Merge pull request CrawfordGroup#13 from kcpearce/add-hints

Finish Early Projects

Author: amjames

.DS_Store

@@ -27,7 +27,7 @@ After downloading the file to your computer (to a file called “geom.dat”, fo
 ## Step 2: Bond Lengths
 Calculate the interatomic distances using the expression:
 
-<img src="./figures/distances.png" width="400">
+<img src="./figures/distances.png" height="40">
 
 where x, y, and z are Cartesian coordinates and i and j denote atomic indices.
 
@@ -40,11 +40,11 @@ where x, y, and z are Cartesian coordinates and i and j denote atomic indices.
 ## Step 3: Bond Angles
 Calculate all possible bond angles. For example, the angle, &phi;<sub>ijk</sub>, between atoms i-j-k, where j is the central atom is given by:
 
-<img src="./figures/bond-angle.png" width="150">
+<img src="./figures/bond-angle.png" height="25">
 
-where the e&#8407;<sub>ij</sub> are unit vectors between the atoms, e.g.,
+where the e<sub>ij</sub> are unit vectors between the atoms, e.g.,
 
-<img src="./figures/unit-vectors.png" width="600">
+<img src="./figures/unit-vectors.png" height="30">
 
 - [Hint 1](./hints/hint3-1.md): Memory allocation for the unit vectors
 - [Hint 2](./hints/hint3-2.md): Avoiding a divide-by-zero
@@ -56,7 +56,7 @@ where the e&#8407;<sub>ij</sub> are unit vectors between the atoms, e.g.,
 ## Step 4: Out-of-Plane Angles
 Calculate all possible out-of-plane angles. For example, the angle &theta;<sub>ijkl</sub> for atom i out of the plane containing atoms j-k-l (with k as the central atom, connected to i) is given by:
 
-<img src="./figures/oop-angle.png" width="200">
+<img src="./figures/oop-angle.png" height="60">
 
 - [Hint 1](./hints/hint4-1.md): Memory allocation?
 - [Hint 2](./hints/hint4-2.md): Cross products
@@ -67,7 +67,7 @@ Calculate all possible out-of-plane angles. For example, the angle &theta;<sub>i
 ## Step 5: Torsion/Dihedral Angles
 Calculate all possible torsional angles. For example, the torsional angle &tau;<sub>ijkl</sub> for the atom connectivity i-j-k-l is given by:
 
-<img src="./figures/torsion-angle.png" width="250">
+<img src="./figures/torsion-angle.png" height="60">
 
 Can you also determine the sign of the torsional angle?
 

Project#01/.DS_Store

@@ -1,96 +1,72 @@
 # Project #2: Harmonic Vibrational Analysis
 
-The purpose of this project is to extend your fundamental C-language programming techniques through a normal coordinate/harmonic vibrational frequency calculation. The theoretical background and a concise set of instructions for this project may be found [here](https://github.com/CrawfordGroup/ProgrammingProjects/blob/master/Project%2302/project2-instructions.pdf)
+The purpose of this project is to extend your fundamental C-language programming techniques through a normal coordinate/harmonic vibrational frequency calculation. The theoretical background and a concise set of instructions for this project may be found [here](./project2-instructions.pdf)
 
 We thank Dr. Yukio Yamaguchi of the University of Georgia for the original version of this project.
 
 ## Step 1: Read the Coordinate Data
 
-The coordinate data are given in a format identical to that for [Project#1](https://github.com/CrawfordGroup/ProgrammingProjects/blob/master/Project%2301). The test case
+The coordinate data are given in a format identical to that for [Project #1](../Project%2301). The test case
 for the remainder of this project is the water molecule, optimized at the SCF/DZP level of theory. You can find the coordinates (in bohr) in the input directory.
 
 ## Step 2: Read the Cartesian Hessian Data
 
 The primary input data for the harmonic vibrational calculation is the Hessian matrix,
 which consists of second derivatives of the energy with respect to atomic positions.
 
-```
-EQUATION
-F_{ij} = \frac{\partial^{2}V}{\partial q_i \partial q_j}
-```
-The Hessian matrix (in units of Eh/a02) can be downloaded here for the H2O test case. 
+<img src="./figures/hessian.png" height="50">
+
+The Hessian matrix (in units of E<sub>h</sub>/a<sub>0</sub><sup>2</sup>) can be downloaded [here](./input/h2o_hessian.txt) for the H<sub>2</sub>O test case. 
 The first integer in the file is the number of atoms (which you should compare to the corresponding value from the geometry file as a test of consistency), 
 while the remaining values have the following format:
 
-```
-EQUATION
-\begin{array}{ccc}
-F_{x_1,x_1} & F_{x_1,y_1} & F_{x_1,z_1} \\
-F_{x_1,x_2} & F_{x_1,y_2} & F_{x_1,z_2} \\
-& \\
-\ldots  & \ldots & \dots \\
-& \\
-F_{x_2,x_1} & F_{x_2,y_1} & F_{x_2,z_1} \\
-& \\
-\ldots & \ldots & \ldots \\
-\end{array}
-```
-
- * Hint 1: Reading the Hessian
+<img src="./figures/hessian-file-format.png" width="200">
+
+ * [Hint 1](./hints/hint1.md): Reading the Hessian
 
 ## Step 3: Mass-Weight the Hessian Matrix
 
 Divide each element of the Hessian matrix by the product of square-roots of the masses of the atoms associated with the given coordinates:
 
-```
-EQUATION
-F^{M}_{ij} = \frac{F_{ij}}{\sqrt{m_i m_j}}
+<img src="./figures/mass-weighted-hessian.png" height="50">
 
-```
 where m<sub>i</sub> represents the mass of the atom corresponding to atom *i*. Use atomic mass units (amu) for the masses, just as 
-for [Project #1](https://github.com/CrawfordGroup/ProgrammingProjects/blob/master/Project%2301).
+for [Project #1](../Project%2301).
 
- * Hint 2: Solution
+ * [Hint 2](./hints/hint2.md): Solution
 
 ## Step 4: Diagonalize the Mass-Weighted Hessian Matrix
 
 Compute the eigenvalues of the mass-weighted Hessian:
 
-```
- EQUATION
-
- F^{M}\mathbf{L} = \mathbf{L}\Lambda
-```
+<img src="./figures/diag-mass-weighted-hessian.png" height="20">
 
 You should consider using the same canned diagonalization function 
-you used in [Project #1](https://github.com/CrawfordGroup/ProgrammingProjects/blob/master/Project%2301).
+you used in [Project #1](../Project%2301).
 
- * Hint 3: Solution
+ * [Hint 3](./hints/hint3.md): Solution
 
 ## Step 5: Compute the Harmonic Vibrational Frequencies
 
 The vibrational frequencies are proportional to the squareroot of the eigenvalues of the mass-weighted Hessian:
 
-```
-EQUATION
-\omega_{i}= {\rm_constant} \times \sqrt{\lambda_i}
-```
+<img src="./figures/vib-freq.png" height="25">
 
 The most common units to use for vibrational frequencies is cm<sup>-1</sup>.
 
- * Hint 4: Solution
+ * [Hint 4](./hints/hint4.md): Solution
 
 ## Reference
-E.B. Willson, J.C. Decius, and P.C. Cross, *Molecular Vibrations*, McGraw-Hill, 1955.
+E.B. Willson, J.C. Decius, and P.C. Cross, __Molecular Vibrations__, McGraw-Hill, 1955.
 
 ## Test Cases
 
- * Water: [Coordinates](https://github.com/CrawfordGroup/ProgrammingProjects/blob/master/Project%2302/input/h2o_geom.txt) 
- | [Hessian](https://github.com/CrawfordGroup/ProgrammingProjects/blob/master/Project%2302/input/h2o_hessian.txt) 
- | [Output](https://github.com/CrawfordGroup/ProgrammingProjects/blob/master/Project%2302/output/h2o_vib_out.txt)
- * Benzene: [Coordinates](https://github.com/CrawfordGroup/ProgrammingProjects/blob/master/Project%2302/input/benzene_geom.txt) 
- | [Hessian](https://github.com/CrawfordGroup/ProgrammingProjects/blob/master/Project%2302/input/benzene_hessian.txt) 
- | [Output](https://github.com/CrawfordGroup/ProgrammingProjects/blob/master/Project%2302/output/benzene_vib_out.txt)
- * 3-chloro-1-butene: [Coordinates](https://github.com/CrawfordGroup/ProgrammingProjects/blob/master/Project%2302/input/3c1b_geom.txt) 
- | [Hessian](https://github.com/CrawfordGroup/ProgrammingProjects/blob/master/Project%2302/input/3c1b_hessian.txt) 
- | [Output](https://github.com/CrawfordGroup/ProgrammingProjects/blob/master/Project%2302/input/3c1b_vib_out.txt)
+ * Water: [Coordinates](./input/h2o_geom.txt) 
+ | [Hessian](./input/h2o_hessian.txt) 
+ | [Output](./output/h2o_vib_out.txt) (see the note at the bottom of [Hint 4](./hints/hint4.md))
+ * Benzene: [Coordinates](./input/benzene_geom.txt) 
+ | [Hessian](./input/benzene_hessian.txt) 
+ | [Output](./output/benzene_vib_out.txt)
+ * 3-chloro-1-butene: [Coordinates](./input/3c1b_geom.txt) 
+ | [Hessian](./input/3c1b_hessian.txt) 
+ | [Output](./output/3c1b_vib_out.txt)

Project#01/README.md

@@ -0,0 +1,17 @@
+The Hessian stored in memory should be a square matrix, while the format of the input file is rectangular.  Understanding the translation between the two takes a bit of thinking.  Here's a partial code block that works for this purpose:
+```c++
+  ...
+  FILE *hessian;
+  int natom;
+  ...
+  
+  double **H = new double* [natom*3];
+  for(int i=0; i < natom*3; i++)
+    H[i] = new double[natom*3];
+ 
+  for(int i=0; i < natom*3; i++) {
+    for(int j=0; j < natom; j++) {
+      fscanf(hessian, "%lf %lf %lf", &H[i][3*j], &H[i][3*j+1], &H[i][3*j+2]);
+    }
+  }
+```

Project#02/README.md

@@ -0,0 +1,14 @@
+For the H<sub>2</sub>O test case, the 9×9 mass-weighted Hessian is:
+```
+           1           2           3           4           5           6           7           8           9
+
+    1   0.0057996   0.0000000   0.0000000  -0.0115523   0.0000000   0.0000000  -0.0115523   0.0000000   0.0000000
+    2   0.0000000   0.0198271   0.0000000   0.0000000  -0.0394937   0.0199304   0.0000000  -0.0394937  -0.0199304
+    3   0.0000000   0.0000000   0.0175112   0.0000000   0.0086617  -0.0348807   0.0000000  -0.0086617  -0.0348807
+    4  -0.0115523   0.0000000   0.0000000   0.0510672   0.0000000   0.0000000  -0.0050452   0.0000000   0.0000000
+    5   0.0000000  -0.0394937   0.0086617   0.0000000   0.1716643  -0.0569527   0.0000000  -0.0143291   0.0224462
+    6   0.0000000   0.0199304  -0.0348807   0.0000000  -0.0569527   0.1258526   0.0000000  -0.0224462   0.0131055
+    7  -0.0115523   0.0000000   0.0000000  -0.0050452   0.0000000   0.0000000   0.0510672   0.0000000   0.0000000
+    8   0.0000000  -0.0394937  -0.0086617   0.0000000  -0.0143291  -0.0224462   0.0000000   0.1716643   0.0569527
+    9   0.0000000  -0.0199304  -0.0348807   0.0000000   0.0224462   0.0131055   0.0000000   0.0569527   0.1258526
+```

Project#02/figures/.DS_Store

@@ -0,0 +1,12 @@
+The eigenvalues of the mass-weighted Hessian for the H<sub>2</sub>O test case (in atomic units) are:
+```
+   8         0.2351542439
+   7         0.2107113210
+   6         0.1317512832
+   5         0.0561123974
+   4         0.0547551476
+   3         0.0518216614
+   2         0.0000000000
+   1         0.0000000000
+   0         0.0000000000
+```

Project#02/figures/diag-mass-weighted-hessian.png

@@ -0,0 +1,13 @@
+The harmonic vibrational frequencies for the H<sub>2</sub>O test case (in cm<sup>-1</sup>) are:
+```
+   8 2492.7600
+   7 2359.6522
+   6 1865.8704
+   5 1217.6808
+   4 1202.8640
+   3 1170.1990
+   2    0.0210
+   1    0.0000
+   0    0.0000
+```
+Note that there are only three zero frequencies in this case when there should be six. This is because the structure used in the computation is not a stationary point on the potential energy surface, and thus the three “rotational frequencies” are non-zero.