From 67c777b50ff969db06f0f938241ce4b802342198 Mon Sep 17 00:00:00 2001
From: Kirk Pearce <kcpearce@vt.edu>
Date: Fri, 31 Mar 2017 11:46:46 -0400
Subject: [PATCH 01/41] Open Pull Request for add-eqns branch

---
 Project#04/figures/test.txt | 0
 1 file changed, 0 insertions(+), 0 deletions(-)
 create mode 100644 Project#04/figures/test.txt

diff --git a/Project#04/figures/test.txt b/Project#04/figures/test.txt
new file mode 100644
index 0000000..e69de29

From 6605547ceb947ec22fad36e2ab9777f1dc56b8b5 Mon Sep 17 00:00:00 2001
From: Kirk Pearce <kcpearce@vt.edu>
Date: Fri, 31 Mar 2017 12:11:07 -0400
Subject: [PATCH 02/41] Test Eqn Changes

---
 Project#03/README.md                          |   2 +-
 .../figures/compound-index-restrictions2.png  | Bin 5328 -> 5343 bytes
 Project#03/figures/eri.png                    | Bin 10791 -> 10853 bytes
 Project#04/figures/noddy-transformation.pdf   | Bin 0 -> 65074 bytes
 Project#04/figures/test.txt                   |   0
 Project#04/figures/two-electron-integrals.pdf | Bin 0 -> 96239 bytes
 6 files changed, 1 insertion(+), 1 deletion(-)
 create mode 100644 Project#04/figures/noddy-transformation.pdf
 delete mode 100644 Project#04/figures/test.txt
 create mode 100644 Project#04/figures/two-electron-integrals.pdf

diff --git a/Project#03/README.md b/Project#03/README.md
index ab949bb..8418944 100644
--- a/Project#03/README.md
+++ b/Project#03/README.md
@@ -69,7 +69,7 @@ where
 Note that the two-electron integrals may be stored efficiently in a one-dimensional array and the above relationship used to map between given 
 &mu;, &nu;, &lambda;, and &sigma; indices and a "compound index" defined as:
 
-<img src="./figures/compound-index-restrictions2.png" height="20">
+<img src="./figures/compound-index-restrictions2.png" height="20">,
 
   * [Hint 1](./hints/hint3-1.md): Compound indices
   * [Hint 2](./hints/hint3-2.md): Pre-Computed Lookup Arrays
diff --git a/Project#03/figures/compound-index-restrictions2.png b/Project#03/figures/compound-index-restrictions2.png
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From 0a4970cd2f06067d0098c12bc904b51366313ac4 Mon Sep 17 00:00:00 2001
From: Kirk Pearce <kcpearce@vt.edu>
Date: Fri, 31 Mar 2017 12:27:50 -0400
Subject: [PATCH 03/41] Project 4 eqns added

---
 Project#03/README.md                          |   2 +-
 Project#04/README.md                          |  27 ++++--------------
 Project#04/figures/eri.png                    | Bin 0 -> 10853 bytes
 Project#04/figures/mp2-energy.png             | Bin 0 -> 12063 bytes
 Project#04/figures/noddy-transform.png        | Bin 0 -> 10615 bytes
 Project#04/figures/noddy-transformation.pdf   | Bin 65074 -> 0 bytes
 Project#04/figures/smart-transform.png        | Bin 0 -> 13241 bytes
 Project#04/figures/two-electron-integrals.pdf | Bin 96239 -> 0 bytes
 Project#04/project4-instructions.pdf          | Bin 0 -> 72790 bytes
 9 files changed, 7 insertions(+), 22 deletions(-)
 create mode 100644 Project#04/figures/eri.png
 create mode 100644 Project#04/figures/mp2-energy.png
 create mode 100644 Project#04/figures/noddy-transform.png
 delete mode 100644 Project#04/figures/noddy-transformation.pdf
 create mode 100644 Project#04/figures/smart-transform.png
 delete mode 100644 Project#04/figures/two-electron-integrals.pdf
 create mode 100644 Project#04/project4-instructions.pdf

diff --git a/Project#03/README.md b/Project#03/README.md
index 8418944..ab949bb 100644
--- a/Project#03/README.md
+++ b/Project#03/README.md
@@ -69,7 +69,7 @@ where
 Note that the two-electron integrals may be stored efficiently in a one-dimensional array and the above relationship used to map between given 
 &mu;, &nu;, &lambda;, and &sigma; indices and a "compound index" defined as:
 
-<img src="./figures/compound-index-restrictions2.png" height="20">,
+<img src="./figures/compound-index-restrictions2.png" height="20">
 
   * [Hint 1](./hints/hint3-1.md): Compound indices
   * [Hint 2](./hints/hint3-2.md): Pre-Computed Lookup Arrays
diff --git a/Project#04/README.md b/Project#04/README.md
index 775a9ec..68f06d1 100644
--- a/Project#04/README.md
+++ b/Project#04/README.md
@@ -2,17 +2,12 @@
 Unlike the Hartree-Fock energy, correlation energies like the MP2 energy are usually expressed in terms of MO-basis quantities (integrals, MO energies). 
 The most expensive part of the calculation is the transformation of the two-electron integrals from the AO to the MO basis representation.  
 The purpose of this project is to understand this transformation and fundamental techniques for its efficient implementation. 
-The theoretical background and a concise set of instructions for this project may be found [[http://sirius.chem.vt.edu/~crawdad/programming/mp2.pdf|here]].
+The theoretical background and a concise set of instructions for this project may be found [here](./project4-instructions.pdf).
 
 ## Step #1: Read the Two-Electron Integrals
 The Mulliken-ordered integrals are defined as:
 
-```
-EQUATION
-(\mu \nu|\lambda \sigma) \equiv \int \phi_\mu({\mathbf r_1})
-\phi_\nu({\mathbf r_1}) r_{12}^{-1} \phi_\lambda({\mathbf r_2})
-\phi_\sigma({\mathbf r_2}) d{\mathbf r_1} d{\mathbf r_2}
-```
+<img src="./figures/eri.png" height="45">
 
 Concise instructions for this step can be found in 
 [Project #3](https://github.com/CrawfordGroup/ProgrammingProjects/tree/master/Project%2303)
@@ -25,11 +20,7 @@ Use the values you computed in the Hartree-Fock program of
 
 The most straightforward expression of the AO/MO integral transformation is
 
-```
-EQUATION
-(pq|rs) = \sum_\mu \sum_\nu \sum_\lambda \sum_\sigma C_\mu^p C_\nu^q
-(\mu \nu|\lambda \sigma) C_\lambda^r C_\sigma^s
-```
+<img src="./figures/noddy-transform.png" height="45">
 
 This approach is easy to implement (hence the word "[noddy](http://www.hackerslang.com/noddy.html)" above), but is expensive due to its N<sup>8</sup> computational order.  
 Nevertheless, you should start with this algorithm to get your code working, and run timings (use the UNIX "time" command) 
@@ -39,10 +30,8 @@ for the test cases below to get an idea of its computational cost.
 ### The Smarter Algorithm
 
 Notice that none of the *C* coefficients in the above expression have any indices in common.  Thus, the summation could be rearranged such that:
-```
-EQUATION 
-(pq|rs) = \sum_\mu C_\mu^p \left[ \sum_\nu C_\nu^q \left[ \sum_\lambda  C_\lambda^r \left[ \sum_\sigma C_\sigma^s (\mu \nu|\lambda \sigma) \right] \right] \right].
-```
+
+<img src="./figures/smart-transform.png" height="45">
 
 This means that each summation within brackets could be carried out separately, 
 starting from the innermost summation over <html>&#963;</html>, if we store the results at each step.  
@@ -57,11 +46,7 @@ After you have the noddy algorithm working and timed, modify it to use this smar
 
 ## Step #4: Compute the MP2 Energy
 
-```
-EQUATION
-E_{\rm MP2} = \sum_{ij} \sum_{ab} \frac{(ia|jb) \left[ 2 (ia|jb) -
-(ib|ja) \right]}{\epsilon_i + \epsilon_j - \epsilon_a - \epsilon_b},
-```
+<img src="./figures/mp2-energy.png" height="45">
 
 where *i* and *j* denote doubly-occupied orbitals and *a* and *b* unoccupied orbitals, and the denominator involves the MO energies.
 
diff --git a/Project#04/figures/eri.png b/Project#04/figures/eri.png
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From c6ca632913d2ef4a3e34889c8d237bc8a57cea27 Mon Sep 17 00:00:00 2001
From: Kirk Pearce <kcpearce@vt.edu>
Date: Fri, 31 Mar 2017 12:29:53 -0400
Subject: [PATCH 04/41] Resize Project 4 eqns

---
 Project#04/README.md | 6 +++---
 1 file changed, 3 insertions(+), 3 deletions(-)

diff --git a/Project#04/README.md b/Project#04/README.md
index 68f06d1..59eb5fc 100644
--- a/Project#04/README.md
+++ b/Project#04/README.md
@@ -20,7 +20,7 @@ Use the values you computed in the Hartree-Fock program of
 
 The most straightforward expression of the AO/MO integral transformation is
 
-<img src="./figures/noddy-transform.png" height="45">
+<img src="./figures/noddy-transform.png" height="60">
 
 This approach is easy to implement (hence the word "[noddy](http://www.hackerslang.com/noddy.html)" above), but is expensive due to its N<sup>8</sup> computational order.  
 Nevertheless, you should start with this algorithm to get your code working, and run timings (use the UNIX "time" command) 
@@ -31,7 +31,7 @@ for the test cases below to get an idea of its computational cost.
 
 Notice that none of the *C* coefficients in the above expression have any indices in common.  Thus, the summation could be rearranged such that:
 
-<img src="./figures/smart-transform.png" height="45">
+<img src="./figures/smart-transform.png" height="75">
 
 This means that each summation within brackets could be carried out separately, 
 starting from the innermost summation over <html>&#963;</html>, if we store the results at each step.  
@@ -46,7 +46,7 @@ After you have the noddy algorithm working and timed, modify it to use this smar
 
 ## Step #4: Compute the MP2 Energy
 
-<img src="./figures/mp2-energy.png" height="45">
+<img src="./figures/mp2-energy.png" height="75">
 
 where *i* and *j* denote doubly-occupied orbitals and *a* and *b* unoccupied orbitals, and the denominator involves the MO energies.
 

From f40d68105d4e58720376a4f676b32a6670aa9128 Mon Sep 17 00:00:00 2001
From: Kirk Pearce <kcpearce@vt.edu>
Date: Fri, 31 Mar 2017 12:30:45 -0400
Subject: [PATCH 05/41] Resize Project 4 eqns

---
 Project#04/README.md | 6 +++---
 1 file changed, 3 insertions(+), 3 deletions(-)

diff --git a/Project#04/README.md b/Project#04/README.md
index 59eb5fc..356ffd1 100644
--- a/Project#04/README.md
+++ b/Project#04/README.md
@@ -20,7 +20,7 @@ Use the values you computed in the Hartree-Fock program of
 
 The most straightforward expression of the AO/MO integral transformation is
 
-<img src="./figures/noddy-transform.png" height="60">
+<img src="./figures/noddy-transform.png" height="50">
 
 This approach is easy to implement (hence the word "[noddy](http://www.hackerslang.com/noddy.html)" above), but is expensive due to its N<sup>8</sup> computational order.  
 Nevertheless, you should start with this algorithm to get your code working, and run timings (use the UNIX "time" command) 
@@ -31,7 +31,7 @@ for the test cases below to get an idea of its computational cost.
 
 Notice that none of the *C* coefficients in the above expression have any indices in common.  Thus, the summation could be rearranged such that:
 
-<img src="./figures/smart-transform.png" height="75">
+<img src="./figures/smart-transform.png" height="60">
 
 This means that each summation within brackets could be carried out separately, 
 starting from the innermost summation over <html>&#963;</html>, if we store the results at each step.  
@@ -46,7 +46,7 @@ After you have the noddy algorithm working and timed, modify it to use this smar
 
 ## Step #4: Compute the MP2 Energy
 
-<img src="./figures/mp2-energy.png" height="75">
+<img src="./figures/mp2-energy.png" height="60">
 
 where *i* and *j* denote doubly-occupied orbitals and *a* and *b* unoccupied orbitals, and the denominator involves the MO energies.
 

From ccdf932439f0292403aa9fe8f505d926f154ee0e Mon Sep 17 00:00:00 2001
From: Kirk Pearce <kcpearce@vt.edu>
Date: Fri, 31 Mar 2017 12:38:00 -0400
Subject: [PATCH 06/41] Change to relative links

---
 Project#04/README.md | 27 +++++++++++----------------
 1 file changed, 11 insertions(+), 16 deletions(-)

diff --git a/Project#04/README.md b/Project#04/README.md
index 356ffd1..5de966d 100644
--- a/Project#04/README.md
+++ b/Project#04/README.md
@@ -1,7 +1,7 @@
 # Project #4: The second-order Moller-Plesset perturbation (MP2) energy
 Unlike the Hartree-Fock energy, correlation energies like the MP2 energy are usually expressed in terms of MO-basis quantities (integrals, MO energies). 
-The most expensive part of the calculation is the transformation of the two-electron integrals from the AO to the MO basis representation.  
-The purpose of this project is to understand this transformation and fundamental techniques for its efficient implementation. 
+The most expensive part of the calculation is the transformation of the two-electron integrals from the AO to the MO basis representation. 
+The purpose of this project is to understand this transformation and fundamental techniques for its efficient implementation.
 The theoretical background and a concise set of instructions for this project may be found [here](./project4-instructions.pdf).
 
 ## Step #1: Read the Two-Electron Integrals
@@ -9,12 +9,10 @@ The Mulliken-ordered integrals are defined as:
 
 <img src="./figures/eri.png" height="45">
 
-Concise instructions for this step can be found in 
-[Project #3](https://github.com/CrawfordGroup/ProgrammingProjects/tree/master/Project%2303)
+Concise instructions for this step can be found in [Project #3](../Project%2303).
 
 ## Step #2: Obtain the MO Coefficients and MO Energies 
-Use the values you computed in the Hartree-Fock program of 
-[Project #3](https://github.com/CrawfordGroup/ProgrammingProjects/tree/master/Project%2303)
+Use the values you computed in the Hartree-Fock program of [Project #3](../Project%2303).
 ## Step #3: Transform the Two-Electron Integrals to the MO Basis 
 ### The Noddy Algorithm 
 
@@ -22,9 +20,7 @@ The most straightforward expression of the AO/MO integral transformation is
 
 <img src="./figures/noddy-transform.png" height="50">
 
-This approach is easy to implement (hence the word "[noddy](http://www.hackerslang.com/noddy.html)" above), but is expensive due to its N<sup>8</sup> computational order.  
-Nevertheless, you should start with this algorithm to get your code working, and run timings (use the UNIX "time" command) 
-for the test cases below to get an idea of its computational cost.
+This approach is easy to implement (hence the word "[noddy](http://www.hackerslang.com/noddy.html)" above), but is expensive due to its N<sup>8</sup> computational order.  Nevertheless, you should start with this algorithm to get your code working, and run timings (use the UNIX "time" command) for the test cases below to get an idea of its computational cost.
 
   * Hint 1: Noddy transformation code
 ### The Smarter Algorithm
@@ -34,8 +30,7 @@ Notice that none of the *C* coefficients in the above expression have any indice
 <img src="./figures/smart-transform.png" height="60">
 
 This means that each summation within brackets could be carried out separately, 
-starting from the innermost summation over <html>&#963;</html>, if we store the results at each step.  
-This reduces the N<sup>8</sup> algorithm above to four N<sup>5</sup> steps.
+starting from the innermost summation over <html>&#963;</html>, if we store the results at each step.  This reduces the N<sup>8</sup> algorithm above to four N<sup>5</sup> steps.
 
 Be careful about the permutational symmetry of the integrals and intermediates at each step: 
 while the AO-basis integrals have eight-fold permutational symmetry, the partially transformed integrals have much less symmetry.
@@ -53,10 +48,10 @@ where *i* and *j* denote doubly-occupied orbitals and *a* and *b* unoccupied orb
   * Hint 3: Occupied versus unoccupied orbitals
 
 ## Test Cases
-The input structures, integrals, etc. for these examples are found in the [input directory](https://github.com/CrawfordGroup/ProgrammingProjects/tree/master/Project%2304/input).
+The input structures, integrals, etc. for these examples are found in the [input directory](./input).
 
-* STO-3G Water | [output](https://github.com/CrawfordGroup/ProgrammingProjects/blob/master/Project%2304/output/h2o/STO-3G/output.txt)
-* DZ Water | [output](https://github.com/CrawfordGroup/ProgrammingProjects/blob/master/Project%2304/output/h2o/DZ/output.txt)
-* DZP Water | [output](https://github.com/CrawfordGroup/ProgrammingProjects/blob/master/Project%2304/output/h2o/DZP/output.txt)
-* STO-3G Methane | [output](https://github.com/CrawfordGroup/ProgrammingProjects/blob/master/Project%2304/output/ch4/STO-3G/output.txt)
+* STO-3G Water | [output](./output/h2o/STO-3G/output.txt)
+* DZ Water | [output](./output/h2o/DZ/output.txt)
+* DZP Water | [output](./output/h2o/DZP/output.txt)
+* STO-3G Methane | [output](./output/ch4/STO-3G/output.txt)
 

From 6ad2c8ca3a509906f827f34a47e6cb1ad78aa6f4 Mon Sep 17 00:00:00 2001
From: Kirk Pearce <kcpearce@vt.edu>
Date: Fri, 31 Mar 2017 12:52:38 -0400
Subject: [PATCH 07/41] Add Project 4 Hints

---
 Project#04/README.md      |  6 ++--
 Project#04/hints/hint1.md | 37 ++++++++++++++++++++++++
 Project#04/hints/hint2.md | 59 +++++++++++++++++++++++++++++++++++++++
 Project#04/hints/hint3.md | 20 +++++++++++++
 4 files changed, 119 insertions(+), 3 deletions(-)
 create mode 100644 Project#04/hints/hint1.md
 create mode 100644 Project#04/hints/hint2.md
 create mode 100644 Project#04/hints/hint3.md

diff --git a/Project#04/README.md b/Project#04/README.md
index 5de966d..fabf300 100644
--- a/Project#04/README.md
+++ b/Project#04/README.md
@@ -22,7 +22,7 @@ The most straightforward expression of the AO/MO integral transformation is
 
 This approach is easy to implement (hence the word "[noddy](http://www.hackerslang.com/noddy.html)" above), but is expensive due to its N<sup>8</sup> computational order.  Nevertheless, you should start with this algorithm to get your code working, and run timings (use the UNIX "time" command) for the test cases below to get an idea of its computational cost.
 
-  * Hint 1: Noddy transformation code
+  * [Hint 1](./hints/hint1.md): Noddy transformation code
 ### The Smarter Algorithm
 
 Notice that none of the *C* coefficients in the above expression have any indices in common.  Thus, the summation could be rearranged such that:
@@ -37,7 +37,7 @@ while the AO-basis integrals have eight-fold permutational symmetry, the partial
 
 After you have the noddy algorithm working and timed, modify it to use this smarter algorithm and time the test cases again.  Enjoy!
 
-  * Hint 2: Smarter transformation code
+  * [Hint 2](./hints/hint2.md): Smarter transformation code
 
 ## Step #4: Compute the MP2 Energy
 
@@ -45,7 +45,7 @@ After you have the noddy algorithm working and timed, modify it to use this smar
 
 where *i* and *j* denote doubly-occupied orbitals and *a* and *b* unoccupied orbitals, and the denominator involves the MO energies.
 
-  * Hint 3: Occupied versus unoccupied orbitals
+  * [Hint 3](./hints/hint3.md): Occupied versus unoccupied orbitals
 
 ## Test Cases
 The input structures, integrals, etc. for these examples are found in the [input directory](./input).
diff --git a/Project#04/hints/hint1.md b/Project#04/hints/hint1.md
new file mode 100644
index 0000000..98c462c
--- /dev/null
+++ b/Project#04/hints/hint1.md
@@ -0,0 +1,37 @@
+Here's a code block that demonstrates how to carry out the AO to MO two-electron integral transformation using a single N<sup>8</sup> step.  Note that the original AO- and transformed MO-basis integrals are stored in a one-dimensional array taking full advantage of their eight-fold permutational symmetry, as described in [Project #3](../Project%2303).
+
+Note how the loop structure in the MO-indices (*i*, *j*, *k*, and *l*) automatically takes into account the permutational symmetry of the fully transformed integrals.
+
+```c++
+  #define INDEX(i,j) ((i>j) ? (((i)*((i)+1)/2)+(j)) : (((j)*((j)+1)/2)+(i)))
+  ...
+  double *TEI_AO, *TEI_MO;
+  int i, j, k, l, ijkl;
+  int p, q, r, s, pq, rs, pqrs;
+  ...
+
+  for(i=0,ijkl=0; i < nao; i++) {
+    for(j=0; j <= i; j++) {
+      for(k=0; k <= i; k++) {
+        for(l=0; l <= (i==k ? j : k); l++,ijkl++) {
+
+          for(p=0; p < nao; p++) {
+            for(q=0; q < nao; q++) {
+              pq = INDEX(p,q);
+              for(r=0; r < nao; r++) {
+                for(s=0; s < nao; s++) {
+                  rs = INDEX(r,s);
+                  pqrs = INDEX(pq,rs);
+
+                  TEI_MO[ijkl] += C[p][i] * C[q][j] * C[r][k] * C[s][l] * TEI_AO[pqrs];
+
+                }
+              }
+            }
+          }
+
+        }
+      }
+    }
+  }
+```
diff --git a/Project#04/hints/hint2.md b/Project#04/hints/hint2.md
new file mode 100644
index 0000000..6a1c632
--- /dev/null
+++ b/Project#04/hints/hint2.md
@@ -0,0 +1,59 @@
+Here's a code block that demonstrates how to carry out the AO to MO two-electron integral transformation using four N<sup>5</sup> steps.  Note that the original AO- and transformed MO-basis integrals are stored in a one-dimensional array taking advantage of their eight-fold permutational symmetry, as described in [Project #3](../Project%2303).  However, the half-transformed integrals, which lack bra-ket permutational symmetry, are stored in a two-dimensional array.
+
+A number of convenient functions [`mmult()`, `init_matrix()`, etc.] used in this code block can be found in the [diag.cc](http://sirius.chem.vt.edu/~crawdad/programming/diag.cc) discussed in [Project #1](../Project%2301).
+
+```c++
+  #define INDEX(i,j) ((i>j) ? (((i)*((i)+1)/2)+(j)) : (((j)*((j)+1)/2)+(i)))
+  ...
+  double **X, **Y, **TMP, *TEI;
+  int i, j, k, l, ij, kl, ijkl, klij;
+
+  ...
+  X = init_matrix(nao, nao);
+  Y = init_matrix(nao, nao);
+
+  TMP = init_matrix((nao*(nao+1)/2),(nao*(nao+1)/2));
+  for(i=0,ij=0; i < nao; i++)
+    for(j=0; j <= i; j++,ij++) {
+      for(k=0,kl=0; k < nao; k++)
+        for(l=0; l <= k; l++,kl++) {
+          ijkl = INDEX(ij,kl);
+          X[k][l] = X[l][k] = TEI[ijkl];
+        }
+      zero_matrix(Y, nao, nao);
+      mmult(C, 1, X, 0, Y, nao, nao, nao);
+      zero_matrix(X, nao, nao);
+      mmult(Y, 0, C, 0, X, nao, nao, nao);
+      for(k=0, kl=0; k < nao; k++)
+        for(l=0; l <= k; l++, kl++)
+          TMP[kl][ij] = X[k][l];
+    }
+
+  zero_array(TEI, (nao*(nao+1)/2)*((nao*(nao+1)/2)+1)/2);
+
+  for(k=0,kl=0; k < nao; k++)
+    for(l=0; l <= k; l++,kl++) {
+      zero_matrix(X, nao, nao);
+      zero_matrix(Y, nao, nao);
+      for(i=0,ij=0; i < nao; i++)
+        for(j=0; j <=i; j++,ij++)
+          X[i][j] = X[j][i] = TMP[kl][ij];
+      zero_matrix(Y, nao, nao);
+      mmult(C, 1, X, 0, Y, nao, nao, nao);
+      zero_matrix(X, nao, nao);
+      mmult(Y, 0, C, 0, X, nao, nao, nao);
+      for(i=0, ij=0; i < nao; i++)
+        for(j=0; j <= i; j++,ij++) {
+          klij = INDEX(kl,ij);
+          TEI[klij] = X[i][j];
+        }
+    }
+
+  ...
+
+  free_matrix(X, nao);
+  free_matrix(Y, nao);
+  free_matrix(TMP, (nao*(nao+1)/2));
+
+```
+
diff --git a/Project#04/hints/hint3.md b/Project#04/hints/hint3.md
new file mode 100644
index 0000000..d6ae3ab
--- /dev/null
+++ b/Project#04/hints/hint3.md
@@ -0,0 +1,20 @@
+The following code block illustrates a simple-minded MP2 energy calculation with the two-electron integrals stored in a one-dimensional array.  The molecular orbitals are assumed to be ordered with all doubly-occupied orbitals first, followed by the unoccupied/virtual orbitals.  Notice the difference in limits of the loops over *i* and *a*:
+
+```c++
+  Emp2 = 0.0;
+  for(i=0; i < ndocc; i++) {
+    for(a=ndocc; a < nao; a++) {
+      ia = INDEX(i,a);
+      for(j=0; j < ndocc; j++) {
+        ja = INDEX(j,a);
+        for(b=ndocc; b < nao; b++) {
+          jb = INDEX(j,b);
+          ib = INDEX(i,b);
+          iajb = INDEX(ia,jb);
+          ibja = INDEX(ib,ja);
+          Emp2 += TEI[iajb] * (2 * TEI[iajb] - TEI[ibja])/(eps[i] + eps[j] - eps[a] - eps[b]);
+        }
+      }
+    }
+  }
+```

From 711984ad90ff6425c5e9811c1845375d1b890872 Mon Sep 17 00:00:00 2001
From: Kirk Pearce <kcpearce@vt.edu>
Date: Fri, 31 Mar 2017 12:55:31 -0400
Subject: [PATCH 08/41] Fix links in hints for Project 4

---
 Project#04/hints/hint1.md | 2 +-
 Project#04/hints/hint2.md | 2 +-
 2 files changed, 2 insertions(+), 2 deletions(-)

diff --git a/Project#04/hints/hint1.md b/Project#04/hints/hint1.md
index 98c462c..01221e1 100644
--- a/Project#04/hints/hint1.md
+++ b/Project#04/hints/hint1.md
@@ -1,4 +1,4 @@
-Here's a code block that demonstrates how to carry out the AO to MO two-electron integral transformation using a single N<sup>8</sup> step.  Note that the original AO- and transformed MO-basis integrals are stored in a one-dimensional array taking full advantage of their eight-fold permutational symmetry, as described in [Project #3](../Project%2303).
+Here's a code block that demonstrates how to carry out the AO to MO two-electron integral transformation using a single N<sup>8</sup> step.  Note that the original AO- and transformed MO-basis integrals are stored in a one-dimensional array taking full advantage of their eight-fold permutational symmetry, as described in [Project #3](../../Project%2303).
 
 Note how the loop structure in the MO-indices (*i*, *j*, *k*, and *l*) automatically takes into account the permutational symmetry of the fully transformed integrals.
 
diff --git a/Project#04/hints/hint2.md b/Project#04/hints/hint2.md
index 6a1c632..6b216e9 100644
--- a/Project#04/hints/hint2.md
+++ b/Project#04/hints/hint2.md
@@ -1,4 +1,4 @@
-Here's a code block that demonstrates how to carry out the AO to MO two-electron integral transformation using four N<sup>5</sup> steps.  Note that the original AO- and transformed MO-basis integrals are stored in a one-dimensional array taking advantage of their eight-fold permutational symmetry, as described in [Project #3](../Project%2303).  However, the half-transformed integrals, which lack bra-ket permutational symmetry, are stored in a two-dimensional array.
+Here's a code block that demonstrates how to carry out the AO to MO two-electron integral transformation using four N<sup>5</sup> steps.  Note that the original AO- and transformed MO-basis integrals are stored in a one-dimensional array taking advantage of their eight-fold permutational symmetry, as described in [Project #3](../../Project%2303).  However, the half-transformed integrals, which lack bra-ket permutational symmetry, are stored in a two-dimensional array.
 
 A number of convenient functions [`mmult()`, `init_matrix()`, etc.] used in this code block can be found in the [diag.cc](http://sirius.chem.vt.edu/~crawdad/programming/diag.cc) discussed in [Project #1](../Project%2301).
 

From f91a09ee7f0b79cacfed8f788a1ed7d84bf69057 Mon Sep 17 00:00:00 2001
From: Kirk Pearce <kcpearce@vt.edu>
Date: Fri, 31 Mar 2017 12:56:23 -0400
Subject: [PATCH 09/41] Fix links in hints for Project 4 again

---
 Project#04/hints/hint2.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/Project#04/hints/hint2.md b/Project#04/hints/hint2.md
index 6b216e9..79a46dd 100644
--- a/Project#04/hints/hint2.md
+++ b/Project#04/hints/hint2.md
@@ -1,6 +1,6 @@
 Here's a code block that demonstrates how to carry out the AO to MO two-electron integral transformation using four N<sup>5</sup> steps.  Note that the original AO- and transformed MO-basis integrals are stored in a one-dimensional array taking advantage of their eight-fold permutational symmetry, as described in [Project #3](../../Project%2303).  However, the half-transformed integrals, which lack bra-ket permutational symmetry, are stored in a two-dimensional array.
 
-A number of convenient functions [`mmult()`, `init_matrix()`, etc.] used in this code block can be found in the [diag.cc](http://sirius.chem.vt.edu/~crawdad/programming/diag.cc) discussed in [Project #1](../Project%2301).
+A number of convenient functions [`mmult()`, `init_matrix()`, etc.] used in this code block can be found in the [diag.cc](http://sirius.chem.vt.edu/~crawdad/programming/diag.cc) discussed in [Project #1](../../Project%2301).
 
 ```c++
   #define INDEX(i,j) ((i>j) ? (((i)*((i)+1)/2)+(j)) : (((j)*((j)+1)/2)+(i)))

From f78f98537cfb7a806650ef9554d4bad4b2898b90 Mon Sep 17 00:00:00 2001
From: Kirk Pearce <kcpearce@vt.edu>
Date: Fri, 31 Mar 2017 13:39:36 -0400
Subject: [PATCH 10/41] Add Project 5 hints and equations

---
 Project#05/README.md | 51 +++++++++++---------------------------------
 1 file changed, 13 insertions(+), 38 deletions(-)

diff --git a/Project#05/README.md b/Project#05/README.md
index 2d3a74a..3b98da9 100644
--- a/Project#05/README.md
+++ b/Project#05/README.md
@@ -6,50 +6,27 @@ amplitudes as well as Fock-matrix elements and antisymmetrized, Dirac-notation t
 (as opposed to the spatial-orbital basis used in the earlier [MP2 Project](https://github.com/CrawfordGroup/ProgrammingProjects/tree/master/Project%2304).
 Thus, the transformation of the AO-basis integrals into the spatial-MO basis must also include their translation into the spin-orbital basis:
 
-```
-EQUATION
-\begin{eqnarray*}
-\langle p q | r s \rangle & \equiv & \int d{\mathbf r}_1 d{\mathbf r}_2 d\omega_1 d\omega_1 \phi_p({\mathbf r}_1)\sigma_p(\omega_1) \phi_q({\mathbf r}_2)\sigma_q(\omega_2) \frac{1}{{\mathbf r}_{12}} \phi_r({\mathbf r}_1)\sigma_r(\omega_1) \phi_s({\mathbf r}_2)\sigma_s(\omega_2) \\
-& \equiv & \int d{\mathbf r}_1 d{\mathbf r}_2 \phi_p({\mathbf r}_1) \phi_q({\mathbf r}_2) \frac{1}{{\mathbf r}_{12}} \phi_r({\mathbf r}_1) \phi_s({\mathbf r}_2) \int d\omega_1 d\omega_1 \sigma_p(\omega_1) \sigma_q(\omega_2) \sigma_r(\omega_1) \sigma_s(\omega_2) \\
-& \equiv & (p r | q s)  \int d\omega_1 d\omega_1 \sigma_p(\omega_1) \sigma_q(\omega_2) \sigma_r(\omega_1) \sigma_s(\omega_2)
-\end{eqnarry*}
-```
-
+<img src="./figures/spin-orbital-eri.png" height="100">
 
 Thus, if you know the ordering of the orbitals (e.g. all occupied orbitals before virtual orbitals, perhaps alternating between alpha and beta spins), it is straightforward to carry out the integration over the spin components (the sigmas) in the above expression.  Thus, each spatial-orbital MO-basis two-electron integral translates to 16 possible spin-orbital integrals, only four of which are non-zero.
 
 Don't forget that you must also create the spin-orbital Fock matrix:
 
-```
-EQUATION
-f_{pq} = h_{pq} + \sum_m^{\rm occ} \langle pm || qm \rangle
-```
+<img src="./figures/spin-orbital-fock.png" height="50">
 
 Suggestion: For simplicity, store the two-electron integrals in a four-dimensional array.  This will greatly facilitate debugging of the complicated CCSD equations.
 
-  * Hint: Sample spatial- to spin-orbital translation code.
+  * [Hint 1](./hints/hint1.md): Sample spatial- to spin-orbital translation code.
 
 ## Step #2: Build the Initial-Guess Cluster Amplitudes 
 For Hartree-Fock reference determinants, the most common initial guess for the cluster amplitudes are the Moller-Plesset first-order perturbed wave function:
-```
-EQUATION
-t_i^a = 0
-```
-
-```
-EQUATION
-t_{ij}^{ab} = \frac{\langle ij || ab \rangle}{\epsilon_i + \epsilon_j - \epsilon_a - \epsilon_b}
 
-```
+<img src="./figures/init-t-amps.png" height="85">
 
 Note that if you have constructed the Fock matrix, two-electron integrals, and initial-guess amplitudes correctly at this point, 
-you should be able to compute the MP2 correlation energy using the simple spin-orbital expression and get identical results to those from 
-[Project #4] (https://github.com/CrawfordGroup/ProgrammingProjects/tree/master/Project%2304):
+you should be able to compute the MP2 correlation energy using the simple spin-orbital expression and get identical results to those from [Project #4](./Project%2304):
 
-```
-EQUATION
-E_{\rm MP2} = \frac{1}{4} \sum_{ijab} \langle ij||ab\rangle t_{ij}^{ab}
-```
+<img src="./figures/mp2-energy.png" height="85">
 
 ## Step #3: Calculate the CC Intermediates 
 Use the spin-orbital Eqs. 3-13 from Stanton's paper to build the two-index (F) and four-index (W) intermediates, as well as the effective doubles (labelled with the Greek letter tau).
@@ -60,18 +37,16 @@ Use Eqs. 1 and 2 from Stanton's paper to compute the updated T<sub>1</sub> and T
 ## Step #5: Check for Convergence and Iterate 
 Calculate the current CC correlation energy:
 
-```
-EQUATION
-E_{\rm CC} = \sum_{ia} f_{ia} t_i^a + \frac{1}{4} \sum_{ijab} \langle ij||ab\rangle t_{ij}^{ab} + \frac{1}{2} \sum_{ijab} \langle ij||ab\rangle t_i^a t_j^b
-```
+<img src="./figures/cc-correlation-energy.png" height="60">
+
 Compare energies and cluster amplitudes (using RMS differences) between iterations to check for convergence to some specified cutoff.  
 If convergence is reached, you're done; if not, return to Step #3 and continue.
 
 ## Test Cases 
 The input structures, integrals, etc. for these examples are found in the 
-[input directory](https://github.com/CrawfordGroup/ProgrammingProjects/tree/master/Project%2305/input).
+[input directory](./input).
 
-* STO-3G Water | [output](https://github.com/CrawfordGroup/ProgrammingProjects/blob/master/Project%2305/output/h2o/STO-3G/output.txt)
-* DZ Water | [output](https://github.com/CrawfordGroup/ProgrammingProjects/blob/master/Project%2305/output/h2o/DZ/output.txt)
-* DZP Water | [output](https://github.com/CrawfordGroup/ProgrammingProjects/blob/master/Project%2305/output/h2o/DZP/output.txt)
-* STO-3G Methane | [output](https://github.com/CrawfordGroup/ProgrammingProjects/blob/master/Project%2305/output/ch4/STO-3G/output.txt)
+* STO-3G Water | [output](./output/h2o/STO-3G/output.txt)
+* DZ Water | [output](./output/h2o/DZ/output.txt)
+* DZP Water | [output](./output/h2o/DZP/output.txt)
+* STO-3G Methane | [output](./output/ch4/STO-3G/output.txt)

From 168bd69eaf9cb109437e3f872205cd2ebe4ee411 Mon Sep 17 00:00:00 2001
From: Kirk Pearce <kcpearce@vt.edu>
Date: Fri, 31 Mar 2017 13:40:20 -0400
Subject: [PATCH 11/41] Add Project 5 hints and eqns

---
 Project#05/figures/cc-correlation-energy.png | Bin 0 -> 13855 bytes
 Project#05/figures/init-t-amps.png           | Bin 0 -> 9035 bytes
 Project#05/figures/mp2-energy.png            | Bin 0 -> 8868 bytes
 Project#05/figures/spin-orbital-eri.png      | Bin 0 -> 37140 bytes
 Project#05/figures/spin-orbital-fock.png     | Bin 0 -> 7769 bytes
 Project#05/hints/hint1.md                    |  19 +++++++++++++++++++
 6 files changed, 19 insertions(+)
 create mode 100644 Project#05/figures/cc-correlation-energy.png
 create mode 100644 Project#05/figures/init-t-amps.png
 create mode 100644 Project#05/figures/mp2-energy.png
 create mode 100644 Project#05/figures/spin-orbital-eri.png
 create mode 100644 Project#05/figures/spin-orbital-fock.png
 create mode 100644 Project#05/hints/hint1.md

diff --git a/Project#05/figures/cc-correlation-energy.png b/Project#05/figures/cc-correlation-energy.png
new file mode 100644
index 0000000000000000000000000000000000000000..0be10b24b98b8ae411a31fea697a0d44f25a8c36
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literal 0
HcmV?d00001

diff --git a/Project#05/hints/hint1.md b/Project#05/hints/hint1.md
new file mode 100644
index 0000000..6483590
--- /dev/null
+++ b/Project#05/hints/hint1.md
@@ -0,0 +1,19 @@
+This block of code takes the one-dimensional array of MO-basis spatial-orbital two-electron integrals we've been using thus far and translates it to a four-dimensional array of antisymmetrized integrals over spin-orbitals.  Note that alpha and beta spin functions alternate here, with even-numbered orbitals corresponding to alpha, and odd-numbered to beta.  The variable `nmo` is the number of spin orbitals, and the `INDEX` function is the same as that [used before](../../Project%2303/hints/hint7-2.md).
+
+```c++
+  /* Translate integrals to spin-orbital basis */
+  for(p=0; p < nmo; p++)
+    for(q=0; q < nmo; q++)
+      for(r=0; r < nmo; r++)
+        for(s=0; s < nmo; s++) {
+          pr = INDEX(p/2,r/2);
+          qs = INDEX(q/2,s/2);
+          prqs = INDEX(pr,qs);
+          value1 = TEI[prqs] * (p%2 == r%2) * (q%2 == s%2);
+          ps = INDEX(p/2,s/2);
+          qr = INDEX(q/2,r/2);
+          psqr = INDEX(ps,qr);
+          value2 = TEI[psqr] * (p%2 == s%2) * (q%2 == r%2);
+          ints[p][q][r][s] = value1 - value2;
+        }
+```

From 862caa9a8a70d96b8611a797205631e945aacc8c Mon Sep 17 00:00:00 2001
From: Kirk Pearce <kcpearce@vt.edu>
Date: Fri, 31 Mar 2017 13:48:56 -0400
Subject: [PATCH 12/41] Resize Project 5 eqns

---
 Project#05/README.md | 18 +++++++++---------
 1 file changed, 9 insertions(+), 9 deletions(-)

diff --git a/Project#05/README.md b/Project#05/README.md
index 3b98da9..eef782f 100644
--- a/Project#05/README.md
+++ b/Project#05/README.md
@@ -3,16 +3,16 @@ The coupled cluster model provides a higher level of accuracy beyond the MP2 app
 ## Step #1: Preparing the Spin-Orbital Basis Integrals 
 Each term in the equations given in the above paper by Stanton et al. depends on the T<sub>1</sub> or T<sub>2</sub> 
 amplitudes as well as Fock-matrix elements and antisymmetrized, Dirac-notation two-electron integrals, all given in the molecular spin-orbital basis 
-(as opposed to the spatial-orbital basis used in the earlier [MP2 Project](https://github.com/CrawfordGroup/ProgrammingProjects/tree/master/Project%2304).
+(as opposed to the spatial-orbital basis used in the earlier [MP2 Project](../Project%2304).
 Thus, the transformation of the AO-basis integrals into the spatial-MO basis must also include their translation into the spin-orbital basis:
 
-<img src="./figures/spin-orbital-eri.png" height="100">
+<img src="./figures/spin-orbital-eri.png" height="150">
 
-Thus, if you know the ordering of the orbitals (e.g. all occupied orbitals before virtual orbitals, perhaps alternating between alpha and beta spins), it is straightforward to carry out the integration over the spin components (the sigmas) in the above expression.  Thus, each spatial-orbital MO-basis two-electron integral translates to 16 possible spin-orbital integrals, only four of which are non-zero.
+Thus, if you know the ordering of the orbitals (e.g. all occupied orbitals before virtual orbitals, perhaps alternating between alpha and beta spins), it is straightforward to carry out the integration over the spin components (the &sigma;'s) in the above expression.  Thus, each spatial-orbital MO-basis two-electron integral translates to 16 possible spin-orbital integrals, only four of which are non-zero.
 
 Don't forget that you must also create the spin-orbital Fock matrix:
 
-<img src="./figures/spin-orbital-fock.png" height="50">
+<img src="./figures/spin-orbital-fock.png" height="60">
 
 Suggestion: For simplicity, store the two-electron integrals in a four-dimensional array.  This will greatly facilitate debugging of the complicated CCSD equations.
 
@@ -21,15 +21,15 @@ Suggestion: For simplicity, store the two-electron integrals in a four-dimension
 ## Step #2: Build the Initial-Guess Cluster Amplitudes 
 For Hartree-Fock reference determinants, the most common initial guess for the cluster amplitudes are the Moller-Plesset first-order perturbed wave function:
 
-<img src="./figures/init-t-amps.png" height="85">
+<img src="./figures/init-t-amps.png" height="100">
 
 Note that if you have constructed the Fock matrix, two-electron integrals, and initial-guess amplitudes correctly at this point, 
-you should be able to compute the MP2 correlation energy using the simple spin-orbital expression and get identical results to those from [Project #4](./Project%2304):
+you should be able to compute the MP2 correlation energy using the simple spin-orbital expression and get identical results to those from [Project #4](../Project%2304):
 
-<img src="./figures/mp2-energy.png" height="85">
+<img src="./figures/mp2-energy.png" height="60">
 
 ## Step #3: Calculate the CC Intermediates 
-Use the spin-orbital Eqs. 3-13 from Stanton's paper to build the two-index (F) and four-index (W) intermediates, as well as the effective doubles (labelled with the Greek letter tau).
+Use the spin-orbital Eqs. 3-13 from Stanton's paper to build the two-index (F) and four-index (W) intermediates, as well as the effective doubles (labelled with the Greek letter &tau;).
 
 ## Step #4: Compute the Updated Cluster Amplitudes 
 Use Eqs. 1 and 2 from Stanton's paper to compute the updated T<sub>1</sub> and T<sub>2</sub> cluster amplitudes.
@@ -39,7 +39,7 @@ Calculate the current CC correlation energy:
 
 <img src="./figures/cc-correlation-energy.png" height="60">
 
-Compare energies and cluster amplitudes (using RMS differences) between iterations to check for convergence to some specified cutoff.  
+Compare energies and cluster amplitudes (using RMS differences) between iterations to check for convergence to some specified cutoff.
 If convergence is reached, you're done; if not, return to Step #3 and continue.
 
 ## Test Cases 

From 8d4d7c92df287b3b9cae69f6f8f72b115dc6fff8 Mon Sep 17 00:00:00 2001
From: Kirk Pearce <kcpearce@vt.edu>
Date: Fri, 31 Mar 2017 13:50:43 -0400
Subject: [PATCH 13/41] Resize Project 5 eqns again

---
 Project#05/README.md | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/Project#05/README.md b/Project#05/README.md
index eef782f..d4ee632 100644
--- a/Project#05/README.md
+++ b/Project#05/README.md
@@ -6,7 +6,7 @@ amplitudes as well as Fock-matrix elements and antisymmetrized, Dirac-notation t
 (as opposed to the spatial-orbital basis used in the earlier [MP2 Project](../Project%2304).
 Thus, the transformation of the AO-basis integrals into the spatial-MO basis must also include their translation into the spin-orbital basis:
 
-<img src="./figures/spin-orbital-eri.png" height="150">
+<img src="./figures/spin-orbital-eri.png" height="160">
 
 Thus, if you know the ordering of the orbitals (e.g. all occupied orbitals before virtual orbitals, perhaps alternating between alpha and beta spins), it is straightforward to carry out the integration over the spin components (the &sigma;'s) in the above expression.  Thus, each spatial-orbital MO-basis two-electron integral translates to 16 possible spin-orbital integrals, only four of which are non-zero.
 
@@ -21,7 +21,7 @@ Suggestion: For simplicity, store the two-electron integrals in a four-dimension
 ## Step #2: Build the Initial-Guess Cluster Amplitudes 
 For Hartree-Fock reference determinants, the most common initial guess for the cluster amplitudes are the Moller-Plesset first-order perturbed wave function:
 
-<img src="./figures/init-t-amps.png" height="100">
+<img src="./figures/init-t-amps.png" height="120">
 
 Note that if you have constructed the Fock matrix, two-electron integrals, and initial-guess amplitudes correctly at this point, 
 you should be able to compute the MP2 correlation energy using the simple spin-orbital expression and get identical results to those from [Project #4](../Project%2304):

From b0b19475a524dc1a8e730e2b854180b8ec98fcdd Mon Sep 17 00:00:00 2001
From: Kirk Pearce <kcpearce@vt.edu>
Date: Sat, 1 Apr 2017 12:48:53 -0400
Subject: [PATCH 14/41] Add Project 6 eqns

---
 Project#06/README.md                          |  55 +++++-------------
 Project#06/figures/D.png                      | Bin 0 -> 7439 bytes
 Project#06/figures/connected-triples.png      | Bin 0 -> 16187 bytes
 Project#06/figures/disconnected-triples.png   | Bin 0 -> 10406 bytes
 Project#06/figures/t-correction.png           | Bin 0 -> 13219 bytes
 .../figures/three-index-permutation.png       | Bin 0 -> 6775 bytes
 Project#06/figures/total-energy.png           | Bin 0 -> 6188 bytes
 7 files changed, 16 insertions(+), 39 deletions(-)
 create mode 100644 Project#06/figures/D.png
 create mode 100644 Project#06/figures/connected-triples.png
 create mode 100644 Project#06/figures/disconnected-triples.png
 create mode 100644 Project#06/figures/t-correction.png
 create mode 100644 Project#06/figures/three-index-permutation.png
 create mode 100644 Project#06/figures/total-energy.png

diff --git a/Project#06/README.md b/Project#06/README.md
index 7fd07b3..3695e08 100644
--- a/Project#06/README.md
+++ b/Project#06/README.md
@@ -1,60 +1,37 @@
 # Project #6: A perturbative triples correction to CCSD: CCSD(T)
 
 The CCSD(T) method is often referred to as the "gold standard" of quantum chemistry for its high accuracy and reliability.  The purpose of this project is to illustrate the fundamental aspects of an efficient implementation of the (T) energy correction.  This project builds upon the results of 
-[Project #5](https://github.com/CrawfordGroup/ProgrammingProjects/tree/master/Project%2305).
+[Project #5](../Project%2305).
 
 The spin-orbital expression for the (T) correction, using the same notation as in 
-[Project #5](https://github.com/CrawfordGroup/ProgrammingProjects/tree/master/Project%2305), is:
-
-```
-EQUATION
-E_{\rm (T)} = \frac{1}{36} \sum_{ijkabc} t_{ijk}^{abc}(c) D_{ijk}^{abc} \left[ t_{ijk}^{abc}(c) + t_{ijk}^{abc}(d) \right],
-```
+[Project #5](../Project%2305), is:
 
+<img src="./figures/t-correction.png" height="70">
 
 where 
 
-```
-EQUATION
-D_{ijk}^{abc} = f_{ii} + f_{jj} + f_{kk} - f_{aa} - f_{bb} - f_{cc},
-```
-
+<img src="./figures/D.png" height="60">
 
 the "disconnected" triples are defined as
 
-```
-EQUATION
-D_{ijk}^{abc} t_{ijk}^{abc}(d) = P(i/jk) P(a/bc) t_i^a \langle jk || bc \rangle,
-```
-
+<img src="./figures/disconnected-triples.png" height="60">
 
 and the "connected" triples as
 
-```
-EQUATION
-D_{ijk}^{abc} t_{ijk}^{abc}(c) = P(i/jk) P(a/bc) \left[ \sum_e t_{jk}^{ae} \langle ei||bc \rangle - \sum_m t_{im}^{bc} \langle ma||jk \rangle \right].
-```
-
+<img src="./figures/connected-triples.png" height="75">
 
 The three-index permutation operator is defined by its action on an algebraic function as
 
-```
-EQUATION
-P(p/qr) f(pqr) = f(pqr) - f(qpr) - f(rqp).
-```
-
+<img src="./figures/three-index-permutation.png" height="60">
 
 The total energy is
 
-```
-EQUATION
-E_{\rm total} = E_{\rm SCF} + E_{\rm CCSD} + E_{\rm (T)}.
-```
-
+<img src="./figures/total-energy.png" height="60">
 
 ## Algorithm #1: Full Storage of Triples
 
 The most straightforward approach to evaluation of the (T) energy correction is to compute and store explicitly the connected and disconnected triples and plug them into the energy expression above. Store the triples amplitudes as six-dimensional arrays over occupied and virtual spin orbitals. 
+
 ## Algorithm #2: Avoided Storage of Triples
 
 Note that each T<sub>3</sub> amplitude depends on all the T<sub>1</sub> and T<sub>2</sub> amplitudes, **not** on other triples.  This suggests that the calculation of the T<sub>3</sub> amplitudes appearing in the energy expression could proceed one amplitude at a time:
@@ -95,13 +72,13 @@ This algorithm reduces the storage (memory and disk) requirements of the (T) cor
   End i loop
 ```
 
-This approach requires more storage (on the order v<sup>3</sup>) than the one-at-a-time approach, but can be considerably faster of modern high-performance computers.  A good discussion of the most efficient algorithm can be found in A. P. Rendell, T. J. Lee, A. Komornicki, //Chem. Phys. Lett.// **178**, 462-470 (1991).
+This approach requires more storage [O(v<sup>3</sup>)] than the one-at-a-time approach, but can be considerably faster for modern high-performance computers.  A good discussion of the most efficient algorithm can be found in A. P. Rendell, T. J. Lee, A. Komornicki, //Chem. Phys. Lett.// **178**, 462-470 (1991).
 
 ## Test Cases
-The input structures, integrals, etc. for these examples may be found in the [input](https://github.com/CrawfordGroup/ProgrammingProjects/tree/master/Project%2306/input) 
-directory, the CCSD Energies can be found in [Project #5](https://github.com/CrawfordGroup/ProgrammingProjects/tree/master/Project%2305).
+The input structures, integrals, etc. for these examples may be found in the [input](./input) 
+directory, the CCSD Energies can be found in [Project #5](../Project%2305).
 
-* STO-3G Water | [Output](https://github.com/CrawfordGroup/ProgrammingProjects/blob/master/Project%2306/output/h2o/STO-3G/output.txt) 
-* DZ Water | [Output](https://github.com/CrawfordGroup/ProgrammingProjects/blob/master/Project%2306/output/h2o/DZ/output.txt) 
-* DZP Water | [Output](https://github.com/CrawfordGroup/ProgrammingProjects/blob/master/Project%2306/output/h2o/DZP/output.txt) 
-* STO-3G Methane | [Output](https://github.com/CrawfordGroup/ProgrammingProjects/blob/master/Project%2306/output/ch4/STO-3G/output.txt) 
+* STO-3G Water | [Output](./output/h2o/STO-3G/output.txt) 
+* DZ Water | [Output](./output/h2o/DZ/output.txt) 
+* DZP Water | [Output](./output/h2o/DZP/output.txt) 
+* STO-3G Methane | [Output](./output/ch4/STO-3G/output.txt) 
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From 2b525e07cc2a1e9f54d204bfb6f7bb2319d7aacb Mon Sep 17 00:00:00 2001
From: Kirk Pearce <kcpearce@vt.edu>
Date: Sat, 1 Apr 2017 12:50:20 -0400
Subject: [PATCH 15/41] Resize Project 6 eqns

---
 Project#06/README.md | 12 ++++++------
 1 file changed, 6 insertions(+), 6 deletions(-)

diff --git a/Project#06/README.md b/Project#06/README.md
index 3695e08..68781b5 100644
--- a/Project#06/README.md
+++ b/Project#06/README.md
@@ -6,27 +6,27 @@ The CCSD(T) method is often referred to as the "gold standard" of quantum chemis
 The spin-orbital expression for the (T) correction, using the same notation as in 
 [Project #5](../Project%2305), is:
 
-<img src="./figures/t-correction.png" height="70">
+<img src="./figures/t-correction.png" height="60">
 
 where 
 
-<img src="./figures/D.png" height="60">
+<img src="./figures/D.png" height="45">
 
 the "disconnected" triples are defined as
 
-<img src="./figures/disconnected-triples.png" height="60">
+<img src="./figures/disconnected-triples.png" height="45">
 
 and the "connected" triples as
 
-<img src="./figures/connected-triples.png" height="75">
+<img src="./figures/connected-triples.png" height="70">
 
 The three-index permutation operator is defined by its action on an algebraic function as
 
-<img src="./figures/three-index-permutation.png" height="60">
+<img src="./figures/three-index-permutation.png" height="40">
 
 The total energy is
 
-<img src="./figures/total-energy.png" height="60">
+<img src="./figures/total-energy.png" height="40">
 
 ## Algorithm #1: Full Storage of Triples
 

From c5e8f5b0fad14bcc079d51eb26f15f3a3cfa8687 Mon Sep 17 00:00:00 2001
From: Kirk Pearce <kcpearce@vt.edu>
Date: Sat, 1 Apr 2017 12:51:24 -0400
Subject: [PATCH 16/41] Resize Project 6 eqns again

---
 Project#06/README.md | 8 ++++----
 1 file changed, 4 insertions(+), 4 deletions(-)

diff --git a/Project#06/README.md b/Project#06/README.md
index 68781b5..8a17915 100644
--- a/Project#06/README.md
+++ b/Project#06/README.md
@@ -10,11 +10,11 @@ The spin-orbital expression for the (T) correction, using the same notation as i
 
 where 
 
-<img src="./figures/D.png" height="45">
+<img src="./figures/D.png" height="40">
 
 the "disconnected" triples are defined as
 
-<img src="./figures/disconnected-triples.png" height="45">
+<img src="./figures/disconnected-triples.png" height="40">
 
 and the "connected" triples as
 
@@ -22,11 +22,11 @@ and the "connected" triples as
 
 The three-index permutation operator is defined by its action on an algebraic function as
 
-<img src="./figures/three-index-permutation.png" height="40">
+<img src="./figures/three-index-permutation.png" height="25">
 
 The total energy is
 
-<img src="./figures/total-energy.png" height="40">
+<img src="./figures/total-energy.png" height="25">
 
 ## Algorithm #1: Full Storage of Triples
 

From 5c4221f0a81d1590c6c78d72868b44b0292cdc54 Mon Sep 17 00:00:00 2001
From: Kirk Pearce <kcpearce@vt.edu>
Date: Sat, 1 Apr 2017 12:52:20 -0400
Subject: [PATCH 17/41] Resize Project 6 eqn again again

---
 Project#06/README.md | 6 +++---
 1 file changed, 3 insertions(+), 3 deletions(-)

diff --git a/Project#06/README.md b/Project#06/README.md
index 8a17915..0e514f7 100644
--- a/Project#06/README.md
+++ b/Project#06/README.md
@@ -10,11 +10,11 @@ The spin-orbital expression for the (T) correction, using the same notation as i
 
 where 
 
-<img src="./figures/D.png" height="40">
+<img src="./figures/D.png" height="30">
 
 the "disconnected" triples are defined as
 
-<img src="./figures/disconnected-triples.png" height="40">
+<img src="./figures/disconnected-triples.png" height="30">
 
 and the "connected" triples as
 
@@ -22,7 +22,7 @@ and the "connected" triples as
 
 The three-index permutation operator is defined by its action on an algebraic function as
 
-<img src="./figures/three-index-permutation.png" height="25">
+<img src="./figures/three-index-permutation.png" height="20">
 
 The total energy is
 

From c2b1106e84b75775af1bb0834d7be5050fe4d6fd Mon Sep 17 00:00:00 2001
From: Kirk Pearce <kcpearce@vt.edu>
Date: Sat, 1 Apr 2017 13:18:17 -0400
Subject: [PATCH 18/41] Resize Project 6 eqn again again again :

---
 Project#06/README.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/Project#06/README.md b/Project#06/README.md
index 0e514f7..3f2d0f9 100644
--- a/Project#06/README.md
+++ b/Project#06/README.md
@@ -18,7 +18,7 @@ the "disconnected" triples are defined as
 
 and the "connected" triples as
 
-<img src="./figures/connected-triples.png" height="70">
+<img src="./figures/connected-triples.png" height="60">
 
 The three-index permutation operator is defined by its action on an algebraic function as
 

From ba3bd83bc1063426fa215fa07067e8d76f4fbabe Mon Sep 17 00:00:00 2001
From: Kirk Pearce <kcpearce@vt.edu>
Date: Sat, 1 Apr 2017 13:18:44 -0400
Subject: [PATCH 19/41] Resize Project 6 eqn again again again :

---
 Project#06/README.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/Project#06/README.md b/Project#06/README.md
index 3f2d0f9..0385e9e 100644
--- a/Project#06/README.md
+++ b/Project#06/README.md
@@ -18,7 +18,7 @@ the "disconnected" triples are defined as
 
 and the "connected" triples as
 
-<img src="./figures/connected-triples.png" height="60">
+<img src="./figures/connected-triples.png" height="65">
 
 The three-index permutation operator is defined by its action on an algebraic function as
 

From 5e263812301d44ba78a61c1e8f1438b954dd3f95 Mon Sep 17 00:00:00 2001
From: Kirk Pearce <kcpearce@vt.edu>
Date: Sat, 1 Apr 2017 13:19:22 -0400
Subject: [PATCH 20/41] Resize Project 6 eqn again again again :

---
 Project#06/README.md | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/Project#06/README.md b/Project#06/README.md
index 0385e9e..effecf3 100644
--- a/Project#06/README.md
+++ b/Project#06/README.md
@@ -18,11 +18,11 @@ the "disconnected" triples are defined as
 
 and the "connected" triples as
 
-<img src="./figures/connected-triples.png" height="65">
+<img src="./figures/connected-triples.png" height="70">
 
 The three-index permutation operator is defined by its action on an algebraic function as
 
-<img src="./figures/three-index-permutation.png" height="20">
+<img src="./figures/three-index-permutation.png" height="25">
 
 The total energy is
 

From 7e95757eb0671f2f79904ea2ca65db27e4c602d0 Mon Sep 17 00:00:00 2001
From: Kirk Pearce <kcpearce@vt.edu>
Date: Sat, 1 Apr 2017 13:49:50 -0400
Subject: [PATCH 21/41] Add Project 8 eqns

---
 Project#08/README.md                     |  84 +++++------------------
 Project#08/figures/Bij.png               | Bin 0 -> 3743 bytes
 Project#08/figures/constraint.png        | Bin 0 -> 3690 bytes
 Project#08/figures/error-matrix.png      | Bin 0 -> 5099 bytes
 Project#08/figures/iter-error-matrix.png | Bin 0 -> 4627 bytes
 Project#08/figures/new-approx-fock.png   | Bin 0 -> 4343 bytes
 Project#08/figures/sys-lin-eqn-ci.png    | Bin 0 -> 22046 bytes
 7 files changed, 19 insertions(+), 65 deletions(-)
 create mode 100644 Project#08/figures/Bij.png
 create mode 100644 Project#08/figures/constraint.png
 create mode 100644 Project#08/figures/error-matrix.png
 create mode 100644 Project#08/figures/iter-error-matrix.png
 create mode 100644 Project#08/figures/new-approx-fock.png
 create mode 100644 Project#08/figures/sys-lin-eqn-ci.png

diff --git a/Project#08/README.md b/Project#08/README.md
index a26270b..b31534c 100644
--- a/Project#08/README.md
+++ b/Project#08/README.md
@@ -1,6 +1,6 @@
 # Project #8: DIIS extrapolation for the SCF procedure
 
-The convergence of the self-consistent field (SCF) algorithm presented in Project #3 is rather slow.  
+The convergence of the self-consistent field (SCF) algorithm presented in [Project #3](../Project%2303) is rather slow.
 Even the STO-3G H<sub>2</sub>O example requires more than 20 iterations to converge the RMS density difference to a relatively loose threshold of 10<sup>-8</sup>.
 One of the most effective methods for speeding up such iterative sequences is the 
 "Direct Inversion in the Iterative Subspace (DIIS)" approach, which was 
@@ -11,110 +11,64 @@ a new guess may be constructed as a simple linear combination of the previous fe
 where the coefficients of such an extrapolation are determined from Lagrangian minimization procedure.
 
 The purpose of this programming project is to implement the Pulay DIIS procedure within the SCF algorithm from 
-[Project #3](https://github.com/CrawfordGroup/ProgrammingProjects/tree/master/Project%2303).
+[Project #3](../Project%2303).
 
 ## Working Equations
 
 Given a series of Fock matrices, <b><i>F<sub>i</sub></i></b> (expressed in the AO basis set), a new approximation to the converged solution may be written as
 
-```
-EQUATION
-{\mathbf F}' = \sum_i c_i {\mathbf F}_i
-```
-
+<img src="./figures/new-approx-fock.png" height="50">
 
 by requiring that the corresponding extrapolated "error" matrix
 
-```
-EQUATION
-{\mathbf e}' = \sum_i c_i {\mathbf e}_i \approx {\mathbf 0}
-```
-
+<img src="./figures/error-matrix.png" height="50">
 
 is approximately the zero matrix in a least-squares sense.  The error matrix for each iteration is given by
 
-```
-EQUATION
-{\mathbf e_i} \equiv {\mathbf F}_i {\mathbf D}_i {\mathbf S} - {\mathbf S} {\mathbf D}_i {\mathbf F}_i,
-```
-
+<img src="./figures/iter-error-matrix.png" height="40">
 
 where <b><i>D<sub>i</sub></i></b> is the AO-basis density matrix used to construct 
-<b><i>F<sub>i</sub></i></b>, and  ** *S* ** is the AO-basis overlap matrix.  Minimization of ** *e'* ** under the constraint that 
-
-```
-EQUATION
-\sum_i c_i = 1
-```
+<b><i>F<sub>i</sub></i></b>, and  <b><i>S</i></b> is the AO-basis overlap matrix.  Minimization of <b><i>e'</i></b> under the constraint that 
 
+<img src="./figures/constraint.png" height="50">
 
 leads to the following system of linear equations for the c<sub>i</sub>:
 
-```
-EQUATION
-\left(
-\begin{array}{ccccc}
-B_{11} & B_{12} & \ldots & B_{1m} & -1 \\
-B_{21} & B_{22} & \ldots & B_{2m} & -1 \\
-\ldots & \ldots & \ldots & \ldots & -1 \\
-B_{m1} & B_{m2} & \ldots & B_{mm} & -1 \\
--1 & -1 & \ldots & -1 & 0 \\
-\end{array}
-\right) \left(
-\begin{array}{c}
-c_1 \\
-c_2 \\
-\ldots \\
-c_m \\
-\lambda
-\end{array}
-\right) = \left(
-\begin{array}{c}
-0 \\
-0 \\
-\ldots \\
-0 \\
--1 \\
-\end{array}
-\right),
-```
-
-where lambda is a Lagrangian multiplier and the elements <b>B<sub>ij</sub></b> are computed as dot products of error matrices:
+<img src="./figures/sys-lin-eqn-ci.png" height="150">
 
-```
-EQUATION
-B_{ij} \equiv {\mathbf e}_i \cdot {\mathbf e}_j.
-```
+where lambda is a Lagrangian multiplier and the elements <i>B<sub>ij</sub></i> are computed as dot products of error matrices:
 
+<img src="./figures/Bij.png" height="40">
 
 ## Step #1: Compute the Error Matrix in Each Iteration
 
-Compute the error matrix for the current iteration using the matrix equation  for **e**<sub>i</sub> given above.  
-Note that the density matrix you use must be the same one used to construct the corresponding Fock matrix.  
+Compute the error matrix for the current iteration using the matrix equation  for <b><i>e<sub>i</sub></i></b> given above.
+Note that the density matrix you use must be the same one used to construct the corresponding Fock matrix.
 If you print the error matrix in each iteration, you should be able to note that it gradually falls to zero as the SCF procedure converges.
 
 ## Step #2: Build the B Matrix and Solve the Linear Equations
 
-After you have computed at least two error matrices, it is possible to begin the DIIS extrapolation procedure.  
+After you have computed at least two error matrices, it is possible to begin the DIIS extrapolation procedure.
 Compute the B matrix above for all of the current error matrices. 
 (Note that the matrix is symmetric, so one need only compute the lower triangle to get the entire matrix).
 
 Next, use a standard linear-equation solver (such as the DGESV function in the LAPACK library, 
 to which PSI4 has an interface named C_DGESV) to compute the <b>c<sub>i</sub></b> coefficients. 
 (NB: If you use the LAPACK solver, or any LAPACK or BLAS function for that matter, you must use matrices allocated as contiguous memory, 
-as discussed in [Project #7](https://github.com/CrawfordGroup/ProgrammingProjects/tree/master/Project%2307#usefulPsiFeatures)).
+as discussed in [Project #7](../Project%2307)).
 
 ## Step #3: Compute the New Fock Matrix
 
-Build a new Fock matrix using the equation for ** *F'* ** above and continue with the SCF procedure as usual.  
-This implies that one must store all of the Fock matrices corresponding to each of the error matrices used in constructing the B matrix above.  Note:
+Build a new Fock matrix using the equation for <b><i>F'</i></b> above and continue with the SCF procedure as usual.
+This implies that one must store all of the Fock matrices corresponding to each of the error matrices used in constructing the B matrix above.
+Note:
 
   * Do **not** use the extrapolated Fock matrix to compute error matrices for subsequent DIIS steps.
   * Keep only a relatively small number of error matrices; six to eight are reasonable for closed-shell SCF calculations. When the chosen number of error
   matrices is exceeded delete the oldest error matrix.
 
-Once the procedure is complete and working correctly, you should notice a considerable reduction in the number of SCF cycles required to achieve convergence.  
-For example, for the STO-3G/H2O test case from [Project #3](https://github.com/CrawfordGroup/ProgrammingProjects/tree/master/Project%2303), 
+Once the procedure is complete and working correctly, you should notice a considerable reduction in the number of SCF cycles required to achieve convergence.
+For example, for the STO-3G/H<sub>2</sub>O test case from [Project #3](../Project%2303), 
 the simple SCF algorithm converges to 10<sup>-12</sup> in the density in 39 iterations:
 ```
  Iter        E(elec)              E(tot)               Delta(E)             RMS(D)
diff --git a/Project#08/figures/Bij.png b/Project#08/figures/Bij.png
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From d0d736549ef2acee987c73f637944986961449b9 Mon Sep 17 00:00:00 2001
From: Kirk Pearce <kcpearce@vt.edu>
Date: Sat, 1 Apr 2017 13:51:04 -0400
Subject: [PATCH 22/41] Add Project 8 eqns again

---
 Project#08/README.md | 12 ++++++------
 1 file changed, 6 insertions(+), 6 deletions(-)

diff --git a/Project#08/README.md b/Project#08/README.md
index b31534c..f2ee4ee 100644
--- a/Project#08/README.md
+++ b/Project#08/README.md
@@ -17,28 +17,28 @@ The purpose of this programming project is to implement the Pulay DIIS procedure
 
 Given a series of Fock matrices, <b><i>F<sub>i</sub></i></b> (expressed in the AO basis set), a new approximation to the converged solution may be written as
 
-<img src="./figures/new-approx-fock.png" height="50">
+<img src="./figures/new-approx-fock.png" height="40">
 
 by requiring that the corresponding extrapolated "error" matrix
 
-<img src="./figures/error-matrix.png" height="50">
+<img src="./figures/error-matrix.png" height="40">
 
 is approximately the zero matrix in a least-squares sense.  The error matrix for each iteration is given by
 
-<img src="./figures/iter-error-matrix.png" height="40">
+<img src="./figures/iter-error-matrix.png" height="20">
 
 where <b><i>D<sub>i</sub></i></b> is the AO-basis density matrix used to construct 
 <b><i>F<sub>i</sub></i></b>, and  <b><i>S</i></b> is the AO-basis overlap matrix.  Minimization of <b><i>e'</i></b> under the constraint that 
 
-<img src="./figures/constraint.png" height="50">
+<img src="./figures/constraint.png" height="40">
 
 leads to the following system of linear equations for the c<sub>i</sub>:
 
-<img src="./figures/sys-lin-eqn-ci.png" height="150">
+<img src="./figures/sys-lin-eqn-ci.png" height="125">
 
 where lambda is a Lagrangian multiplier and the elements <i>B<sub>ij</sub></i> are computed as dot products of error matrices:
 
-<img src="./figures/Bij.png" height="40">
+<img src="./figures/Bij.png" height="20">
 
 ## Step #1: Compute the Error Matrix in Each Iteration
 

From 1e0cfba35c0d5956bf290deee49825460234744e Mon Sep 17 00:00:00 2001
From: Kirk Pearce <kcpearce@vt.edu>
Date: Sat, 1 Apr 2017 13:51:49 -0400
Subject: [PATCH 23/41] Add Project 8 eqns again

---
 Project#08/README.md | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/Project#08/README.md b/Project#08/README.md
index f2ee4ee..f32de8c 100644
--- a/Project#08/README.md
+++ b/Project#08/README.md
@@ -30,7 +30,7 @@ is approximately the zero matrix in a least-squares sense.  The error matrix for
 where <b><i>D<sub>i</sub></i></b> is the AO-basis density matrix used to construct 
 <b><i>F<sub>i</sub></i></b>, and  <b><i>S</i></b> is the AO-basis overlap matrix.  Minimization of <b><i>e'</i></b> under the constraint that 
 
-<img src="./figures/constraint.png" height="40">
+<img src="./figures/constraint.png" height="45">
 
 leads to the following system of linear equations for the c<sub>i</sub>:
 
@@ -38,7 +38,7 @@ leads to the following system of linear equations for the c<sub>i</sub>:
 
 where lambda is a Lagrangian multiplier and the elements <i>B<sub>ij</sub></i> are computed as dot products of error matrices:
 
-<img src="./figures/Bij.png" height="20">
+<img src="./figures/Bij.png" height="25">
 
 ## Step #1: Compute the Error Matrix in Each Iteration
 

From 642c6833cc2ab4d7a04b3d416a8524c1ec7f80d2 Mon Sep 17 00:00:00 2001
From: Kirk Pearce <kcpearce@vt.edu>
Date: Sat, 1 Apr 2017 14:11:12 -0400
Subject: [PATCH 24/41] Add Project 9 hints and eqns

---
 Project#09/README.md | 71 +++++++++++++++++++-------------------------
 Project#09/hint1.md  | 28 +++++++++++++++++
 2 files changed, 58 insertions(+), 41 deletions(-)
 create mode 100644 Project#09/hint1.md

diff --git a/Project#09/README.md b/Project#09/README.md
index f3a37a7..c8fe857 100644
--- a/Project#09/README.md
+++ b/Project#09/README.md
@@ -3,11 +3,11 @@
 The use of point-group symmetry in quantum chemical calculations can lead to exceptionally efficient programs, 
 both in terms of their memory/disk usage and their computing times.  If a molecule has even one element of symmetry, 
 for example, this can reduce the number of wave function parameters that must be stored by up to a factor of two (the order of the point group) 
-and the computing effort by up to a factor of four (the square of the order of the point group).   
+and the computing effort by up to a factor of four (the square of the order of the point group).
 Most quantum chemical programs can take advantage of Abelian symmetry -- more accurately, 
 D<sub>2h</sub> and its subgroups -- which means that higher symmetry structures can yield up to a factor of 64 reduction in the computational costs.
 
-The purpose of this project is to take advantage of basic point-group symmetry in an SCF calculation.  
+The purpose of this project is to take advantage of basic point-group symmetry in an SCF calculation.
 While the most efficient programs will store only the symmetry-allowed sets of two-electron integrals, 
 for the purposes of this project we'll deal only with the one-electron (two-index) components.
 
@@ -15,22 +15,19 @@ for the purposes of this project we'll deal only with the one-electron (two-inde
 
 The vanishing integral rule of point-group theory may be stated as follows for Abelian groups:
 
-```
-EQUATION
-\langle \phi_i | \hat{A} |\phi_j \rangle = 0 \ \ \ \ {\rm if} \ \ \ \ \Gamma_A \neq \Gamma_{\phi_i} \otimes \Gamma_{\phi_j},
-```
+<img src="./figures/point-group-rule.png" height="20">
 
-where <html>&Gamma;<sub>X</sub></html> denotes the irreducible representation (irrep) of entity X.  
+where <html>&Gamma;<sub>X</sub></html> denotes the irreducible representation (irrep) of entity X.
 That is, unless the direct product of the irreps of the functions or operators in the integrand contains the totally symmetric irrep, the integral must be zero.
 
 The vanishing integral rule of group theory may be readily applied to the various integrals and wave-function parameters in the SCF procedure 
-** *if* ** the AO-basis functions over which the they are evaluated have been properly symmetrized 
+<b><i>if</i></b> the AO-basis functions over which the they are evaluated have been properly symmetrized 
 [i.e. used to form symmetry-adapted linear combinations, normally referred to as symmetry orbitals (SOs)].
 
-For example, consider the H<sub>2</sub>O test case with the STO-3G basis set as given in Project #3.  The AO 
+For example, consider the H<sub>2</sub>O test case with the STO-3G basis set as given in [Project #3](../Project%2303).  The AO 
 ([contracted Gaussian](http://sirius.chem.vt.edu/wiki/lib/exe/fetch.php?media=basis_sets.pdf)) basis functions 
 on the oxygen atom consist of 1s, 2s, 2p<sub>x</sub>, 2p<sub>y</sub>, and 2p<sub>z</sub> orbitals, 
-which those for each hydrogen atom consist of only the 1s, giving a total of seven AOs.  
+which those for each hydrogen atom consist of only the 1s, giving a total of seven AOs.
 If we take oxygen to be the first atom and ignore the C<sub>2v</sub> point group symmetry of the molecule, 
 the resulting overlap integral matrix looks like:
 
@@ -48,20 +45,20 @@ the resulting overlap integral matrix looks like:
 The first five row/column indices correspond to the basis functions on the oxygen (in the order listed above),
 while rows/columns six and seven correspond to the basis functions on each hydrogen. 
 Thus, the 1-2 element is the overlap integral between the 1s and 2s orbitals on the oxygen, 
-while element 3-7 is the overlap integral between the oxygen 2p<sub>x</sub> orbital and the second hydrogen 1s orbital.  
+while element 3-7 is the overlap integral between the oxygen 2p<sub>x</sub> orbital and the second hydrogen 1s orbital.
 
 Points to note about this matrix:
-  - The 1s and 2s orbitals on oxygen are not orthogonal to one another, but both are orthogonal to all three p-type orbitals.  
+  - The 1s and 2s orbitals on oxygen are not orthogonal to one another, but both are orthogonal to all three p-type orbitals.
   This occurs because, in Cartesian Gaussian basis sets, 
   orthogonality is generally obtained only between functions of different angular momenta on the same atomic center.
   - The 1s orbitals on the hydrogens have the same overlap integral (to within a sign) 
-  with all the orbitals on the oxygen because they are ** *symmetry equivalent* ** .  
+  with all the orbitals on the oxygen because they are <b><i>symmetry equivalent</i></b>.
   However, the 1s orbitals alone do not transform as irreps of the C<sub>2v</sub> point group.
-  - The hydrogen 1s orbitals are orthogonal to the 2p<sub>z</sub> orbital on the oxygen (cf. elements 5-6 and 5-7).  
+  - The hydrogen 1s orbitals are orthogonal to the 2p<sub>z</sub> orbital on the oxygen (cf. elements 5-6 and 5-7).
   This is because the water molecule was arbitrarily oriented in the xy-plane to compute the above integrals; 
   thus the 2p<sub>z</sub> orbital is perpendicular to the molecular plane containing the hydrogen 1s orbitals.
 
-What happens if we turn on symmetry, i.e. take linear combinations of the AO to obtain SOs?  
+What happens if we turn on symmetry, i.e. take linear combinations of the AO to obtain SOs?
 If we orient the molecule oriented in the yz-plane, so that the z-axis corresponds to the C<sub>2</sub> rotation axis, 
 we find that the AOs transform as the following irreps:
 
@@ -75,7 +72,7 @@ we find that the AOs transform as the following irreps:
 |  H<sub>1</sub> 1s + H<sub>2</sub> 1s  |  A<sub>1</sub>  |
 |  H<sub>1</sub> 1s - H<sub>2</sub> 1s  |  B<sub>2</sub>  |
 
-Thus, after symmetrization, there are four A<sub>1</sub> SOs, one B<sub>1</sub> SO, and two B<sub>2</sub> SOs.  
+Thus, after symmetrization, there are four A<sub>1</sub> SOs, one B<sub>1</sub> SO, and two B<sub>2</sub> SOs.
 (The STO-3G basis set contains no A<sub>2</sub> orbitals.  One must include at least d-type functions on the oxygen and/or 
 p-type functions on the hydrogens to obtain A<sub>2</sub> orbitals.)  Note that a normalization factor of 1/sqrt(2) should be included for the H 1s SOs above.
 
@@ -107,13 +104,13 @@ The PSI checkpoint file contains a great deal of useful symmetry information. To
 
 ## Storing One-Electron Integrals with Symmetry
 
-The block-diagonal structure above implies that one need only store and use the sub-matrices of the integrals rather than the full matrix.  
+The block-diagonal structure above implies that one need only store and use the sub-matrices of the integrals rather than the full matrix.
 One way of doing this is to store the one-electron integrals as an array of smaller matrices, 
 the size of which is dictated by the number of SOs per irrep.  For example, here's a code snippet that will extract the one-electron integrals 
-from the appropriate file (as was done in [Project #7](https://github.com/CrawfordGroup/ProgrammingProjects/tree/master/Project%2307) ), 
+from the appropriate file (as was done in [Project #7](../Project%2307) ), 
 but stores them in an array of matrices:
 
-```cpp
+```c++
   int ntri, nirreps;
   double *scratch;
   double ***S;
@@ -138,9 +135,9 @@ but stores them in an array of matrices:
 ```
 
 Notice how S is now a `double ***` -- i.e. an array of matrices.  This code uses the same `INDEX()` function found in 
-[Project #3](https://github.com/CrawfordGroup/ProgrammingProjects/tree/master/Project%2303), 
-and the sopi array is the integer array of SOs per irrep obtained using the `chkpt_rd_sopi()` function.  
-The `symm_offset` array is useful for converting between ** *relative* ** orbital indices and ** *absolute* ** orbital indices.  
+[Project #3](https://github.com/CrawfordGroup/ProgrammingProjects/tree/master/Project%2303),
+and the sopi array is the integer array of SOs per irrep obtained using the `chkpt_rd_sopi()` function.
+The `symm_offset` array is useful for converting between <b><i>relative</i></b> orbital indices and <b><i>absolute</i></b> orbital indices.
 (For example, orbital 0 in the B<sub>2</sub> irrep block corresponds to orbital 5 
 in the full list of orbitals; the first number is the relative index, the second is the absolute index.)
 
@@ -149,28 +146,23 @@ in the full list of orbitals; the first number is the relative index, the second
 If we store the integral, MO coefficient, density, and Fock matrices in the block-diagonal structure described above, 
 we may compute matrix products of these quantities separately for each diagonal block.  For example, the matrix product may be written as:
 
-```
-EQUATION
-{\mathbf C} = {\mathbf A} {\mathbf B} = \bigoplus_h {\mathbf C}_h = \bigoplus_h {\mathbf A}_h {\mathbf B}_h,
-```
+<img src="./figures/matrix-product.png" height="25">
 
-where *h* denotes a particular irrep of the point group and <b><i>C</i></b><sub>h</sub> denotes the *h*-th irrep subblock of the full matrix ** *C* ** .  
+where *h* denotes a particular irrep of the point group and <b><i>C</i></b><sub>h</sub> denotes the *h*-th irrep subblock of the full matrix <b><i>C</i></b>.
 (The notation of the large plus with a circle around it indicates a direct sum.)
 
 ## Occupied Orbitals and the Density Matrix
 
 The calculation of the density matrix requires information about the number of occupied orbitals:
 
-```
-EQUATION
-D_{\mu\nu} = \sum_m^{\rm occ.} \left( {\mathbf C} \right)_\mu^m \left( {\mathbf C} \right)_\nu^m
-```
+<img src="./figures/density-matrix.png" height="25">
+
 In a calculation without symmetry, one need only know the number of electrons (equivalently, the atomic numbers of the atoms and the overall molecular charge) 
 to evaluate this expression.  However, if the density and MO coefficient matrices are stored in symmetry-blocked form, 
-as above, one needs to know the number of occupied MOs in each irrep.  
+as above, one needs to know the number of occupied MOs in each irrep.
 
 For the case of the C<sub>2v</sub> water molecule, which has two <html>&sigma;</html> O-H bonds, two oxygen lone pairs, 
-and one oxygen core orbital, the orbital occupations are not difficult to determine, given the above information about the number of SOs in each irrep.  
+and one oxygen core orbital, the orbital occupations are not difficult to determine, given the above information about the number of SOs in each irrep.
 Since the MOs are constructed as linear combinations of the SOs, and only SOs of the same irrep can combine, we may rationalize the occupied orbitals of 
 H<sub>2</sub>O as follows:
 
@@ -184,9 +176,9 @@ H<sub>2</sub>O as follows:
 
 Thus, for the water molecule, there are 3 A<sub>1</sub>, 1 B<sub>1</sub>, and 1 B<sub>2</sub> occupied orbitals.
 
-Determining the occupations automatically is sometimes a difficult task, especially for molecules with unusual bonding patterns.  
+Determining the occupations automatically is sometimes a difficult task, especially for molecules with unusual bonding patterns.
 Many SCF programs will simply take the core Hamiltonian matrix (often used as the initial guess for the Fock matrix), 
-diagonalize it (in the othogonal AO basis) in each irrep block, and identify the lowest eigenvalues in each irrep as occupied.  
+diagonalize it (in the othogonal AO basis) in each irrep block, and identify the lowest eigenvalues in each irrep as occupied.
 However, this approach often fails for many molecules (especially those with open shells) because of near-degeneracies 
 that may not be adequately described by the simple core Hamiltonian.  Thus, some programs will make use of more 
 sophisticated guesses at the Fock matrix, e.g. Hueckel orbitals or other (semi)empirical forms.
@@ -195,12 +187,9 @@ sophisticated guesses at the Fock matrix, e.g. Hueckel orbitals or other (semi)e
 
 The only portion of the SCF procedure involving two-electron integrals is the Fock-matrix build:
 
-```
-EQUATION
-F_{\mu\nu} = H^{\rm core}_{\mu\nu} + \sum_{\lambda\sigma}^{\rm AO} D^{i-1}_{\lambda\sigma} \left[ 2 (\mu\nu | \lambda\sigma) - (\mu\lambda|\nu\sigma) \right],
-```
+<img src="./figures/fock-build.png" height="30">
 
-where the notation used in [Project #3](https://github.com/CrawfordGroup/ProgrammingProjects/tree/master/Project%2303) is retained above.  
+where the notation used in [Project #3](../Project%2303) is retained above.
 If the Fock matrix and density matrix are stored in a symmetry-blocked manner similar to the overlap matrix above, 
 then one may limit the corresponding loops only to the symmetry-allowed combinations, e.g. the irreps of <html>&mu;</html> and 
 <html>&nu;</html> must match for F<html><sub>&mu;&nu;</sub></html> to be non-zero and the irreps of <html>&lambda;</html> 
@@ -208,4 +197,4 @@ and <html>&sigma;</html> must match for D<html><sub>&lambda;&sigma;</sub></html>
 Note, however, that, because of the symmetries in the two-electron integrals, 
 the irrep of <html>&lambda;</html> and <html>&sigma;</html> do not necessarily have to match that of <html>&mu;</html> and <html>&nu;</html>.
 
-  * Hint: Fock-Build Code
+  * [Hint](./hints/hint1.md): Fock-Build Code
diff --git a/Project#09/hint1.md b/Project#09/hint1.md
new file mode 100644
index 0000000..670bf7d
--- /dev/null
+++ b/Project#09/hint1.md
@@ -0,0 +1,28 @@
+```c++
+    for(h=0; h < nirreps; h++) {
+      for(i=0; i < sopi[h]; i++) {
+        I = symm_offset[h] + i;
+        for(j=0; j < sopi[h]; j++) {
+          J = symm_offset[h] + j;
+          F[h][i][j] = H[h][i][j];
+
+          for(g=0; g < nirreps; g++) {
+            for(k=0; k < sopi[g]; k++) {
+              K = symm_offset[g] + k;
+              for(l=0; l < sopi[g]; l++) {
+                L = symm_offset[g] + l;
+
+                ij = INDEX(I,J); kl = INDEX(K,L);
+                ijkl = INDEX(ij,kl);
+                ik = INDEX(I,K); jl = INDEX(J,L);
+                ikjl = INDEX(ik,jl);
+
+                F[h][i][j] += D[g][k][l] * (2.0 * TEI[ijkl] - TEI[ikjl]);
+              }
+            }
+
+          }
+        }
+      }
+    }
+```

From e0986fb7bcc76bb3143da692c0c5d7530b7f76f3 Mon Sep 17 00:00:00 2001
From: Kirk Pearce <kcpearce@vt.edu>
Date: Sat, 1 Apr 2017 14:12:28 -0400
Subject: [PATCH 25/41] Add Project 9 hints and eqns

---
 Project#09/figures/density-matrix.png   | Bin 0 -> 7231 bytes
 Project#09/figures/fock-build.png       | Bin 0 -> 11884 bytes
 Project#09/figures/matrix-product.png   | Bin 0 -> 7176 bytes
 Project#09/figures/point-group-rule.png | Bin 0 -> 7824 bytes
 Project#09/{ => hints}/hint1.md         |   0
 5 files changed, 0 insertions(+), 0 deletions(-)
 create mode 100644 Project#09/figures/density-matrix.png
 create mode 100644 Project#09/figures/fock-build.png
 create mode 100644 Project#09/figures/matrix-product.png
 create mode 100644 Project#09/figures/point-group-rule.png
 rename Project#09/{ => hints}/hint1.md (100%)

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diff --git a/Project#09/hint1.md b/Project#09/hints/hint1.md
similarity index 100%
rename from Project#09/hint1.md
rename to Project#09/hints/hint1.md

From 4187c915afe752fbcb0049692ba43116c4d3cce9 Mon Sep 17 00:00:00 2001
From: Kirk Pearce <kcpearce@vt.edu>
Date: Sat, 1 Apr 2017 14:13:58 -0400
Subject: [PATCH 26/41] Resize Project 9 eqns

---
 Project#09/README.md | 8 ++++----
 1 file changed, 4 insertions(+), 4 deletions(-)

diff --git a/Project#09/README.md b/Project#09/README.md
index c8fe857..1a6e4f6 100644
--- a/Project#09/README.md
+++ b/Project#09/README.md
@@ -15,7 +15,7 @@ for the purposes of this project we'll deal only with the one-electron (two-inde
 
 The vanishing integral rule of point-group theory may be stated as follows for Abelian groups:
 
-<img src="./figures/point-group-rule.png" height="20">
+<img src="./figures/point-group-rule.png" height="40">
 
 where <html>&Gamma;<sub>X</sub></html> denotes the irreducible representation (irrep) of entity X.
 That is, unless the direct product of the irreps of the functions or operators in the integrand contains the totally symmetric irrep, the integral must be zero.
@@ -146,7 +146,7 @@ in the full list of orbitals; the first number is the relative index, the second
 If we store the integral, MO coefficient, density, and Fock matrices in the block-diagonal structure described above, 
 we may compute matrix products of these quantities separately for each diagonal block.  For example, the matrix product may be written as:
 
-<img src="./figures/matrix-product.png" height="25">
+<img src="./figures/matrix-product.png" height="50">
 
 where *h* denotes a particular irrep of the point group and <b><i>C</i></b><sub>h</sub> denotes the *h*-th irrep subblock of the full matrix <b><i>C</i></b>.
 (The notation of the large plus with a circle around it indicates a direct sum.)
@@ -155,7 +155,7 @@ where *h* denotes a particular irrep of the point group and <b><i>C</i></b><sub>
 
 The calculation of the density matrix requires information about the number of occupied orbitals:
 
-<img src="./figures/density-matrix.png" height="25">
+<img src="./figures/density-matrix.png" height="50">
 
 In a calculation without symmetry, one need only know the number of electrons (equivalently, the atomic numbers of the atoms and the overall molecular charge) 
 to evaluate this expression.  However, if the density and MO coefficient matrices are stored in symmetry-blocked form, 
@@ -187,7 +187,7 @@ sophisticated guesses at the Fock matrix, e.g. Hueckel orbitals or other (semi)e
 
 The only portion of the SCF procedure involving two-electron integrals is the Fock-matrix build:
 
-<img src="./figures/fock-build.png" height="30">
+<img src="./figures/fock-build.png" height="60">
 
 where the notation used in [Project #3](../Project%2303) is retained above.
 If the Fock matrix and density matrix are stored in a symmetry-blocked manner similar to the overlap matrix above, 

From 060ca242d224816fdda19e0fee6667e2200c0d20 Mon Sep 17 00:00:00 2001
From: Kirk Pearce <kcpearce@vt.edu>
Date: Sat, 1 Apr 2017 14:14:43 -0400
Subject: [PATCH 27/41] Resize Project 9 eqns again

---
 Project#09/README.md | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/Project#09/README.md b/Project#09/README.md
index 1a6e4f6..1a01dbd 100644
--- a/Project#09/README.md
+++ b/Project#09/README.md
@@ -15,7 +15,7 @@ for the purposes of this project we'll deal only with the one-electron (two-inde
 
 The vanishing integral rule of point-group theory may be stated as follows for Abelian groups:
 
-<img src="./figures/point-group-rule.png" height="40">
+<img src="./figures/point-group-rule.png" height="30">
 
 where <html>&Gamma;<sub>X</sub></html> denotes the irreducible representation (irrep) of entity X.
 That is, unless the direct product of the irreps of the functions or operators in the integrand contains the totally symmetric irrep, the integral must be zero.
@@ -146,7 +146,7 @@ in the full list of orbitals; the first number is the relative index, the second
 If we store the integral, MO coefficient, density, and Fock matrices in the block-diagonal structure described above, 
 we may compute matrix products of these quantities separately for each diagonal block.  For example, the matrix product may be written as:
 
-<img src="./figures/matrix-product.png" height="50">
+<img src="./figures/matrix-product.png" height="40">
 
 where *h* denotes a particular irrep of the point group and <b><i>C</i></b><sub>h</sub> denotes the *h*-th irrep subblock of the full matrix <b><i>C</i></b>.
 (The notation of the large plus with a circle around it indicates a direct sum.)

From 80af4704036b8a5e52083ac22fc087a6b22acd9c Mon Sep 17 00:00:00 2001
From: Kirk Pearce <kcpearce@vt.edu>
Date: Mon, 3 Apr 2017 09:56:36 -0400
Subject: [PATCH 28/41] Add Project 10 eqns

---
 Project#10/README.md                  |  53 ++++----------------------
 Project#10/figures/Bij.png            | Bin 0 -> 3743 bytes
 Project#10/figures/error-vector.png   | Bin 0 -> 3652 bytes
 Project#10/figures/new-t-amps.png     | Bin 0 -> 4803 bytes
 Project#10/figures/sys-lin-eqn-ci.png | Bin 0 -> 22046 bytes
 5 files changed, 8 insertions(+), 45 deletions(-)
 create mode 100644 Project#10/figures/Bij.png
 create mode 100644 Project#10/figures/error-vector.png
 create mode 100644 Project#10/figures/new-t-amps.png
 create mode 100644 Project#10/figures/sys-lin-eqn-ci.png

diff --git a/Project#10/README.md b/Project#10/README.md
index 808e9f7..85bfc70 100644
--- a/Project#10/README.md
+++ b/Project#10/README.md
@@ -12,10 +12,7 @@ zero as the equations converge.  In the SCF procedure, we use the atomic-orbital
 representation of the occupied-virtual block of the Fock matrix. In the CC method, we
 could choose the difference between successive sets of cluster amplitudes:
 
-```
-EQUATION
-{\mathbf e}_i = {\mathbf T}_{i+1} - {\mathbf T}_i,
-```
+<img src="./figures/error-vector.png" height="25">
 
 where <b>T</b><sub>i</sub> represents a vector containing all the cluster amplitudes for the *i*-th iteration.
 
@@ -23,8 +20,8 @@ There are three important points to note about this choice of error vector:
   * Defining the error vectors as differences between successive sets of amplitudes
     implies that one cannot begin the DIIS extrapolation until at least three
     iterations are complete.
-  * Just as in the SCF DIIS procedure, error vectors should only be computed using *
-    **non-extrapolated** * sets of cluster amplitudes.
+  * Just as in the SCF DIIS procedure, error vectors should only be computed using
+    <b><i>non-extrapolated</i></b> sets of cluster amplitudes.
   * The set of cluster amplitudes can require substantial memory for larger
     molecules; hence, the choice of the number of error vectors used in the CC DIIS
     procedure is potentially dependent on the available storage (disk and memory).
@@ -32,48 +29,15 @@ There are three important points to note about this choice of error vector:
 ## Extrapolation
 Given the above definition of the error vectors, the set of linear equations to be solved is the same as that for the SCF DIIS procedure:
 
-```
-EQUATION
-\left(
-\begin{array}{ccccc}
-B_{11} & B_{12} & \ldots & B_{1m} & -1 \\
-B_{21} & B_{22} & \ldots & B_{2m} & -1 \\
-\ldots & \ldots & \ldots & \ldots & -1 \\
-B_{m1} & B_{m2} & \ldots & B_{mm} & -1 \\
--1 & -1 & \ldots & -1 & 0 \\
-\end{array}
-\right) \left(
-\begin{array}{c}
-c_1 \\
-c_2 \\
-\ldots \\
-c_m \\
-\lambda
-\end{array}
-\right) = \left(
-\begin{array}{c}
-0 \\
-0 \\
-\ldots \\
-0 \\
--1 \\
-\end{array}
-\right),
-```
+<img src="./figures/sys-lin-eqn-ci.png" height="125">
 
-where lambda is a Lagrangian multiplier and the elements <i>B<sub>ij</sub></i> are computed as dot products of error matrices:
+where &lambda; is a Lagrangian multiplier and the elements <i>B<sub>ij</sub></i> are computed as dot products of error matrices:
 
-```
-EQUATION
-B_{ij} \equiv {\mathbf e}_i \cdot {\mathbf e}_j.
-```
+<img src="./figures/Bij.png" height="25">
 
 A new set of cluster amplitudes is then obtained as a linear combinations of older amplitudes using the coefficients from the linear equations above:
 
-```
-EQUATION
-{\mathbf T}_{\rm new} = \sum_i c_i {\mathbf T}_i
-```
+<img src="./figures/new-t-amps.png" height="25">
 
 Again: The extrapolated cluster amplitudes should be used only in the CC amplitude equations, not to compute subsequent error vectors.
 
@@ -81,8 +45,7 @@ Again: The extrapolated cluster amplitudes should be used only in the CC amplitu
 Once the procedure is working, you should observe a considerable reduction in the
 number of iterations required to converge the CC amplitude equations to a given
 tolerance.  For example, without DIIS extrapolation, the STO-3G H<sub>2</sub>O test
-case from [Project #5](https://github.com/CrawfordGroup/ProgrammingProjects/tree/master/Project%2305) 
-converges in 38 iterations to a
+case from [Project #5](../Project%2305) converges in 38 iterations to a
 precision of 10<sup>-12</sup>:
 
 ```
diff --git a/Project#10/figures/Bij.png b/Project#10/figures/Bij.png
new file mode 100644
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From fc63bcbdf79a0588dac6be662a1412892eb51f9e Mon Sep 17 00:00:00 2001
From: Kirk Pearce <kcpearce@vt.edu>
Date: Mon, 3 Apr 2017 09:57:53 -0400
Subject: [PATCH 29/41] Resize Project 10 eqns

---
 Project#10/README.md | 6 +++---
 1 file changed, 3 insertions(+), 3 deletions(-)

diff --git a/Project#10/README.md b/Project#10/README.md
index 85bfc70..48e85fe 100644
--- a/Project#10/README.md
+++ b/Project#10/README.md
@@ -12,7 +12,7 @@ zero as the equations converge.  In the SCF procedure, we use the atomic-orbital
 representation of the occupied-virtual block of the Fock matrix. In the CC method, we
 could choose the difference between successive sets of cluster amplitudes:
 
-<img src="./figures/error-vector.png" height="25">
+<img src="./figures/error-vector.png" height="20">
 
 where <b>T</b><sub>i</sub> represents a vector containing all the cluster amplitudes for the *i*-th iteration.
 
@@ -33,11 +33,11 @@ Given the above definition of the error vectors, the set of linear equations to
 
 where &lambda; is a Lagrangian multiplier and the elements <i>B<sub>ij</sub></i> are computed as dot products of error matrices:
 
-<img src="./figures/Bij.png" height="25">
+<img src="./figures/Bij.png" height="20">
 
 A new set of cluster amplitudes is then obtained as a linear combinations of older amplitudes using the coefficients from the linear equations above:
 
-<img src="./figures/new-t-amps.png" height="25">
+<img src="./figures/new-t-amps.png" height="50">
 
 Again: The extrapolated cluster amplitudes should be used only in the CC amplitude equations, not to compute subsequent error vectors.
 

From 6d0bc92f1b42640dc618e9f0b9fffed794e5bbfb Mon Sep 17 00:00:00 2001
From: Kirk Pearce <kcpearce@vt.edu>
Date: Mon, 3 Apr 2017 10:15:56 -0400
Subject: [PATCH 30/41] Add Project 11 eqns

---
 Project#08/README.md                       |   2 +-
 Project#11/README.md                       |  38 +++++++--------------
 Project#11/figures/compound-indices.png    | Bin 0 -> 7610 bytes
 Project#11/figures/fock-contribution-1.png | Bin 0 -> 6214 bytes
 Project#11/figures/fock-contribution-2.png | Bin 0 -> 5509 bytes
 Project#11/figures/fock-contribution-3.png | Bin 0 -> 10580 bytes
 Project#11/figures/fock-matrix.png         | Bin 0 -> 11536 bytes
 Project#11/figures/index-restrictions.png  | Bin 0 -> 5344 bytes
 8 files changed, 13 insertions(+), 27 deletions(-)
 create mode 100644 Project#11/figures/compound-indices.png
 create mode 100644 Project#11/figures/fock-contribution-1.png
 create mode 100644 Project#11/figures/fock-contribution-2.png
 create mode 100644 Project#11/figures/fock-contribution-3.png
 create mode 100644 Project#11/figures/fock-matrix.png
 create mode 100644 Project#11/figures/index-restrictions.png

diff --git a/Project#08/README.md b/Project#08/README.md
index f32de8c..dc85873 100644
--- a/Project#08/README.md
+++ b/Project#08/README.md
@@ -38,7 +38,7 @@ leads to the following system of linear equations for the c<sub>i</sub>:
 
 where lambda is a Lagrangian multiplier and the elements <i>B<sub>ij</sub></i> are computed as dot products of error matrices:
 
-<img src="./figures/Bij.png" height="25">
+<img src="./figures/Bij.png" height="20">
 
 ## Step #1: Compute the Error Matrix in Each Iteration
 
diff --git a/Project#11/README.md b/Project#11/README.md
index e963872..333f859 100644
--- a/Project#11/README.md
+++ b/Project#11/README.md
@@ -34,10 +34,7 @@ iteration rather than stored on disk.))
 
 At the heart of the SCF procedure is the expensive Fock-matrix term:
 
-```
-EQUATION
-F_{ij} = H^{\rm core}_{ij} + \sum_{kl}^{\rm AO} D_{kl} \left[ 2 (ij|kl) - (ik|jl) \right],
-```
+<img src="./figures/fock-matrix.png" height="50">
 
 where we use *i*, *j*, *k*, and *l* to denote AO-basis indices.  As described
 in [Project #3](https://github.com/CrawfordGroup/ProgrammingProjects/tree/master/Project%2303), a simple algorithm for
@@ -117,44 +114,33 @@ each individual integral to the Fock matrix:
   iwl_buf_close(&InBuf, 1);
 ```
 
-Thus a given integral, (ij|kl), would contribute to **//at least two//** Fock matrix elements as:
+Thus a given integral, (ij|kl), would contribute to <b><i>at least two</i></b> Fock matrix elements as:
+
+<img src="./figures/fock-contribution-1.png" height="25">
 
-```
-EQUATION
-F_{ij} \leftarrow +2 * D_{kl} (ij|kl)
-```
 and
 
-```
-EQUATION
-F_{ik} \leftarrow - D_{jl} (ij|kl).
-```
+<img src="./figures/fock-contribution-2.png" height="25">
 
 ## Handling Permutational Symmetry
 
 The most difficult aspect of the out-of-core algorithm is the fact that file contains only the permutationally unique integrals,
 *(ij|kl)*, such that:
-```
-EQUATION
-i \geq j,\ \ \ \ \ k \geq l,\ \ \ \ \ \ {\rm and}\ \ \ \ \ \ ij \geq kl,
-```
+
+<img src="./figures/index-restrictions.png" height="25">
+
 where
 
-```
-EQUATION
-ij \equiv i(i+1)/2 + j\ \ \ \ \ \ {\rm and} \ \ \ \ \ \ kl \equiv k(k+1)/2 + l.
-```
+<img src="./figures/compound-indices.png" height="25">
+
 Thus, when determining the contribution of a given integral to various elements
 of the Fock matrix, one must consider all possible unique permutations of the
 indices, *i*, *j*, *k*, and *l*.  Note, however, that coincidences among the
 indices can limit the number of possibilities.  For example, if one encountered
-the integral *(22|11)*, it would contribute to a total of * **four** * Fock
+the integral *(22|11)*, it would contribute to a total of <b><i>four</i></b> Fock
 matrix elements, viz.
 
-```
-EQUATION
-F_{22} \leftarrow 2 D_{11} (22|11),\ \ \ F_{21} \leftarrow - D_{21} (22|11),\ \ \ F_{11} \leftarrow 2 D_{22} (22|11),\ \ \ {\rm and}\ \ \ F_{12} \leftarrow -D_{12} (22|11).
-```
+<img src="./figures/fock-contribution-3.png" height="25">
 
 All such cases must be included in the algorithm to obtain a correct Fock matrix.
 
diff --git a/Project#11/figures/compound-indices.png b/Project#11/figures/compound-indices.png
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HcmV?d00001


From 9478d7302294641c57d4018400e89e0e99cbb33c Mon Sep 17 00:00:00 2001
From: Kirk Pearce <kcpearce@vt.edu>
Date: Mon, 3 Apr 2017 10:17:26 -0400
Subject: [PATCH 31/41] Resize Project 11 eqns

---
 Project#11/README.md | 12 ++++++------
 1 file changed, 6 insertions(+), 6 deletions(-)

diff --git a/Project#11/README.md b/Project#11/README.md
index 333f859..b11405f 100644
--- a/Project#11/README.md
+++ b/Project#11/README.md
@@ -34,7 +34,7 @@ iteration rather than stored on disk.))
 
 At the heart of the SCF procedure is the expensive Fock-matrix term:
 
-<img src="./figures/fock-matrix.png" height="50">
+<img src="./figures/fock-matrix.png" height="75">
 
 where we use *i*, *j*, *k*, and *l* to denote AO-basis indices.  As described
 in [Project #3](https://github.com/CrawfordGroup/ProgrammingProjects/tree/master/Project%2303), a simple algorithm for
@@ -116,22 +116,22 @@ each individual integral to the Fock matrix:
 
 Thus a given integral, (ij|kl), would contribute to <b><i>at least two</i></b> Fock matrix elements as:
 
-<img src="./figures/fock-contribution-1.png" height="25">
+<img src="./figures/fock-contribution-1.png" height="20">
 
 and
 
-<img src="./figures/fock-contribution-2.png" height="25">
+<img src="./figures/fock-contribution-2.png" height="20">
 
 ## Handling Permutational Symmetry
 
 The most difficult aspect of the out-of-core algorithm is the fact that file contains only the permutationally unique integrals,
 *(ij|kl)*, such that:
 
-<img src="./figures/index-restrictions.png" height="25">
+<img src="./figures/index-restrictions.png" height="20">
 
 where
 
-<img src="./figures/compound-indices.png" height="25">
+<img src="./figures/compound-indices.png" height="20">
 
 Thus, when determining the contribution of a given integral to various elements
 of the Fock matrix, one must consider all possible unique permutations of the
@@ -140,7 +140,7 @@ indices can limit the number of possibilities.  For example, if one encountered
 the integral *(22|11)*, it would contribute to a total of <b><i>four</i></b> Fock
 matrix elements, viz.
 
-<img src="./figures/fock-contribution-3.png" height="25">
+<img src="./figures/fock-contribution-3.png" height="20">
 
 All such cases must be included in the algorithm to obtain a correct Fock matrix.
 

From d48b86d45040bf630d3d7911db4c794153b3edb1 Mon Sep 17 00:00:00 2001
From: Kirk Pearce <kcpearce@vt.edu>
Date: Mon, 3 Apr 2017 10:22:15 -0400
Subject: [PATCH 32/41] Resize Project 11 eqns again

---
 Project#11/README.md | 18 ++++++++----------
 1 file changed, 8 insertions(+), 10 deletions(-)

diff --git a/Project#11/README.md b/Project#11/README.md
index b11405f..cccbb82 100644
--- a/Project#11/README.md
+++ b/Project#11/README.md
@@ -26,19 +26,18 @@ physical memory available on workstations or supercomputer nodes.
 The purpose of this project is to consider and implement an "out-of-core" SCF
 algorithm, that is, an approach that minimizes the core-memory requirements of
 the program by reading the two-electron repulsion integrals from disk in
-batches when only they are needed.((An alternative approach is the so-called
+batches when only they are needed (An alternative approach is the so-called
 "direct" SCF, in which the two-electron integrals are re-computed in each SCF
-iteration rather than stored on disk.))
+iteration rather than stored on disk)
 
 ## The Fock Matrix Build
 
 At the heart of the SCF procedure is the expensive Fock-matrix term:
 
-<img src="./figures/fock-matrix.png" height="75">
+<img src="./figures/fock-matrix.png" height="60">
 
 where we use *i*, *j*, *k*, and *l* to denote AO-basis indices.  As described
-in [Project #3](https://github.com/CrawfordGroup/ProgrammingProjects/tree/master/Project%2303), a simple algorithm for
-evaluating this matrix is:
+in [Project #3](../Project%2303), a simple algorithm for evaluating this matrix is:
 
 ```cpp
 for(i=0; i < nao; i++)
@@ -60,10 +59,9 @@ for(i=0; i < nao; i++)
 
 This algorithm hinges on the fact that all the two-electron integrals are
 immediately available in the TEI array (which takes advantage of permutational
-symmetry).  In 
-[Project #7](https://github.com/CrawfordGroup/ProgrammingProjects/tree/master/Project%2307) 
-you made use of the following code, which reads the integrals into the TEI
-array in batches (sometimes referred to in PSI as "buffers"):
+symmetry).  In [Project #7](../Project%2307) you made use of the following code, 
+which reads the integrals into the TEI array in batches (sometimes referred to 
+in PSI as "buffers"):
 
 ```cpp
   iwl_buf_init(&InBuf, PSIF_SO_TEI, 1e-14, 1, 0);
@@ -145,5 +143,5 @@ matrix elements, viz.
 All such cases must be included in the algorithm to obtain a correct Fock matrix.
 
 ## Additional Reading
-  * J. Almlof, K. Faegri, and K. Korsell, "Principles for a Direct SCF Approach to LCAO-MO //Ab Initio// Calculations," //J. Comp. Chem.// **3**, 385-399 (1982).
+  * J. Almlof, K. Faegri, and K. Korsell, "Principles for a Direct SCF Approach to LCAO-MO *Ab Initio* Calculations," *J. Comp. Chem.* **3**, 385-399 (1982).
 

From b93da1f189ecf64b93adf7fca1eedf5baed96fb6 Mon Sep 17 00:00:00 2001
From: Kirk Pearce <kcpearce@vt.edu>
Date: Mon, 3 Apr 2017 10:57:49 -0400
Subject: [PATCH 33/41] Resize Project 11 eqns again agin

---
 Project#11/README.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/Project#11/README.md b/Project#11/README.md
index cccbb82..944698a 100644
--- a/Project#11/README.md
+++ b/Project#11/README.md
@@ -129,7 +129,7 @@ The most difficult aspect of the out-of-core algorithm is the fact that file con
 
 where
 
-<img src="./figures/compound-indices.png" height="20">
+<img src="./figures/compound-indices.png" height="22.5">
 
 Thus, when determining the contribution of a given integral to various elements
 of the Fock matrix, one must consider all possible unique permutations of the

From 9fe6cd63d15f5734bb989be31a9ced4f7f66d417 Mon Sep 17 00:00:00 2001
From: Kirk Pearce <kcpearce@vt.edu>
Date: Mon, 3 Apr 2017 11:39:32 -0400
Subject: [PATCH 34/41] Add Project 12 hints and eqns

---
 Project#12/README.md                          | 221 +-----
 Project#12/figures/A-matrix.png               | Bin 0 -> 10160 bytes
 Project#12/figures/B-matrix.png               | Bin 0 -> 5292 bytes
 Project#12/figures/excited-det-schrod-eqn.png | Bin 0 -> 9372 bytes
 .../figures/four-possible-determinants.png    | Bin 0 -> 8043 bytes
 Project#12/figures/identical-ci-coeff.png     | Bin 0 -> 4887 bytes
 Project#12/figures/inverse-ci-coeff.png       | Bin 0 -> 5011 bytes
 .../figures/matrix-eigenvalue-problem.png     | Bin 0 -> 4804 bytes
 Project#12/figures/matrix-elements.png        | Bin 0 -> 10084 bytes
 Project#12/figures/simpler-hamiltonian.png    | Bin 0 -> 12353 bytes
 .../figures/singlet-triplet-combinations.png  | Bin 0 -> 19505 bytes
 .../figures/singly-excited-determinant.png    | Bin 0 -> 11412 bytes
 Project#12/figures/smarter-tdhf-1.png         | Bin 0 -> 4783 bytes
 Project#12/figures/smarter-tdhf-2.png         | Bin 0 -> 4614 bytes
 Project#12/figures/smarter-tdhf-3.png         | Bin 0 -> 6120 bytes
 Project#12/figures/smarter-tdhf-4.png         | Bin 0 -> 6328 bytes
 Project#12/figures/smarter-tdhf-5.png         | Bin 0 -> 6694 bytes
 Project#12/figures/smarter-tdhf-6.png         | Bin 0 -> 7115 bytes
 .../figures/spatial-orbital-expression.png    | Bin 0 -> 13257 bytes
 Project#12/figures/spin-factored-eqn.png      | Bin 0 -> 20412 bytes
 Project#12/figures/tdhf-eqn.png               | Bin 0 -> 9075 bytes
 Project#12/hints/hint1.md                     | 174 +++++
 Project#12/hints/hint2.md                     |  15 +
 Project#12/hints/hint3.md                     | 666 ++++++++++++++++++
 24 files changed, 880 insertions(+), 196 deletions(-)
 create mode 100644 Project#12/figures/A-matrix.png
 create mode 100644 Project#12/figures/B-matrix.png
 create mode 100644 Project#12/figures/excited-det-schrod-eqn.png
 create mode 100644 Project#12/figures/four-possible-determinants.png
 create mode 100644 Project#12/figures/identical-ci-coeff.png
 create mode 100644 Project#12/figures/inverse-ci-coeff.png
 create mode 100644 Project#12/figures/matrix-eigenvalue-problem.png
 create mode 100644 Project#12/figures/matrix-elements.png
 create mode 100644 Project#12/figures/simpler-hamiltonian.png
 create mode 100644 Project#12/figures/singlet-triplet-combinations.png
 create mode 100644 Project#12/figures/singly-excited-determinant.png
 create mode 100644 Project#12/figures/smarter-tdhf-1.png
 create mode 100644 Project#12/figures/smarter-tdhf-2.png
 create mode 100644 Project#12/figures/smarter-tdhf-3.png
 create mode 100644 Project#12/figures/smarter-tdhf-4.png
 create mode 100644 Project#12/figures/smarter-tdhf-5.png
 create mode 100644 Project#12/figures/smarter-tdhf-6.png
 create mode 100644 Project#12/figures/spatial-orbital-expression.png
 create mode 100644 Project#12/figures/spin-factored-eqn.png
 create mode 100644 Project#12/figures/tdhf-eqn.png
 create mode 100644 Project#12/hints/hint1.md
 create mode 100644 Project#12/hints/hint2.md
 create mode 100644 Project#12/hints/hint3.md

diff --git a/Project#12/README.md b/Project#12/README.md
index 2725558..5f8cf54 100644
--- a/Project#12/README.md
+++ b/Project#12/README.md
@@ -6,7 +6,7 @@ and time-dependent Hartree-Fock (TDHF), which is also known as the random-phase
 approximation (RPA). The purpose of this project is to develop simple
 implementations of these two methods and to understand the basic differences
 between them.  This project assumes you have completed at least the
-[CCSD programming project](https://github.com/CrawfordGroup/ProgrammingProjects/tree/master/Project%2305).
+[CCSD programming project](../Project%2305).
 
 You can read more about these methods in 
 [J.B. Foresman, M. Head-Gordon, J.A. Pople, and M.J. Frisch, J. Phys. Chem. 96, 135-141 (1992)](http://pubs.acs.org/doi/pdf/10.1021/j100180a030) (CIS) 
@@ -18,39 +18,27 @@ The fundamental idea behind CIS is the representation of the excited-state wave
 functions as linear combinations of singly excited determinants relative to the
 Hartree-Fock reference wave function, *viz.*
 
-```
-EQUATION
-\[ |\Psi(m)\rangle = \sum_{jb} c_j^b(m) \{ a^+_b a_j \} |\Phi_0\rangle = \sum_{jb} c_j^b(m) |\phi_j^b\rangle, \]
-```
+<img src="./figures/singly-excited-determinant.png" height="60">
 
 where *m* identifies the various excited states, and we will use *i* and *j*
 (*a* and *b*) to denote occupied (unoccupied) spin-orbitals. Inserting this
 into the Schr<html>&ouml;</html>dinger equation and left-projecting onto a
 particular singly excited determinant gives
 
-```
-EQUATION
-\sum_{jb} \langle \phi_i^a|H|\phi_j^b\rangle c_j^b(m) = E(m) c_i^a(m).
-```
+<img src="./figures/excited-det-schrod-eqn.png" height="60">
 
 If we recognize that we have one of these equations for every combination of
 *i* and *a* spin-orbitals, then this equation may be viewed as a matrix
 eigenvalue problem:
 
-```
-EQUATION
-\[ {\mathbf {\underline{\underline H}}}\ {\mathbf {\underline c}}(m) = E(m) {\mathbf {\underline c}}(m). \]
-```
+<img src="./figures/matrix-eigenvalue-problem.png" height="60">
 
 To solve this equation, we need an expression for the matrix elements in terms
 of things we already know, i.e. Fock matrix elements and two-electron
 integrals.  This can be done using either algebraic or diagrammatic techniques
 to obtain (in the spin-orbital notation of previous projects):
 
-```
-EQUATION
-\langle \phi_i^a | H | \phi_j^b \rangle = H_{ia,jb} = f_{ab} \delta_{ij} - f_{ij} \delta_{ab} + \langle aj || ib \rangle.
-```
+<img src="./figures/matrix-elements.png" height="60">
 
 Our task is then relatively simple: Build the Hamiltonian matrix (expressed in
 the basis of all singly excited determinants) using the above expression and
@@ -60,7 +48,7 @@ occupied orbitals times the number of unoccupied orbitals.  For our
 STO-3G/H<sub>2</sub>O test case, with its ten occupied and four unoccupied
 spin-orbitals, the matrix will be 40 x 40.
 
-  * Hint: CIS Hamiltonian for STO-3G H<sub>2</sub>O
+  * [Hint](./hints/hint1.md): CIS Hamiltonian for STO-3G H<sub>2</sub>O
 
 Make sure you can compute the correct CIS excitation energies for each of the
 four test cases provided at the end of the page.  Before you move on to the
@@ -84,99 +72,11 @@ from a simple two-electron/two-orbital example (such as the *1s 2s* excited
 state configuration of the He atom).  One can easily show that the four
 possible determinants arising from this configuration,
 
-```
-EQUATION
-\[ \left|
-\begin{array}{cc}
-\uparrow &  \\
-\hline
-\uparrow & \\
-\hline
-\end{array}
-\right\rangle,\ \ 
-\left|
-\begin{array}{cc}
-& \downarrow \\
-\hline
-& \downarrow \\
-\hline
-\end{array}
-\right\rangle,\ \ 
-\left|
-\begin{array}{cc}
-\uparrow &  \\
-\hline
-& \downarrow\\
-\hline
-\end{array}
-\right\rangle,\ \ 
-\left|
-\begin{array}{cc}
- & \downarrow \\
-\hline
-\uparrow & \\
-\hline
-\end{array}
-\right\rangle,\ \ \]
-```
+<img src="./figures/four-possible-determinants.png" height="60">
 
 are components of one singlet and one triplet in the following combinations:
 
-```
-EQUATION
-\[ |\Phi^3_1\rangle = \left|
-\begin{array}{cc}
-\uparrow &  \\
-\hline
-\uparrow & \\
-\hline
-\end{array}
-\right\rangle,\ \ 
-|\Phi^3_{-1}\rangle = \left|
-\begin{array}{cc}
-& \downarrow \\
-\hline
-& \downarrow\\
-\hline
-\end{array}
-\right\rangle,\ \ 
-|\Phi^3_{0}\rangle = 
-\frac{1}{\sqrt{2}}\left\{
-\left|
-\begin{array}{cc}
-\uparrow & \\
-\hline
-& \downarrow\\
-\hline
-\end{array}
-\right\rangle -
-\left|
-\begin{array}{cc}
-& \downarrow \\
-\hline
-\uparrow & \\
-\hline
-\end{array}
-\right\rangle\right\},\ \ {\rm and}\ \ 
-|\Phi^1_{0}\rangle = 
-\frac{1}{\sqrt{2}}\left\{
-\left|
-\begin{array}{cc}
-\uparrow & \\
-\hline
-& \downarrow\\
-\hline
-\end{array}
-\right\rangle +
-\left|
-\begin{array}{cc}
-& \downarrow \\
-\hline
-\uparrow & \\
-\hline
-\end{array}
-\right\rangle\right\}, \]
-```
+<img src="./figures/singlet-triplet-combinations.png" height="60">
 
 where the superscript is the spin multiplicity (*2S+1*) and the subscript is
 the *M<sub>S</sub>* value of the wave function.  So, if we wanted to compute
@@ -192,13 +92,7 @@ expression and the equation for the CIS Hamiltonian matrix elements in the
 previous section, we may write a spin-factored equation for the
 <html>&alpha;</html> coefficients as
 
-```
-EQUATION
-c_{i_\alpha}^{a_\alpha}(m) E(m) = \sum_{j_\alpha b_\alpha} 
-\left[ f_{a_\alpha b_\alpha} \delta_{i_\alpha j_\alpha} - f_{i_\alpha j_\alpha} \delta_{a_\alpha b_\alpha} + \langle a_\alpha j_\alpha || i_\alpha b_\alpha \rangle \right]
-c_{j_\alpha}^{b_\alpha}(m) + \sum_{j_\beta b_\beta} \langle a_\alpha j_\beta ||
-i_\alpha b_\beta \rangle c_{j_\beta}^{b_\beta}(m).
-```
+<img src="./figures/pin-factored-eqn.png" height="60">
 
 Note that the mix-spin cases (where *j=*<html>&alpha;</html> and
 *b=*<html>&beta;</html> or *vice versa*) do not contribute since the Fock
@@ -207,39 +101,27 @@ carry out spin integration on the integrals in the above expression and assume
 that the <html>&alpha;</html> and <html>&beta;</html> CI coefficients are
 identical for the same spatial orbitals, i.e.,
 
-```
-EQUATION
-c_{i_\alpha}^{a_\alpha}(m) = c_{i_\beta}^{a_\beta}(m).
-```
+<img src="./figures/identical-ci-coeff.png" height="60">
 
-we obtain the ***spatial orbital*** expression
+we obtain the <b><i>spatial orbital</i></b> expression
 
-```
-EQUATION
-c_i^a(m) E(m) = \sum_{jb} \left[ f_{ab} \delta_{ij} - f_{ij} \delta_{ab} + 2 <aj|ib> - <aj|bi> \right] c_j^b(m).
-```
+<img src="./figures/spatial-orbital-expression.png" height="60">
 
 The part in brackets above is an expression for the spatial-orbital CIS
 Hamiltonian, spin-adapted for singlet excited states, and diagonalization of
 this matrix will yield only the singlet eigenvalues you obtained from your
 spin-orbital matrix earlier.
 
-  * Hint: Spin-adapted CIS singlet Hamiltonian for STO-3G H<sub>2</sub>O
+  * [Hint](./hints/hint2.md): Spin-adapted CIS singlet Hamiltonian for STO-3G H<sub>2</sub>O
 
 How about the triplets?  We use exactly the same spin-factorization, but
 instead require
 
-```
-EQUATION
-c_{i_\alpha}^{a_\alpha}(m) = -c_{i_\beta}^{a_\beta}(m).
-```
+<img src="./figures/inverse-ci-coeff.png" height="60">
 
 This yields a slightly simpler Hamiltonian:
 
-```
-EQUATION
-c_i^a(m) E(m) = \sum_{jb} \left[ f_{ab} \delta_{ij} - f_{ij} \delta_{ab} - <aj|bi> \right] c_j^b(m),
-```
+<img src="./figures/simpler-hamiltonian.png" height="60">
 
 which, upon diagonalization, will yield only the triplet eigenvalues (but each only occurring once) from your earlier diagonalziation.
 
@@ -266,47 +148,23 @@ TDHF/RPA wave function expansion in terms of orbital rotations instead of
 Slater determinants, but that's a discussion for another day.)  The TDHF/RPA
 eigenvalue equations take the form
 
-```
-EQUATION
-\left(
-\begin{array}{cc}
-{\mathbf {\underline{\underline A}}} & {\mathbf {\underline{\underline B}}} \\
--{\mathbf {\underline{\underline B}}} & - {\mathbf {\underline{\underline A}}} \\
-\end{array}
-\right) \left(
-\begin{array}{c}
-{\mathbf {\underline X}} \\
-{\mathbf {\underline Y}} \\
-\end{array}
-\right) = E \left(
-\begin{array}{c}
-{\mathbf {\underline X}} \\
-{\mathbf {\underline Y}} \\
-\end{array}
-\right).
-```
+<img src="./figures/tdhf-eqn.png" height="60">
 
 The definition of the ***A*** matrix is just the CIS matrix itself, *viz.*
 
-```
-EQUATION
-A_{ia,jb} \equiv \langle \phi_i^a | H | \phi_j^b \rangle = f_{ab} \delta_{ij} - f_{ij} \delta_{ab} + \langle aj || ib \rangle,
-```
+<img src="./figures/A-matrix.png" height="60">
 
 while **X** and **Y** are the parameters of single excitations and
 de-excitations, respectively, and the ***B*** matrix is simply
 
-```
-EQUATION
-B_{ia,jb} \equiv \langle ab || ij \rangle.
-```
+<img src="./figures/B-matrix.png" height="60">
 
 Thus, the row/column dimension of the TDHF/RPA Hamiltonian is twice that of the
 CIS Hamiltonian, and the matrix is non-symmetric (so you must be careful about
 the diagonalization function you choose).  Do you obtain twice as many
 excitation energies?
 
-  * Hint: TDHF/RPA Hamiltonian for STO-3G H<sub>2</sub>O
+  * [Hint](./hints/hint3.md): TDHF/RPA Hamiltonian for STO-3G H<sub>2</sub>O
 
 ## A Better Approach to Solving the TDHF/RPA Eigenvalue Equations
 
@@ -316,57 +174,28 @@ Hamiltonian storage cost), one can rearrange the eigenvalue equations.  First
 write eigenvalue equation two separate equations, each in terms of the
 submatrices **A** and **B**:
 
-```
-EQUATION
-\[ {\mathbf {\underline{\underline A}}} {\mathbf {\underline X}} + {\mathbf {\underline{\underline B}}} {\mathbf {\underline Y}} = E {\mathbf {\underline X}} \] 
-```
+<img src="./figures/smarter-tdhf-1.png" height="60">
 
 and
 
-```
-EQUATION
-\[ 
--{\mathbf {\underline{\underline B}}} {\mathbf {\underline X}} - {\mathbf {\underline{\underline A}}} {\mathbf {\underline Y}} = E {\mathbf {\underline Y}}. \]
-```
+<img src="./figures/smarter-tdhf-2.png" height="60">
 
 Now take +/- combinations of these equations to obtain
 
-```
-EQUATION
-\[ 
-({\mathbf {\underline{\underline A}}} - {\mathbf {\underline{\underline B}}})
-({\mathbf {\underline X}} - {\mathbf {\underline Y}}) 
-= E ({\mathbf {\underline X}} + {\mathbf {\underline Y}}) \]
-```
+<img src="./figures/smarter-tdhf-3.png" height="60">
 
 and
 
-```
-EQUATION
-\[ ({\mathbf {\underline{\underline A}}} + {\mathbf {\underline{\underline B}}})
-({\mathbf {\underline X}} + {\mathbf {\underline Y}}) 
-= E ({\mathbf {\underline X}} - {\mathbf {\underline Y}}). \]
-```
+<img src="./figures/smarter-tdhf-4.png" height="60">
 
 Solve for ***(X+Y)*** in the second equation:
 
-```
-EQUATION
-\[ ({\mathbf {\underline X}} + {\mathbf {\underline Y}}) 
-= E ({\mathbf {\underline{\underline A}}} + {\mathbf {\underline{\underline B}}})^{-1} 
-({\mathbf {\underline X}} - {\mathbf {\underline Y}}). \]
-```
+<img src="./figures/smarter-tdhf-5.png" height="60">
 
 Insert this result into the first equation, rearrange a bit, and finally
 obtain:
 
-```
-EQUATION
-\[ ({\mathbf {\underline{\underline A}}} + {\mathbf {\underline{\underline B}}}) 
-({\mathbf {\underline{\underline A}}} - {\mathbf {\underline{\underline B}}})
-({\mathbf {\underline X}} - {\mathbf {\underline Y}}) 
-= E^2 ({\mathbf {\underline X}} - {\mathbf {\underline Y}}). \]
-```
+<img src="./figures/smarter-tdhf-6.png" height="60">
 
 This is an eigenvalue equation of the same dimension as the CIS eigenvalue
 equation (number of occupied orbitals times number of unoccupied orbitals),
diff --git a/Project#12/figures/A-matrix.png b/Project#12/figures/A-matrix.png
new file mode 100644
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diff --git a/Project#12/hints/hint1.md b/Project#12/hints/hint1.md
new file mode 100644
index 0000000..a05b2d7
--- /dev/null
+++ b/Project#12/hints/hint1.md
@@ -0,0 +1,174 @@
+Here is the CIS matrix for the same H<sub>2</sub>O STO-3G test case we've been using throughout these projects.  Note that the row/column ordering follows *i* and *a* combinations with *a* as the faster-running index (i.e., *a* indexes the inner loop).  Thus the first row/column corresponds to *i=0* and *a=0* and the second row/column to *i=0* and *a=1*.  Notice that the matrix is diagonally dominant --- that is, the largest element on any row/column combination is the diagonal entry --- and that all the diagonal elements are positive.
+```
+           1           2           3           4           5           6           7           8           9          10
+
+    1  19.9847006   0.0000000   0.0000000   0.0000000   0.0000000   0.0261967   0.0000000  -0.0000000   0.0172405   0.0000000
+    2   0.0000000  19.9585039   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0074145
+    3   0.0000000   0.0000000  20.0308559   0.0000000   0.0000000  -0.0000000   0.0000000   0.0195218   0.0000000   0.0000000
+    4   0.0000000   0.0000000   0.0000000  20.0113341   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000  -0.0000000
+    5   0.0000000   0.0000000   0.0000000   0.0000000  19.9585039   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000
+    6   0.0261967   0.0000000  -0.0000000   0.0000000   0.0000000  19.9847006   0.0000000   0.0000000   0.0098259   0.0000000
+    7   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000  20.0113341   0.0000000   0.0000000   0.0000000
+    8  -0.0000000   0.0000000   0.0195218   0.0000000   0.0000000   0.0000000   0.0000000  20.0308559   0.0000000   0.0000000
+    9   0.0172405   0.0000000   0.0000000   0.0000000   0.0000000   0.0098259   0.0000000   0.0000000   1.1883321   0.0000000
+   10   0.0000000   0.0074145   0.0000000  -0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   1.0931793
+   11   0.0000000   0.0000000   0.0280239   0.0000000   0.0000000   0.0000000   0.0000000   0.0191033   0.0000000   0.0000000
+   12   0.0000000  -0.0000000   0.0000000   0.0089205   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000
+   13   0.0000000   0.0000000   0.0000000   0.0000000   0.0074145   0.0000000  -0.0000000   0.0000000   0.0000000   0.0000000
+   14   0.0098259   0.0000000   0.0000000   0.0000000   0.0000000   0.0172405   0.0000000   0.0000000   0.0951527   0.0000000
+   15   0.0000000   0.0000000   0.0000000   0.0000000  -0.0000000   0.0000000   0.0089205   0.0000000   0.0000000   0.0000000
+   16   0.0000000   0.0000000   0.0191033   0.0000000   0.0000000   0.0000000   0.0000000   0.0280239  -0.0000000   0.0000000
+   17  -0.0000000   0.0000000  -0.0046225   0.0000000   0.0000000  -0.0000000   0.0000000   0.0030386   0.0000000   0.0000000
+   18   0.0000000  -0.0000000   0.0000000  -0.0076611   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000  -0.0000000
+   19  -0.0134310   0.0000000  -0.0000000   0.0000000   0.0000000  -0.0057699   0.0000000   0.0000000  -0.0178319   0.0000000
+   20   0.0000000  -0.0076611   0.0000000  -0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000  -0.0867137
+   21   0.0000000   0.0000000   0.0000000   0.0000000  -0.0000000   0.0000000  -0.0076611   0.0000000   0.0000000   0.0000000
+   22  -0.0000000   0.0000000   0.0030386   0.0000000   0.0000000  -0.0000000   0.0000000  -0.0046225   0.0000000   0.0000000
+   23   0.0000000   0.0000000   0.0000000   0.0000000  -0.0076611   0.0000000  -0.0000000   0.0000000   0.0000000   0.0000000
+   24  -0.0057699   0.0000000   0.0000000   0.0000000   0.0000000  -0.0134310   0.0000000  -0.0000000   0.0688817   0.0000000
+   25   0.0197793   0.0000000  -0.0000000   0.0000000   0.0000000   0.0017709   0.0000000   0.0000000  -0.0117148   0.0000000
+   26   0.0000000   0.0180084   0.0000000  -0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0485372
+   27   0.0000000   0.0000000   0.0149879   0.0000000   0.0000000   0.0000000   0.0000000   0.0112477   0.0000000   0.0000000
+   28   0.0000000  -0.0000000   0.0000000   0.0037402   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000  -0.0000000
+   29   0.0000000   0.0000000   0.0000000   0.0000000   0.0180084   0.0000000  -0.0000000   0.0000000   0.0000000   0.0000000
+   30   0.0017709   0.0000000   0.0000000   0.0000000   0.0000000   0.0197793   0.0000000  -0.0000000  -0.0602520   0.0000000
+   31   0.0000000   0.0000000   0.0000000   0.0000000  -0.0000000   0.0000000   0.0037402   0.0000000   0.0000000   0.0000000
+   32   0.0000000   0.0000000   0.0112477   0.0000000   0.0000000   0.0000000   0.0000000   0.0149879   0.0000000   0.0000000
+   33  -0.0000000   0.0000000   0.0000000   0.0000000   0.0000000  -0.0000000   0.0000000  -0.0000000   0.0000000   0.0000000
+   34   0.0000000  -0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000
+   35   0.0000000   0.0000000  -0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000
+   36   0.0000000   0.0000000   0.0000000  -0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000
+   37   0.0000000   0.0000000   0.0000000   0.0000000  -0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000
+   38  -0.0000000   0.0000000  -0.0000000   0.0000000   0.0000000  -0.0000000   0.0000000   0.0000000  -0.0000000   0.0000000
+   39   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000  -0.0000000   0.0000000   0.0000000   0.0000000
+   40   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000  -0.0000000  -0.0000000   0.0000000
+
+          11          12          13          14          15          16          17          18          19          20
+
+    1   0.0000000   0.0000000   0.0000000   0.0098259   0.0000000   0.0000000  -0.0000000   0.0000000  -0.0134310   0.0000000
+    2   0.0000000  -0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000  -0.0000000   0.0000000  -0.0076611
+    3   0.0280239   0.0000000   0.0000000   0.0000000   0.0000000   0.0191033  -0.0046225   0.0000000  -0.0000000   0.0000000
+    4   0.0000000   0.0089205   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000  -0.0076611   0.0000000  -0.0000000
+    5   0.0000000   0.0000000   0.0074145   0.0000000  -0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000
+    6   0.0000000   0.0000000   0.0000000   0.0172405   0.0000000   0.0000000  -0.0000000   0.0000000  -0.0057699   0.0000000
+    7   0.0000000   0.0000000  -0.0000000   0.0000000   0.0089205   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000
+    8   0.0191033   0.0000000   0.0000000   0.0000000   0.0000000   0.0280239   0.0030386   0.0000000   0.0000000   0.0000000
+    9   0.0000000   0.0000000   0.0000000   0.0951527   0.0000000  -0.0000000   0.0000000   0.0000000  -0.0178319   0.0000000
+   10   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000  -0.0000000   0.0000000  -0.0867137
+   11   1.2566569   0.0000000   0.0000000  -0.0000000   0.0000000   0.0678165  -0.0564572   0.0000000   0.0000000   0.0000000
+   12   0.0000000   1.1888405   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000  -0.0867137   0.0000000   0.0000000
+   13   0.0000000   0.0000000   1.0931793   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000
+   14  -0.0000000   0.0000000   0.0000000   1.1883321   0.0000000   0.0000000   0.0000000   0.0000000   0.0688817   0.0000000
+   15   0.0000000   0.0000000   0.0000000   0.0000000   1.1888405   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000
+   16   0.0678165   0.0000000   0.0000000   0.0000000   0.0000000   1.2566569   0.0302565   0.0000000  -0.0000000   0.0000000
+   17  -0.0564572   0.0000000   0.0000000   0.0000000   0.0000000   0.0302565   0.5507646   0.0000000   0.0000000   0.0000000
+   18   0.0000000  -0.0867137   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.4757743   0.0000000   0.0000000
+   19   0.0000000   0.0000000   0.0000000   0.0688817   0.0000000  -0.0000000   0.0000000   0.0000000   0.7122157   0.0000000
+   20   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.5527668
+   21   0.0000000   0.0000000  -0.0000000   0.0000000  -0.0867137   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000
+   22   0.0302565   0.0000000   0.0000000   0.0000000   0.0000000  -0.0564572   0.0749902   0.0000000   0.0000000   0.0000000
+   23   0.0000000   0.0000000  -0.0867137   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000
+   24  -0.0000000   0.0000000   0.0000000  -0.0178319   0.0000000   0.0000000   0.0000000   0.0000000   0.1594489   0.0000000
+   25   0.0000000   0.0000000   0.0000000  -0.0602520   0.0000000   0.0000000  -0.0000000   0.0000000  -0.0383679   0.0000000
+   26   0.0000000  -0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0615255
+   27  -0.0011401   0.0000000   0.0000000   0.0000000   0.0000000   0.0171886   0.0073884   0.0000000  -0.0000000   0.0000000
+   28   0.0000000  -0.0183287   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0615255   0.0000000  -0.0000000
+   29   0.0000000   0.0000000   0.0485372   0.0000000  -0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000
+   30   0.0000000   0.0000000   0.0000000  -0.0117148   0.0000000   0.0000000  -0.0000000   0.0000000  -0.0998933   0.0000000
+   31   0.0000000   0.0000000  -0.0000000   0.0000000  -0.0183287   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000
+   32   0.0171886   0.0000000   0.0000000   0.0000000   0.0000000  -0.0011401  -0.0541371   0.0000000   0.0000000   0.0000000
+   33   0.0000000   0.0000000   0.0000000  -0.0000000   0.0000000  -0.0000000  -0.0000000   0.0000000  -0.0000000   0.0000000
+   34   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000  -0.0000000   0.0000000   0.0000000
+   35   0.0000000   0.0000000   0.0000000  -0.0000000   0.0000000   0.0000000   0.0000000   0.0000000  -0.0000000   0.0000000
+   36   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000  -0.0000000
+   37   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000
+   38  -0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000  -0.0000000   0.0000000  -0.0000000   0.0000000
+   39   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000
+   40   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000  -0.0000000   0.0000000  -0.0000000   0.0000000
+
+          21          22          23          24          25          26          27          28          29          30
+
+    1   0.0000000  -0.0000000   0.0000000  -0.0057699   0.0197793   0.0000000   0.0000000   0.0000000   0.0000000   0.0017709
+    2   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0180084   0.0000000  -0.0000000   0.0000000   0.0000000
+    3   0.0000000   0.0030386   0.0000000   0.0000000  -0.0000000   0.0000000   0.0149879   0.0000000   0.0000000   0.0000000
+    4   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000  -0.0000000   0.0000000   0.0037402   0.0000000   0.0000000
+    5  -0.0000000   0.0000000  -0.0076611   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0180084   0.0000000
+    6   0.0000000  -0.0000000   0.0000000  -0.0134310   0.0017709   0.0000000   0.0000000   0.0000000   0.0000000   0.0197793
+    7  -0.0076611   0.0000000  -0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000  -0.0000000   0.0000000
+    8   0.0000000  -0.0046225   0.0000000  -0.0000000   0.0000000   0.0000000   0.0112477   0.0000000   0.0000000  -0.0000000
+    9   0.0000000   0.0000000   0.0000000   0.0688817  -0.0117148   0.0000000   0.0000000   0.0000000   0.0000000  -0.0602520
+   10   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0485372   0.0000000  -0.0000000   0.0000000   0.0000000
+   11   0.0000000   0.0302565   0.0000000  -0.0000000   0.0000000   0.0000000  -0.0011401   0.0000000   0.0000000   0.0000000
+   12   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000  -0.0000000   0.0000000  -0.0183287   0.0000000   0.0000000
+   13  -0.0000000   0.0000000  -0.0867137   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0485372   0.0000000
+   14   0.0000000   0.0000000   0.0000000  -0.0178319  -0.0602520   0.0000000   0.0000000   0.0000000   0.0000000  -0.0117148
+   15  -0.0867137   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000  -0.0000000   0.0000000
+   16   0.0000000  -0.0564572   0.0000000   0.0000000   0.0000000   0.0000000   0.0171886   0.0000000   0.0000000   0.0000000
+   17   0.0000000   0.0749902   0.0000000   0.0000000  -0.0000000   0.0000000   0.0073884   0.0000000   0.0000000  -0.0000000
+   18   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0615255   0.0000000   0.0000000
+   19   0.0000000   0.0000000   0.0000000   0.1594489  -0.0383679   0.0000000  -0.0000000   0.0000000   0.0000000  -0.0998933
+   20   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0615255   0.0000000  -0.0000000   0.0000000   0.0000000
+   21   0.4757743   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000
+   22   0.0000000   0.5507646   0.0000000   0.0000000  -0.0000000   0.0000000  -0.0541371   0.0000000   0.0000000  -0.0000000
+   23   0.0000000   0.0000000   0.5527668   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0615255   0.0000000
+   24   0.0000000   0.0000000   0.0000000   0.7122157  -0.0998933   0.0000000   0.0000000   0.0000000   0.0000000  -0.0383679
+   25   0.0000000  -0.0000000   0.0000000  -0.0998933   0.4658980   0.0000000   0.0000000   0.0000000   0.0000000   0.0955699
+   26   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.3703281   0.0000000  -0.0000000   0.0000000   0.0000000
+   27   0.0000000  -0.0541371   0.0000000   0.0000000   0.0000000   0.0000000   0.5152458   0.0000000   0.0000000   0.0000000
+   28   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000  -0.0000000   0.0000000   0.4442931   0.0000000   0.0000000
+   29   0.0000000   0.0000000   0.0615255   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.3703281   0.0000000
+   30   0.0000000  -0.0000000   0.0000000  -0.0383679   0.0955699   0.0000000   0.0000000   0.0000000   0.0000000   0.4658980
+   31   0.0615255   0.0000000  -0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000  -0.0000000   0.0000000
+   32   0.0000000   0.0073884   0.0000000  -0.0000000   0.0000000   0.0000000   0.0709527   0.0000000   0.0000000   0.0000000
+   33   0.0000000  -0.0000000   0.0000000  -0.0000000   0.0000000   0.0000000  -0.0000000   0.0000000   0.0000000   0.0000000
+   34   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000  -0.0000000   0.0000000   0.0000000
+   35   0.0000000  -0.0000000   0.0000000  -0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000
+   36   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000  -0.0000000   0.0000000   0.0000000   0.0000000   0.0000000
+   37  -0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000
+   38   0.0000000  -0.0000000   0.0000000  -0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000
+   39   0.0000000   0.0000000  -0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000  -0.0000000   0.0000000
+   40   0.0000000   0.0000000   0.0000000  -0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000
+
+          31          32          33          34          35          36          37          38          39          40
+
+    1   0.0000000   0.0000000  -0.0000000   0.0000000   0.0000000   0.0000000   0.0000000  -0.0000000   0.0000000   0.0000000
+    2   0.0000000   0.0000000   0.0000000  -0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000
+    3   0.0000000   0.0112477   0.0000000   0.0000000  -0.0000000   0.0000000   0.0000000  -0.0000000   0.0000000   0.0000000
+    4   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000  -0.0000000   0.0000000   0.0000000   0.0000000   0.0000000
+    5  -0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000  -0.0000000   0.0000000   0.0000000   0.0000000
+    6   0.0000000   0.0000000  -0.0000000   0.0000000   0.0000000   0.0000000   0.0000000  -0.0000000   0.0000000   0.0000000
+    7   0.0037402   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000  -0.0000000   0.0000000
+    8   0.0000000   0.0149879  -0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000  -0.0000000
+    9   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000  -0.0000000   0.0000000  -0.0000000
+   10   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000
+   11   0.0000000   0.0171886   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000  -0.0000000   0.0000000   0.0000000
+   12   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000
+   13  -0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000
+   14   0.0000000   0.0000000  -0.0000000   0.0000000  -0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000
+   15  -0.0183287   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000
+   16   0.0000000  -0.0011401  -0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000
+   17   0.0000000  -0.0541371  -0.0000000   0.0000000   0.0000000   0.0000000   0.0000000  -0.0000000   0.0000000  -0.0000000
+   18   0.0000000   0.0000000   0.0000000  -0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000
+   19   0.0000000   0.0000000  -0.0000000   0.0000000  -0.0000000   0.0000000   0.0000000  -0.0000000   0.0000000  -0.0000000
+   20   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000  -0.0000000   0.0000000   0.0000000   0.0000000   0.0000000
+   21   0.0615255   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000  -0.0000000   0.0000000   0.0000000   0.0000000
+   22   0.0000000   0.0073884  -0.0000000   0.0000000  -0.0000000   0.0000000   0.0000000  -0.0000000   0.0000000   0.0000000
+   23  -0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000  -0.0000000   0.0000000
+   24   0.0000000  -0.0000000  -0.0000000   0.0000000  -0.0000000   0.0000000   0.0000000  -0.0000000   0.0000000  -0.0000000
+   25   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000
+   26   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000  -0.0000000   0.0000000   0.0000000   0.0000000   0.0000000
+   27   0.0000000   0.0709527  -0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000
+   28   0.0000000   0.0000000   0.0000000  -0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000
+   29  -0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000  -0.0000000   0.0000000
+   30   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000
+   31   0.4442931   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000  -0.0000000   0.0000000   0.0000000   0.0000000
+   32   0.0000000   0.5152458   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000  -0.0000000   0.0000000   0.0000000
+   33   0.0000000   0.0000000   0.3218586   0.0000000   0.0000000   0.0000000   0.0000000   0.0346031   0.0000000  -0.0000000
+   34   0.0000000   0.0000000   0.0000000   0.2872555   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000
+   35   0.0000000   0.0000000   0.0000000   0.0000000   0.3910304   0.0000000   0.0000000  -0.0000000   0.0000000   0.0250414
+   36   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.3659890   0.0000000   0.0000000   0.0000000   0.0000000
+   37  -0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.2872555   0.0000000   0.0000000   0.0000000
+   38   0.0000000  -0.0000000   0.0346031   0.0000000  -0.0000000   0.0000000   0.0000000   0.3218586   0.0000000   0.0000000
+   39   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.3659890   0.0000000
+   40   0.0000000   0.0000000  -0.0000000   0.0000000   0.0250414   0.0000000   0.0000000   0.0000000   0.0000000   0.3910304
+```
diff --git a/Project#12/hints/hint2.md b/Project#12/hints/hint2.md
new file mode 100644
index 0000000..f14b3fc
--- /dev/null
+++ b/Project#12/hints/hint2.md
@@ -0,0 +1,15 @@
+Here is the CIS spin-adapted singlet matrix for the H<sub>2</sub>O STO-3G test case.  Note that the row/column ordering follows *i* and *a* combinations with *a* as the faster-running index (i.e., *a* indexes the inner loop).  Thus the first row/column corresponds to *i=0* and *a=0* and the second row/column to *i=0* and *a=1*.  Notice that the matrix is diagonally dominant --- that is, the largest element on any row/column combination is the diagonal entry --- and that all the diagonal elements are positive.
+```
+           1           2           3           4           5           6           7           8           9          10
+
+    1  19.9585039   0.0000000   0.0074145  -0.0000000  -0.0000000  -0.0076611   0.0180084  -0.0000000  -0.0000000   0.0000000
+    2   0.0000000  20.0113341  -0.0000000   0.0089205  -0.0076611  -0.0000000  -0.0000000   0.0037402   0.0000000  -0.0000000
+    3   0.0074145  -0.0000000   1.0931793   0.0000000  -0.0000000  -0.0867137   0.0485372  -0.0000000   0.0000000   0.0000000
+    4  -0.0000000   0.0089205   0.0000000   1.1888405  -0.0867137   0.0000000  -0.0000000  -0.0183287   0.0000000   0.0000000
+    5  -0.0000000  -0.0076611  -0.0000000  -0.0867137   0.4757743   0.0000000   0.0000000   0.0615255  -0.0000000   0.0000000
+    6  -0.0076611  -0.0000000  -0.0867137   0.0000000   0.0000000   0.5527668   0.0615255  -0.0000000   0.0000000  -0.0000000
+    7   0.0180084  -0.0000000   0.0485372  -0.0000000   0.0000000   0.0615255   0.3703281  -0.0000000   0.0000000  -0.0000000
+    8  -0.0000000   0.0037402  -0.0000000  -0.0183287   0.0615255  -0.0000000  -0.0000000   0.4442931  -0.0000000   0.0000000
+    9  -0.0000000   0.0000000   0.0000000   0.0000000  -0.0000000   0.0000000   0.0000000  -0.0000000   0.2872555   0.0000000
+   10   0.0000000  -0.0000000   0.0000000   0.0000000   0.0000000  -0.0000000  -0.0000000   0.0000000   0.0000000   0.3659890
+```
diff --git a/Project#12/hints/hint3.md b/Project#12/hints/hint3.md
new file mode 100644
index 0000000..13b0e43
--- /dev/null
+++ b/Project#12/hints/hint3.md
@@ -0,0 +1,666 @@
+Here is the TDHF/RPA version of the H<sub>2</sub>O STO-3G Hamitlonian.  The row/column ordering is the same as for CIS above, but there two blocks of *i/a* pairs, one for each of the ***A*** and ***B*** submatrices.  This matrix is also diagonally dominant, but not all the diagonal elements are positive.
+```
+           1           2           3           4           5           6           7           8           9          10
+
+    1  19.9847006   0.0000000   0.0000000   0.0000000   0.0000000   0.0261967   0.0000000  -0.0000000   0.0172405   0.0000000
+    2   0.0000000  19.9585039   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0074145
+    3   0.0000000   0.0000000  20.0308559   0.0000000   0.0000000  -0.0000000   0.0000000   0.0195218   0.0000000   0.0000000
+    4   0.0000000   0.0000000   0.0000000  20.0113341   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000  -0.0000000
+    5   0.0000000   0.0000000   0.0000000   0.0000000  19.9585039   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000
+    6   0.0261967   0.0000000  -0.0000000   0.0000000   0.0000000  19.9847006   0.0000000   0.0000000   0.0098259   0.0000000
+    7   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000  20.0113341   0.0000000   0.0000000   0.0000000
+    8  -0.0000000   0.0000000   0.0195218   0.0000000   0.0000000   0.0000000   0.0000000  20.0308559   0.0000000   0.0000000
+    9   0.0172405   0.0000000   0.0000000   0.0000000   0.0000000   0.0098259   0.0000000   0.0000000   1.1883321   0.0000000
+   10   0.0000000   0.0074145   0.0000000  -0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   1.0931793
+   11   0.0000000   0.0000000   0.0280239   0.0000000   0.0000000   0.0000000   0.0000000   0.0191033   0.0000000   0.0000000
+   12   0.0000000  -0.0000000   0.0000000   0.0089205   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000
+   13   0.0000000   0.0000000   0.0000000   0.0000000   0.0074145   0.0000000  -0.0000000   0.0000000   0.0000000   0.0000000
+   14   0.0098259   0.0000000   0.0000000   0.0000000   0.0000000   0.0172405   0.0000000   0.0000000   0.0951527   0.0000000
+   15   0.0000000   0.0000000   0.0000000   0.0000000  -0.0000000   0.0000000   0.0089205   0.0000000   0.0000000   0.0000000
+   16   0.0000000   0.0000000   0.0191033   0.0000000   0.0000000   0.0000000   0.0000000   0.0280239  -0.0000000   0.0000000
+   17  -0.0000000   0.0000000  -0.0046225   0.0000000   0.0000000  -0.0000000   0.0000000   0.0030386   0.0000000   0.0000000
+   18   0.0000000  -0.0000000   0.0000000  -0.0076611   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000  -0.0000000
+   19  -0.0134310   0.0000000  -0.0000000   0.0000000   0.0000000  -0.0057699   0.0000000   0.0000000  -0.0178319   0.0000000
+   20   0.0000000  -0.0076611   0.0000000  -0.0000000   0.0000000   0.0000000   0.0000000   0.0000000   0.0000000  -0.0867137
+   21   0.0000000   0.0000000   0.0000000   0.0000000  -0.0000000   0.0000000  -0.0076611   0.0000000   0.0000000   0.0000000
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+   76  -0.0000000  -0.0000000  -0.0000000  -0.0000000  -0.0000000  -0.3659890  -0.0000000  -0.0000000  -0.0000000  -0.0000000
+   77   0.0000000  -0.0000000  -0.0000000  -0.0000000  -0.0000000  -0.0000000  -0.2872555  -0.0000000  -0.0000000  -0.0000000
+   78  -0.0000000   0.0000000  -0.0346031  -0.0000000   0.0000000  -0.0000000  -0.0000000  -0.3218586  -0.0000000  -0.0000000
+   79  -0.0000000  -0.0000000  -0.0000000  -0.0000000  -0.0000000  -0.0000000  -0.0000000  -0.0000000  -0.3659890  -0.0000000
+   80  -0.0000000  -0.0000000   0.0000000  -0.0000000  -0.0250414  -0.0000000  -0.0000000  -0.0000000  -0.0000000  -0.3910304
+```

From 0f0fae634a942634f7cc79cc00e0146469b915e3 Mon Sep 17 00:00:00 2001
From: Kirk Pearce <kcpearce@vt.edu>
Date: Mon, 3 Apr 2017 11:48:04 -0400
Subject: [PATCH 35/41] Resize Project 12 eqns

---
 Project#12/README.md | 48 ++++++++++++++++++++++----------------------
 1 file changed, 24 insertions(+), 24 deletions(-)

diff --git a/Project#12/README.md b/Project#12/README.md
index 5f8cf54..1db4824 100644
--- a/Project#12/README.md
+++ b/Project#12/README.md
@@ -18,27 +18,27 @@ The fundamental idea behind CIS is the representation of the excited-state wave
 functions as linear combinations of singly excited determinants relative to the
 Hartree-Fock reference wave function, *viz.*
 
-<img src="./figures/singly-excited-determinant.png" height="60">
+<img src="./figures/singly-excited-determinant.png" height="50">
 
 where *m* identifies the various excited states, and we will use *i* and *j*
 (*a* and *b*) to denote occupied (unoccupied) spin-orbitals. Inserting this
 into the Schr<html>&ouml;</html>dinger equation and left-projecting onto a
 particular singly excited determinant gives
 
-<img src="./figures/excited-det-schrod-eqn.png" height="60">
+<img src="./figures/excited-det-schrod-eqn.png" height="50">
 
 If we recognize that we have one of these equations for every combination of
 *i* and *a* spin-orbitals, then this equation may be viewed as a matrix
 eigenvalue problem:
 
-<img src="./figures/matrix-eigenvalue-problem.png" height="60">
+<img src="./figures/matrix-eigenvalue-problem.png" height="30">
 
 To solve this equation, we need an expression for the matrix elements in terms
 of things we already know, i.e. Fock matrix elements and two-electron
 integrals.  This can be done using either algebraic or diagrammatic techniques
 to obtain (in the spin-orbital notation of previous projects):
 
-<img src="./figures/matrix-elements.png" height="60">
+<img src="./figures/matrix-elements.png" height="30">
 
 Our task is then relatively simple: Build the Hamiltonian matrix (expressed in
 the basis of all singly excited determinants) using the above expression and
@@ -72,7 +72,7 @@ from a simple two-electron/two-orbital example (such as the *1s 2s* excited
 state configuration of the He atom).  One can easily show that the four
 possible determinants arising from this configuration,
 
-<img src="./figures/four-possible-determinants.png" height="60">
+<img src="./figures/four-possible-determinants.png" height="50">
 
 are components of one singlet and one triplet in the following combinations:
 
@@ -92,7 +92,7 @@ expression and the equation for the CIS Hamiltonian matrix elements in the
 previous section, we may write a spin-factored equation for the
 <html>&alpha;</html> coefficients as
 
-<img src="./figures/pin-factored-eqn.png" height="60">
+<img src="./figures/spin-factored-eqn.png" height="30">
 
 Note that the mix-spin cases (where *j=*<html>&alpha;</html> and
 *b=*<html>&beta;</html> or *vice versa*) do not contribute since the Fock
@@ -101,11 +101,11 @@ carry out spin integration on the integrals in the above expression and assume
 that the <html>&alpha;</html> and <html>&beta;</html> CI coefficients are
 identical for the same spatial orbitals, i.e.,
 
-<img src="./figures/identical-ci-coeff.png" height="60">
+<img src="./figures/identical-ci-coeff.png" height="30">
 
 we obtain the <b><i>spatial orbital</i></b> expression
 
-<img src="./figures/spatial-orbital-expression.png" height="60">
+<img src="./figures/spatial-orbital-expression.png" height="50">
 
 The part in brackets above is an expression for the spatial-orbital CIS
 Hamiltonian, spin-adapted for singlet excited states, and diagonalization of
@@ -117,11 +117,11 @@ spin-orbital matrix earlier.
 How about the triplets?  We use exactly the same spin-factorization, but
 instead require
 
-<img src="./figures/inverse-ci-coeff.png" height="60">
+<img src="./figures/inverse-ci-coeff.png" height="30">
 
 This yields a slightly simpler Hamiltonian:
 
-<img src="./figures/simpler-hamiltonian.png" height="60">
+<img src="./figures/simpler-hamiltonian.png" height="50">
 
 which, upon diagonalization, will yield only the triplet eigenvalues (but each only occurring once) from your earlier diagonalziation.
 
@@ -148,16 +148,16 @@ TDHF/RPA wave function expansion in terms of orbital rotations instead of
 Slater determinants, but that's a discussion for another day.)  The TDHF/RPA
 eigenvalue equations take the form
 
-<img src="./figures/tdhf-eqn.png" height="60">
+<img src="./figures/tdhf-eqn.png" height="50">
 
 The definition of the ***A*** matrix is just the CIS matrix itself, *viz.*
 
-<img src="./figures/A-matrix.png" height="60">
+<img src="./figures/A-matrix.png" height="30">
 
 while **X** and **Y** are the parameters of single excitations and
 de-excitations, respectively, and the ***B*** matrix is simply
 
-<img src="./figures/B-matrix.png" height="60">
+<img src="./figures/B-matrix.png" height="30">
 
 Thus, the row/column dimension of the TDHF/RPA Hamiltonian is twice that of the
 CIS Hamiltonian, and the matrix is non-symmetric (so you must be careful about
@@ -174,28 +174,28 @@ Hamiltonian storage cost), one can rearrange the eigenvalue equations.  First
 write eigenvalue equation two separate equations, each in terms of the
 submatrices **A** and **B**:
 
-<img src="./figures/smarter-tdhf-1.png" height="60">
+<img src="./figures/smarter-tdhf-1.png" height="30">
 
 and
 
-<img src="./figures/smarter-tdhf-2.png" height="60">
+<img src="./figures/smarter-tdhf-2.png" height="30">
 
 Now take +/- combinations of these equations to obtain
 
-<img src="./figures/smarter-tdhf-3.png" height="60">
+<img src="./figures/smarter-tdhf-3.png" height="30">
 
 and
 
-<img src="./figures/smarter-tdhf-4.png" height="60">
+<img src="./figures/smarter-tdhf-4.png" height="30">
 
 Solve for ***(X+Y)*** in the second equation:
 
-<img src="./figures/smarter-tdhf-5.png" height="60">
+<img src="./figures/smarter-tdhf-5.png" height="30">
 
 Insert this result into the first equation, rearrange a bit, and finally
 obtain:
 
-<img src="./figures/smarter-tdhf-6.png" height="60">
+<img src="./figures/smarter-tdhf-6.png" height="30">
 
 This is an eigenvalue equation of the same dimension as the CIS eigenvalue
 equation (number of occupied orbitals times number of unoccupied orbitals),
@@ -208,12 +208,12 @@ approach, for all four test cases below.
 
 ## Test Cases
 The input structures, integrals, etc. for these examples may be found in the 
-[input directory](https://github.com/CrawfordGroup/ProgrammingProjects/tree/master/Project%2312/input).
+[input directory](./input).
 
 | Test Case | CIS | RPA (Method 1) | RPA (Method 2) |
 |-----------|-----|----------------|----------------|
-| STO-3G Water | [output](https://github.com/CrawfordGroup/ProgrammingProjects/tree/master/Project%2312/output/h2o/STO-3G/output_cis.txt) | [output](https://github.com/CrawfordGroup/ProgrammingProjects/tree/master/Project%2312/output/h2o/STO-3G/output_rpa1.txt) | [output](https://github.com/CrawfordGroup/ProgrammingProjects/tree/master/Project%2312/output/h2o/STO-3G/output_rpa2.txt) |
-| DZ Water | [output](https://github.com/CrawfordGroup/ProgrammingProjects/tree/master/Project%2312/output/h2o/DZ/output_cis.txt) | [output](https://github.com/CrawfordGroup/ProgrammingProjects/tree/master/Project%2312/output/h2o/DZ/output_rpa1.txt) | [output](https://github.com/CrawfordGroup/ProgrammingProjects/tree/master/Project%2312/output/h2o/DZ/output_rpa2.txt) |
-| DZP Water | [output](https://github.com/CrawfordGroup/ProgrammingProjects/tree/master/Project%2312/output/h2o/DZP/output_cis.txt) | [output](https://github.com/CrawfordGroup/ProgrammingProjects/tree/master/Project%2312/output/h2o/DZP/output_rpa1.txt) | [output](https://github.com/CrawfordGroup/ProgrammingProjects/tree/master/Project%2312/output/h2o/DZP/output_rpa2.txt) |
-| STO-3G Methane | [output](https://github.com/CrawfordGroup/ProgrammingProjects/tree/master/Project%2312/output/ch4/STO-3G/output_cis.txt) | [output](https://github.com/CrawfordGroup/ProgrammingProjects/tree/master/Project%2312/output/ch4/STO-3G/output_rpa1.txt) | [output](https://github.com/CrawfordGroup/ProgrammingProjects/tree/master/Project%2312/output/ch4/STO-3G/output_rpa2.txt) |
+| STO-3G Water | [output](./output/h2o/STO-3G/output_cis.txt) | [output](./output/h2o/STO-3G/output_rpa1.txt) | [output](./output/h2o/STO-3G/output_rpa2.txt) |
+| DZ Water | [output](./output/h2o/DZ/output_cis.txt) | [output](./output/h2o/DZ/output_rpa1.txt) | [output](./output/h2o/DZ/output_rpa2.txt) |
+| DZP Water | [output](./output/h2o/DZP/output_cis.txt) | [output](./output/h2o/DZP/output_rpa1.txt) | [output](./output/h2o/DZP/output_rpa2.txt) |
+| STO-3G Methane | [output](./output/ch4/STO-3G/output_cis.txt) | [output](./output/ch4/STO-3G/output_rpa1.txt) | [output](./output/ch4/STO-3G/output_rpa2.txt) |
 

From 42bb490393c9a3b0d295c32d9f09f38600c249a7 Mon Sep 17 00:00:00 2001
From: Kirk Pearce <kcpearce@vt.edu>
Date: Mon, 3 Apr 2017 11:55:14 -0400
Subject: [PATCH 36/41] Resize Project 12 eqns again

---
 Project#12/README.md                        |  26 +++++++++++---------
 Project#12/figures/singlet-combinations.png | Bin 0 -> 10696 bytes
 Project#12/figures/triplet-combinations.png | Bin 0 -> 14519 bytes
 3 files changed, 14 insertions(+), 12 deletions(-)
 create mode 100644 Project#12/figures/singlet-combinations.png
 create mode 100644 Project#12/figures/triplet-combinations.png

diff --git a/Project#12/README.md b/Project#12/README.md
index 1db4824..8c98391 100644
--- a/Project#12/README.md
+++ b/Project#12/README.md
@@ -31,7 +31,7 @@ If we recognize that we have one of these equations for every combination of
 *i* and *a* spin-orbitals, then this equation may be viewed as a matrix
 eigenvalue problem:
 
-<img src="./figures/matrix-eigenvalue-problem.png" height="30">
+<img src="./figures/matrix-eigenvalue-problem.png" height="25">
 
 To solve this equation, we need an expression for the matrix elements in terms
 of things we already know, i.e. Fock matrix elements and two-electron
@@ -76,7 +76,9 @@ possible determinants arising from this configuration,
 
 are components of one singlet and one triplet in the following combinations:
 
-<img src="./figures/singlet-triplet-combinations.png" height="60">
+<img src="./figures/triplet-combinations.png" height="60">
+
+<img src="./figures/singlet-combinations.png" height="60">
 
 where the superscript is the spin multiplicity (*2S+1*) and the subscript is
 the *M<sub>S</sub>* value of the wave function.  So, if we wanted to compute
@@ -89,10 +91,10 @@ wanted only triplets, we could require that the <html>&alpha;</html> and
 
 Let's begin with the singlets.  Starting from the spin-orbital eigenvalue
 expression and the equation for the CIS Hamiltonian matrix elements in the
-previous section, we may write a spin-factored equation for the
-<html>&alpha;</html> coefficients as
+previous section, we may write a spin-factored equation for the <html>&alpha;</html> 
+coefficients as
 
-<img src="./figures/spin-factored-eqn.png" height="30">
+<img src="./figures/spin-factored-eqn.png" height="60">
 
 Note that the mix-spin cases (where *j=*<html>&alpha;</html> and
 *b=*<html>&beta;</html> or *vice versa*) do not contribute since the Fock
@@ -157,7 +159,7 @@ The definition of the ***A*** matrix is just the CIS matrix itself, *viz.*
 while **X** and **Y** are the parameters of single excitations and
 de-excitations, respectively, and the ***B*** matrix is simply
 
-<img src="./figures/B-matrix.png" height="30">
+<img src="./figures/B-matrix.png" height="25">
 
 Thus, the row/column dimension of the TDHF/RPA Hamiltonian is twice that of the
 CIS Hamiltonian, and the matrix is non-symmetric (so you must be careful about
@@ -174,28 +176,28 @@ Hamiltonian storage cost), one can rearrange the eigenvalue equations.  First
 write eigenvalue equation two separate equations, each in terms of the
 submatrices **A** and **B**:
 
-<img src="./figures/smarter-tdhf-1.png" height="30">
+<img src="./figures/smarter-tdhf-1.png" height="25">
 
 and
 
-<img src="./figures/smarter-tdhf-2.png" height="30">
+<img src="./figures/smarter-tdhf-2.png" height="25">
 
 Now take +/- combinations of these equations to obtain
 
-<img src="./figures/smarter-tdhf-3.png" height="30">
+<img src="./figures/smarter-tdhf-3.png" height="25">
 
 and
 
-<img src="./figures/smarter-tdhf-4.png" height="30">
+<img src="./figures/smarter-tdhf-4.png" height="25">
 
 Solve for ***(X+Y)*** in the second equation:
 
-<img src="./figures/smarter-tdhf-5.png" height="30">
+<img src="./figures/smarter-tdhf-5.png" height="25">
 
 Insert this result into the first equation, rearrange a bit, and finally
 obtain:
 
-<img src="./figures/smarter-tdhf-6.png" height="30">
+<img src="./figures/smarter-tdhf-6.png" height="25">
 
 This is an eigenvalue equation of the same dimension as the CIS eigenvalue
 equation (number of occupied orbitals times number of unoccupied orbitals),
diff --git a/Project#12/figures/singlet-combinations.png b/Project#12/figures/singlet-combinations.png
new file mode 100644
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From e16b3df176c9de7e4fb1008bc35c3ffe9f9cb1ce Mon Sep 17 00:00:00 2001
From: Kirk Pearce <kcpearce@vt.edu>
Date: Mon, 3 Apr 2017 11:59:39 -0400
Subject: [PATCH 37/41] Resize Project 12 eqns again

---
 Project#12/README.md | 24 ++++++++++++------------
 1 file changed, 12 insertions(+), 12 deletions(-)

diff --git a/Project#12/README.md b/Project#12/README.md
index 8c98391..c0e76cd 100644
--- a/Project#12/README.md
+++ b/Project#12/README.md
@@ -76,25 +76,25 @@ possible determinants arising from this configuration,
 
 are components of one singlet and one triplet in the following combinations:
 
-<img src="./figures/triplet-combinations.png" height="60">
+<img src="./figures/triplet-combinations.png" height="50">
 
-<img src="./figures/singlet-combinations.png" height="60">
+<img src="./figures/singlet-combinations.png" height="50">
 
 where the superscript is the spin multiplicity (*2S+1*) and the subscript is
 the *M<sub>S</sub>* value of the wave function.  So, if we wanted to compute
 only the eigenvalues and eigenfunctions corresponding spin singlets in our CIS
-calculation, we could introduce the restriction on our CIS coefficients that
-<html>&alpha;</html> and <html>&beta;</html> excitations involving the same
-*spatial* orbitals must be identical (including the sign).  Similarly, if we
-wanted only triplets, we could require that the <html>&alpha;</html> and
-<html>&beta;</html> excitations have the same magnitude but opposite signs.
+calculation, we could introduce the restriction on our CIS coefficients that <html>&alpha;</html> 
+and <html>&beta;</html> excitations involving the same <i>spatial</i> orbitals 
+must be identical (including the sign).  Similarly, if we wanted only triplets, 
+we could require that the <html>&alpha;</html> and <html>&beta;</html> excitations 
+have the same magnitude but opposite signs.
 
 Let's begin with the singlets.  Starting from the spin-orbital eigenvalue
 expression and the equation for the CIS Hamiltonian matrix elements in the
 previous section, we may write a spin-factored equation for the <html>&alpha;</html> 
 coefficients as
 
-<img src="./figures/spin-factored-eqn.png" height="60">
+<img src="./figures/spin-factored-eqn.png" height="55">
 
 Note that the mix-spin cases (where *j=*<html>&alpha;</html> and
 *b=*<html>&beta;</html> or *vice versa*) do not contribute since the Fock
@@ -176,19 +176,19 @@ Hamiltonian storage cost), one can rearrange the eigenvalue equations.  First
 write eigenvalue equation two separate equations, each in terms of the
 submatrices **A** and **B**:
 
-<img src="./figures/smarter-tdhf-1.png" height="25">
+<img src="./figures/smarter-tdhf-1.png" height="22.5">
 
 and
 
-<img src="./figures/smarter-tdhf-2.png" height="25">
+<img src="./figures/smarter-tdhf-2.png" height="22.5">
 
 Now take +/- combinations of these equations to obtain
 
-<img src="./figures/smarter-tdhf-3.png" height="25">
+<img src="./figures/smarter-tdhf-3.png" height="22.5">
 
 and
 
-<img src="./figures/smarter-tdhf-4.png" height="25">
+<img src="./figures/smarter-tdhf-4.png" height="22.5">
 
 Solve for ***(X+Y)*** in the second equation:
 

From efd34cf097e570e84a6dbc7da0e1e6b9fcfd1b4b Mon Sep 17 00:00:00 2001
From: Kirk Pearce <kcpearce@vt.edu>
Date: Mon, 3 Apr 2017 13:09:52 -0400
Subject: [PATCH 38/41] Add Project 13 eqns

---
 .../figures/guess-vector-hamiltonian.png      | Bin 0 -> 8534 bytes
 Project#13/README.md                          |  48 ++++++------------
 Project#13/figures/coefficients.png           | Bin 0 -> 5403 bytes
 Project#13/figures/correction-vectors.png     | Bin 0 -> 7752 bytes
 .../figures/diag-subspace-hamiltonian.png     | Bin 0 -> 5914 bytes
 Project#13/figures/final-eigenvectors.png     | Bin 0 -> 5399 bytes
 .../figures/guess-vector-hamiltonian.png      | Bin 0 -> 8534 bytes
 Project#13/figures/residual-vectors.png       | Bin 0 -> 7505 bytes
 Project#13/figures/spin-adapted-cis-eqn.png   | Bin 0 -> 15434 bytes
 9 files changed, 15 insertions(+), 33 deletions(-)
 create mode 100644 Project#12/figures/guess-vector-hamiltonian.png
 create mode 100644 Project#13/figures/coefficients.png
 create mode 100644 Project#13/figures/correction-vectors.png
 create mode 100644 Project#13/figures/diag-subspace-hamiltonian.png
 create mode 100644 Project#13/figures/final-eigenvectors.png
 create mode 100644 Project#13/figures/guess-vector-hamiltonian.png
 create mode 100644 Project#13/figures/residual-vectors.png
 create mode 100644 Project#13/figures/spin-adapted-cis-eqn.png

diff --git a/Project#12/figures/guess-vector-hamiltonian.png b/Project#12/figures/guess-vector-hamiltonian.png
new file mode 100644
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diff --git a/Project#13/README.md b/Project#13/README.md
index 1b5532e..35ca464 100644
--- a/Project#13/README.md
+++ b/Project#13/README.md
@@ -9,7 +9,7 @@ algorithm to allow computation of several eigenvalues simultaneously rather
 than one at a time<sup id="r2">[2](#f2)</sup>. The
 purpose of this project is to illustrate the use of what is now called the
 Davidson-Liu algorithm in the context of a 
-[CIS computation](https://github.com/CrawfordGroup/ProgrammingProjects/tree/master/Project%2312).
+[CIS computation](../Project%2312).
 
 ## The Basic Algorithm
 
@@ -29,26 +29,19 @@ from a well-chosen subspace of the full determinantal space.
 
 Compute a representation of the Hamiltonian within the space of guess vectors,
 
-```
-EQUATION
-G_{ij} \equiv \langle {\mathbf b}_i | H | {\mathbf b}_j \rangle = 
-\langle {\mathbf b}_i | H {\mathbf b}_j \rangle = \langle {\mathbf b}_i | {\mathbf \sigma}_j \rangle,\ \ 1\leq i,j \leq L
-```
+<img src="./figures/guess-vector-hamiltonian.png" height="50">
+
 and then diagonalize this so-called "subspace Hamiltonian",
 
-```
-EQUATION
-{\mathbf G} \alpha^k = \lambda^k \alpha^k,\ \ k=1,2,\cdots,M
-```
+<img src="./figures/diag-subspace-hamiltonian.png" height="50">
+
 where *M* is the number of roots of interest. The current estimate of each of
 the *M* eigenvectors we want is a linear combination of the guess vectors,
 with the &alpha;<sup>k</sup> subspace eigenvectors providing the
 coefficients, *viz.*
 
-```
-EQUATION
-{\mathbf c}^k = \sum_i^L \alpha_i^k {\mathbf b}_i.
-```
+<img src="./figures/coefficients.png" height="50">
+
 The dimension of ***G*** is typically very small (perhaps a dozen times the
 number of guess vectors, *L*), so one can used a standard diagonalization
 package (such as DSYEV in LAPACK) for this task.  Note that the most expensive
@@ -62,16 +55,12 @@ elements must be computed "on the fly" during the computation of each
 
 Build a set of "correction vectors",
 
-```
-EQUATION
-\delta^k_I \equiv \left(\lambda^k - H_{II}\right)^{-1} r_I^k, \ \ I=1,2,\cdots,N
-```
+<img src="./figures/correction-vectors.png" height="50">
+
 where the "residual" vectors are defined as
 
-```
-EQUATION
-{\mathbf r}^k \equiv \sum_{i=1}^L \alpha^k_i\left( {\mathbf H} - \lambda^k \right) {\mathbf b}_i,
-```
+<img src="./figures/residual-vectors.png" height="50">
+
 and *N* is the dimension of the Hamiltonian (i.e. the number of determinants).
 The inverse appearing in the definition of the correction vectors is commonly
 referred to as the "preconditioner". Notice that the residual vectors are so
@@ -93,12 +82,9 @@ Return to step #2 and continue.
 
 We will focus on the spin-adapted singlet formulation of CIS, 
 for which the <b>&sigma;</b> = <b>H c</b>equation was given in 
-[Project 12](https://github.com/CrawfordGroup/ProgrammingProjects/tree/master/Project%2312):
+[Project 12](../Project%2312):
 
-```
-EQUATION
-\sigma(m)_{ia} = \sum_{jb} H_{ia,jb} c_j^b(m) = \sum_{jb} \left[ f_{ab} \delta_{ij} - f_{ij} \delta_{ab} + 2 <aj|ib> - <aj|bi> \right] c_j^b(m).
-```
+<img src="./figures/spin-adapted-cis-eqn.png" height="50">
 
 ## Unit Guess Vectors
 
@@ -106,7 +92,7 @@ What should we choose for guess vectors?  As noted above, the simplest choice
 is probably a set of unit vectors, one for each eigenvalue you want.  But in
 what position of the vector should we put the 1?  For a hint, look at the
 structure of the 
-[spin-adapted singlet CIS Hamiltonian](https://github.com/CrawfordGroup/ProgrammingProjects/tree/master/Project%2312/hints/hint1.2)  
+[spin-adapted singlet CIS Hamiltonian](../Project%2312/hints/hint2.md)  
 for the H<sub>2</sub>O STO-3G test case and note that it is
 strongly diagonally dominant.  Thus, if the diagonal elements are reasonable
 approximations to the true eigenvalues, and we want to compute only the lowest
@@ -120,11 +106,7 @@ dimension to something more manageable before continuing the Davidson-Liu
 algorithm.  A typical choice is to collapse to the current best set of guesses
 using the equation given above for the current final eigenvectors:
 
-```
-EQUATION
-{\mathbf c}^k = \sum_i^L \alpha_i^k {\mathbf b}_i.
-```
-
+<img src="./figures/final-eigenvectors.png" height="50">
 
 #### References
  - <b id="f1">1</b>: E.R. Davidson, "The iterative calculation of a few of the lowest eigenvalues and corresponding eigenvectors of large real-symmetric matrices," *J. Comput. Phys.* **17**, 87 (1975).[up](#r1)
diff --git a/Project#13/figures/coefficients.png b/Project#13/figures/coefficients.png
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From 66f5b856e43f2b129ef05cbdf44c49a5997758da Mon Sep 17 00:00:00 2001
From: Kirk Pearce <kcpearce@vt.edu>
Date: Mon, 3 Apr 2017 13:11:49 -0400
Subject: [PATCH 39/41] Resize Project 13 eqns

---
 Project#13/README.md | 12 ++++++------
 1 file changed, 6 insertions(+), 6 deletions(-)

diff --git a/Project#13/README.md b/Project#13/README.md
index 35ca464..5bc711d 100644
--- a/Project#13/README.md
+++ b/Project#13/README.md
@@ -29,18 +29,18 @@ from a well-chosen subspace of the full determinantal space.
 
 Compute a representation of the Hamiltonian within the space of guess vectors,
 
-<img src="./figures/guess-vector-hamiltonian.png" height="50">
+<img src="./figures/guess-vector-hamiltonian.png" height="25">
 
 and then diagonalize this so-called "subspace Hamiltonian",
 
-<img src="./figures/diag-subspace-hamiltonian.png" height="50">
+<img src="./figures/diag-subspace-hamiltonian.png" height="25">
 
 where *M* is the number of roots of interest. The current estimate of each of
 the *M* eigenvectors we want is a linear combination of the guess vectors,
 with the &alpha;<sup>k</sup> subspace eigenvectors providing the
 coefficients, *viz.*
 
-<img src="./figures/coefficients.png" height="50">
+<img src="./figures/coefficients.png" height="60">
 
 The dimension of ***G*** is typically very small (perhaps a dozen times the
 number of guess vectors, *L*), so one can used a standard diagonalization
@@ -55,11 +55,11 @@ elements must be computed "on the fly" during the computation of each
 
 Build a set of "correction vectors",
 
-<img src="./figures/correction-vectors.png" height="50">
+<img src="./figures/correction-vectors.png" height="30">
 
 where the "residual" vectors are defined as
 
-<img src="./figures/residual-vectors.png" height="50">
+<img src="./figures/residual-vectors.png" height="60">
 
 and *N* is the dimension of the Hamiltonian (i.e. the number of determinants).
 The inverse appearing in the definition of the correction vectors is commonly
@@ -106,7 +106,7 @@ dimension to something more manageable before continuing the Davidson-Liu
 algorithm.  A typical choice is to collapse to the current best set of guesses
 using the equation given above for the current final eigenvectors:
 
-<img src="./figures/final-eigenvectors.png" height="50">
+<img src="./figures/final-eigenvectors.png" height="60">
 
 #### References
  - <b id="f1">1</b>: E.R. Davidson, "The iterative calculation of a few of the lowest eigenvalues and corresponding eigenvectors of large real-symmetric matrices," *J. Comput. Phys.* **17**, 87 (1975).[up](#r1)

From 58f1752eaeb1732c2b89144a1fe62024bae009f9 Mon Sep 17 00:00:00 2001
From: Kirk Pearce <kcpearce@vt.edu>
Date: Mon, 3 Apr 2017 13:13:28 -0400
Subject: [PATCH 40/41] Resize Project 13 eqns again

---
 Project#13/README.md | 8 ++++----
 1 file changed, 4 insertions(+), 4 deletions(-)

diff --git a/Project#13/README.md b/Project#13/README.md
index 5bc711d..2402dbb 100644
--- a/Project#13/README.md
+++ b/Project#13/README.md
@@ -29,7 +29,7 @@ from a well-chosen subspace of the full determinantal space.
 
 Compute a representation of the Hamiltonian within the space of guess vectors,
 
-<img src="./figures/guess-vector-hamiltonian.png" height="25">
+<img src="./figures/guess-vector-hamiltonian.png" height="22.5">
 
 and then diagonalize this so-called "subspace Hamiltonian",
 
@@ -109,6 +109,6 @@ using the equation given above for the current final eigenvectors:
 <img src="./figures/final-eigenvectors.png" height="60">
 
 #### References
- - <b id="f1">1</b>: E.R. Davidson, "The iterative calculation of a few of the lowest eigenvalues and corresponding eigenvectors of large real-symmetric matrices," *J. Comput. Phys.* **17**, 87 (1975).[up](#r1)
- - <b id="f2">2</b>: B.Liu, "The simultaneous expansion method for the iterative solution of several of the lowest eigenvalues and corresponding eigenvectors of large real-symmetric matrices," Technical Report LBL-8158, Lawrence Berkeley Laboratory, University of California, Berkeley, 1978.[up](#r2)
- - <b id="f3">3</b>: C.D. Sherrill, "The Configuration Interaction Method: Advances in Highly Correlated Approaches," *Adv. Quantum Chem.* **34**, 143 (1998). [up](#r3)
+ - <b id="f1">1</b>: E.R. Davidson, "The iterative calculation of a few of the lowest eigenvalues and corresponding eigenvectors of large real-symmetric matrices," *J. Comput. Phys.* **17**, 87 (1975). [(return to text)](#r1)
+ - <b id="f2">2</b>: B.Liu, "The simultaneous expansion method for the iterative solution of several of the lowest eigenvalues and corresponding eigenvectors of large real-symmetric matrices," Technical Report LBL-8158, Lawrence Berkeley Laboratory, University of California, Berkeley, 1978. [(return to text)](#r2)
+ - <b id="f3">3</b>: C.D. Sherrill, "The Configuration Interaction Method: Advances in Highly Correlated Approaches," *Adv. Quantum Chem.* **34**, 143 (1998). [(return to text)](#r3)

From 12b5a00a4d9c643c707ec9dcdf806ed9ab3cf9f9 Mon Sep 17 00:00:00 2001
From: Kirk Pearce <kcpearce@vt.edu>
Date: Mon, 3 Apr 2017 13:14:49 -0400
Subject: [PATCH 41/41] Resize Project 13 eqns again again

---
 Project#13/README.md | 5 ++---
 1 file changed, 2 insertions(+), 3 deletions(-)

diff --git a/Project#13/README.md b/Project#13/README.md
index 2402dbb..27d1be7 100644
--- a/Project#13/README.md
+++ b/Project#13/README.md
@@ -81,7 +81,7 @@ Return to step #2 and continue.
 ## CIS Sigma Equation
 
 We will focus on the spin-adapted singlet formulation of CIS, 
-for which the <b>&sigma;</b> = <b>H c</b>equation was given in 
+for which the <b>&sigma;</b> = <b>H c</b> equation was given in 
 [Project 12](../Project%2312):
 
 <img src="./figures/spin-adapted-cis-eqn.png" height="50">
@@ -91,8 +91,7 @@ for which the <b>&sigma;</b> = <b>H c</b>equation was given in
 What should we choose for guess vectors?  As noted above, the simplest choice
 is probably a set of unit vectors, one for each eigenvalue you want.  But in
 what position of the vector should we put the 1?  For a hint, look at the
-structure of the 
-[spin-adapted singlet CIS Hamiltonian](../Project%2312/hints/hint2.md)  
+structure of the [spin-adapted singlet CIS Hamiltonian](../Project%2312/hints/hint2.md)  
 for the H<sub>2</sub>O STO-3G test case and note that it is
 strongly diagonally dominant.  Thus, if the diagonal elements are reasonable
 approximations to the true eigenvalues, and we want to compute only the lowest