Q, and G N-representability conditions and the minimal "sto-3g" basis set.  The calculation, which completes in seconds, contains a wealth of information in the form of a convenient Maple table that we assign to the variable data.

data := Variational2RDM(atom);

table(%id = 18446744313704784158)

(1.3)

 

The table contains the total ground-state energy of the beryllium atom in the atomic unit of energy (hartrees)

data[e_tot];

HFloat(-14.40370016681039)

(1.4)

 

We also have the atomic orbitals (AOs) employed in the calculation

data[aolabels];

Vector(5, {(1) = "0 Be 1s", (2) = "0 Be 2s", (3) = "0 Be 2px", (4) = "0 Be 2py", (5) = "0 Be 2pz"})

(1.5)

 

as well as the Mulliken populations of these orbitals

data[populations];

Vector(5, {(1) = 1.9995807710723152, (2) = 1.7913484714571852, (3) = 0.6969023822632789e-1, (4) = 0.6969026475511847e-1, (5) = 0.6969029119010149e-1})

(1.6)

 

We see that 2 electrons are located in the 1s orbital, 1.8 electrons in the 2s orbital, and about 0.2 electrons in the 2p orbitals.  By default the calculation also returns the 1-RDM

data[rdm1];

Matrix(5, 5, {(1, 1) = 1.9999258249189755, (1, 2) = -0.37784860208539793e-2, (1, 3) = 0., (1, 4) = 0., (1, 5) = 0., (2, 1) = -0.37784860208539793e-2, (2, 2) = 1.7910034176105256, (2, 3) = 0., (2, 4) = 0., (2, 5) = 0., (3, 1) = 0., (3, 2) = 0., (3, 3) = 0.6969023822632789e-1, (3, 4) = 0., (3, 5) = 0., (4, 1) = 0., (4, 2) = 0., (4, 3) = 0., (4, 4) = 0.6969026475511847e-1, (4, 5) = 0., (5, 1) = 0., (5, 2) = 0., (5, 3) = 0., (5, 4) = 0., (5, 5) = 0.6969029119010149e-1})

(1.7)

 

The eigenvalues of the 1-RDM are the natural orbital occupations

LinearAlgebra:-Eigenvalues(data[rdm1]);

Vector(5, {(1) = 1.9999941387490443+0.*I, (2) = 1.7909351037804568+0.*I, (3) = 0.6969023822632789e-1+0.*I, (4) = 0.6969026475511847e-1+0.*I, (5) = 0.6969029119010149e-1+0.*I})

(1.8)

 

We can display the density of the 2s-like 2nd natural orbital using the DensityPlot3D command providing the atom, the data, and the orbitalindex keyword

DensityPlot3D(atom,data,orbitalindex=2);

 

 

Similarly,  using the DensityPlot3D command, we can readily display the 2p-like 3rd natural orbital

DensityPlot3D(atom,data,orbitalindex=3);

 

 

By using Maple keyword arguments in the Variational2RDM command, we can readily change the basis set, use point-group symmetry, add active orbitals with or without self-consistent-field, change the N-representability conditions, as well as explore many other options.  Having reenacted one of the first variational 2-RDM calculations ever, let's examine a more complicated molecule.

 

Explosive TNT

 

We consider the molecule TNT that is used as an explosive. Using the command MolecularGeometry, we can import the experimental geometry of TNT from the online PubChem database.

mol := MolecularGeometry("TNT");

[["O", .5454, -3.514, 0.12e-2], ["O", .5495, 3.5137, 0.8e-3], ["O", 2.4677, -2.4539, -0.5e-3], ["O", 2.4705, 2.4513, 0.3e-3], ["O", -3.5931, -1.0959, 0.4e-3], ["O", -3.5922, 1.0993, 0.6e-3], ["N", 1.2142, -2.454, 0.2e-3], ["N", 1.217, 2.4527, 0], ["N", -2.9846, 0.15e-2, 0.1e-3], ["C", 1.2253, -0.6e-3, -0.9e-3], ["C", .5271, -1.2082, -0.8e-3], ["C", .5284, 1.2078, -0.8e-3], ["C", -1.5646, 0.8e-3, -0.4e-3], ["C", -.8678, -1.2074, -0.6e-3], ["C", -.8666, 1.2084, -0.6e-3], ["C", 2.7239, -0.16e-2, 0.11e-2], ["H", -1.4159, -2.1468, -0.3e-3], ["H", -1.4137, 2.1483, -0.3e-3], ["H", 3.1226, .2418, -.9891], ["H", 3.0863, .6934, .7662], ["H", 3.3154, -.8111, .4109]]

(1.9)

 

The command PlotMolecule generates a 3D ball-and-stick plot of the molecule

PlotMolecule(mol);

 

 

We perform a variational calculation of the 2-RDM of TNT in an active space of 10 electrons and 10 orbitals by setting the keyword active to the list [10,10].  The keyword casscf is set to true to optimize the active orbitals during the calculation.  The keyword basis is used to set the basis set to a minimal basis set sto-3g for illustration.   

data := Variational2RDM(mol, active=[10,10], casscf=true, basis="sto-3g");

table(%id = 18446744493271367454)

(1.10)

 

The ground-state energy of TNT in hartrees is

data[e_tot];

HFloat(-868.8629631593426)

(1.11)

 

Unlike beryllium, the electric dipole moment of TNT in debyes is nonzero

data[dipole];

Vector(3, {(1) = .5158925019252739, (2) = -0.5985274393363119e-1, (3) = .1277528280025474})

(1.12)

 

We can easily visualize the dipole moment relative to the molecule's ball-and-stick model with the DipolePlot command

DipolePlot(mol,method=Variational2RDM, active=[10,10], casscf=true, basis="sto-3g");

 

 

The 1-RDM is returned by default

data[rdm1];

_rtable[18446744313709602566]

(1.13)

 

The natural molecular-orbital (MO) occupations are the eigenvalues of the 1-RDM

data[mo_occ];

_rtable[18446744313709600150]

(1.14)

 

All of the occupations can be viewed at once by converting the Vector to a list

convert(data[mo_occ], list);

[HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(1.9110133620349001), HFloat(1.8984139688344246), HFloat(1.6231436866358906), HFloat(1.6158489471020905), HFloat(1.6145310163161273), HFloat(0.38920731792133734), HFloat(0.387039366894289), HFloat(0.37786347287813526), HFloat(0.09734187094597906), HFloat(0.08559699476985069), HFloat(0.0), HFloat(0.0), HFloat(0.0), HFloat(0.0), HFloat(0.0), HFloat(0.0), HFloat(0.0), HFloat(0.0), HFloat(0.0), HFloat(0.0), HFloat(0.0), HFloat(0.0), HFloat(0.0), HFloat(0.0), HFloat(0.0), HFloat(0.0), HFloat(0.0), HFloat(0.0), HFloat(0.0), HFloat(0.0), HFloat(0.0), HFloat(0.0)]

(1.15)

 

We can visualize these occupations with the MOOccupationsPlot command

MOOccupationsPlot(mol,method=Variational2RDM, active=[10,10], casscf=true, basis="sto-3g");

 

 

The occupations, we observe, show significant deviations from 0 and 2, indicating that the electrons have substantial correlation beyond the mean-field (Hartree-Fock) limit.  The blue lines indicate the first N/2 spatial orbitals where N is the total number of electrons while the red lines indicate the remaining spatial orbitals.  We can visualize the highest "occupied" molecular orbital (58) with the DensityPlot3D command

DensityPlot3D(mol,data, orbitalindex=58);

 

 

Similarly, we can visualize the lowest "unoccupied" molecular orbital (59) with the DensityPlot3D command

DensityPlot3D(mol,data, orbitalindex=59);

 

 

Comparison of orbitals 58 and 59 reveals an increase in the number of nodes (changes in the phase of the orbitals denoted by green and purple), which reflects an increase in the energy of the orbital.

 

Looking Ahead

 

The Maple Quantum Chemistry Toolbox 2019, an new Add-on for Maple 2019 from RDMChem, provides a easy-to-use, research-grade environment for the computation of the energies and properties of atoms and molecules.  In this blog we discussed its origins in graduate research at Harvard, its reproduction of an early 2-RDM calculation of beryllium, and its application to the explosive molecule TNT.  We have illustrated only some of the many features and electronic structure methods of the Maple Quantum Chemistry package.  There is much more chemistry and physics to explore.  Enjoy!    

 

Selected References

 

[1] D. A. Mazziotti, Chem. Rev. 112, 244 (2012). "Two-electron Reduced Density Matrix as the Basic Variable in Many-Electron Quantum Chemistry and Physics"

[2]  Reduced-Density-Matrix Mechanics: With Application to Many-Electron Atoms and Molecules (Adv. Chem. Phys.) ; D. A. Mazziotti, Ed.; Wiley: New York, 2007; Vol. 134.

[3] A. J. Coleman and V. I. Yukalov, Reduced Density Matrices: Coulson’s Challenge (Springer-Verlag,  New York, 2000).

[4] D. A. Mazziotti, Phys. Rev. Lett. 106, 083001 (2011). "Large-scale Semidefinite Programming for Many-electron Quantum Mechanics"

[5] A. W. Schlimgen, C. W. Heaps, and D. A. Mazziotti, J. Phys. Chem. Lett. 7, 627-631 (2016). "Entangled Electrons Foil Synthesis of Elusive Low-Valent Vanadium Oxo Complex"

[6] J. M. Montgomery and D. A. Mazziotti, J. Phys. Chem. A 122, 4988-4996 (2018). "Strong Electron Correlation in Nitrogenase Cofactor, FeMoco"

 

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