General notes: You will have to write codes to complete this assignment. While I would prefer it if you wrote the codes using fortran 90 or 77, you are welcome to use whichever language you would like. Please send me electronic copies of all programs you write with instructions regarding how they should be compiled.
Question 1. The electronic density of states, N(E) describes the distribution of electronic energy levels as a function of energy. A plot of the density of states often exhibits bands – groups of states at similar energies that are separated by large regions of energy in which no states reside. These bands can correspond to either occupied states (e.g. valence bands) or unoccupied states (e.g. conduction bands). Integrating the density of states over the range of energies containing a band tells us how many electrons are within that band (for occupied state) or could be placed within that band (for unoccupied states).
A set of data corresponding to the density of states of Al2O3 is provided in Al2O3_DOS.txt. Use it to do the following:
Note you should identify three bands for this system. Two of these will be at energies below 0 eV and contain occupied states. The third band, at energies above 0 eV, contains unoccupied states.
Question 2. In class, we discussed linear regression in the context of fitting data to polynomials. However, this procedure can be applied to cases where the data represent forms that are not best described by a polynomial.
A set of (x,y) data corresponding to a Gaussian function ( y c= exp(−α(x x− 0)2) is provided the file gaussian.dat. Use it to:
Question 3. n-heptane has a large number of degrees of freedom that correspond to rotations about C-C bonds. This molecule is shown below, with rotatable bonds of interest to this question labeled 1 through 4.
CH3 CH2 1 CH2 2 CH2 3 CH2 4 CH2 CH3
The energy of the system as a can be described, in kJ/mol, in terms of these torsions as:
E= 4.184⎡⎢⎣∑i=41 ⎡⎣0.5cos(φi )+0.1cos(2(φπi − ))+0.5cos(3φi )⎦⎤⎤⎦⎥
where ϕi is the torsion angle of bond i in radians, and is defined by the relative locations of the carbon atoms defining the torsion angle. For example, ϕ1 = 0 when the carbon atoms in the leftmost methyl group and central methylene group are eclipsed when looking down bond 1.
The energy function described above will have several local minima and one global minimum. With this in mind:
where R is the ideal gas constant in SI units, and T is the temperature in Kelvin. Run a series of simulations using T = 1, 10, 100, and 1000 K, with all simulations starting from some common point on the energy surface that does not correspond to the energy well containing the global minimum.
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