Computational Chemistry Modeling Using MATLAB
Computational chemistry involves using computer simulations to solve chemical problems. MATLAB provides a versatile environment to model and analyze complex systems in chemistry, such as molecular dynamics, reaction kinetics, and quantum mechanics.
Basic Modeling in Computational Chemistry
1. Molecular Dynamics
Molecular dynamics simulations involve solving Newton’s equations of motion for a system of particles. The force on each particle is calculated based on potential energy functions.
% Molecular dynamics simulation
% Force between two particles using Lennard-Jones potential
sigma = 3.4; % Angstrom
epsilon = 0.0103; % eV
r = linspace(2, 10, 100); % Distance between particles
force = 24 * epsilon * ((2 * (sigma ./ r).^12) - (sigma ./ r).^6) ./ r;
plot(r, force);
xlabel('Distance (Å)');
ylabel('Force (eV/Å)');
title('Lennard-Jones Force Curve');
2. Reaction Kinetics
Reaction kinetics involve solving differential equations that describe the rate of chemical reactions. MATLAB’s ODE solvers can be used for this purpose.
% Reaction kinetics example: A -> B with rate constant k
k = 0.1; % Rate constant
tspan = [0 100]; % Time span
A0 = 1; % Initial concentration of A
[t, A] = ode45(@(t, A) -k * A, tspan, A0);
B = A0 - A; % Concentration of B
plot(t, A, 'r', t, B, 'b');
xlabel('Time (s)');
ylabel('Concentration');
legend('A', 'B');
title('Reaction Kinetics: A -> B');
3. Quantum Mechanics
MATLAB can be used to solve the Schrödinger equation, which describes the quantum behavior of particles. For example, solving the equation for a particle in a 1D box:
% Solving Schrödinger equation for a particle in a box
L = 1; % Length of the box
n = 1:5; % Quantum numbers
x = linspace(0, L, 100);
psi = sqrt(2 / L) * sin(pi * n' .* x / L);
plot(x, psi);
xlabel('Position (x)');
ylabel('Wavefunction (\psi)');
title('Wavefunctions for Particle in a Box');
4. Optimization in Computational Chemistry
Optimizing molecular geometries or reaction pathways involves minimizing the potential energy surface. MATLAB’s optimization toolbox can be used for this.
% Geometry optimization example
energy = @(x) x.^4 - 4 * x.^3 + 4 * x.^2; % Potential energy surface
[x_min, f_min] = fminsearch(energy, 0.5);
disp(['Minimum energy: ', num2str(f_min), ' at x = ', num2str(x_min)]);
Advance Examples in Computational Chemistry Problems Using MATLAB
Example 1: Solving the Schrödinger Equation for a 2D Quantum Harmonic Oscillator
The 2D quantum harmonic oscillator is a fundamental problem in quantum mechanics. The wavefunction satisfies the equation:
\[ -\frac{\hbar^2}{2m} \nabla^2 \psi + \frac{1}{2} m \omega^2 (x^2 + y^2) \psi = E \psi \]
% Constants
hbar = 1; m = 1; omega = 1;
N = 50; L = 5;
x = linspace(-L, L, N); y = x;
[X, Y] = meshgrid(x, y);
V = 0.5 * m * omega^2 * (X.^2 + Y.^2);
% Constructing the Hamiltonian
dx = x(2) - x(1);
T = -hbar^2 / (2 * m) * (diag(ones(N-1,1),1) + diag(ones(N-1,1),-1) - 2 * diag(ones(N,1))) / dx^2;
H = kron(eye(N), T) + kron(T, eye(N)) + diag(V(:));
% Solving for eigenvalues and eigenfunctions
[psi, E] = eigs(H, 5, 'smallestabs');
psi1 = reshape(psi(:,1), N, N);
surf(X, Y, abs(psi1).^2);
xlabel('x'); ylabel('y'); zlabel('|\psi|^2');
title('Ground State of 2D Quantum Harmonic Oscillator');
Example 2: Potential Energy Surface and Reaction Pathway Optimization
This example computes the potential energy surface for a diatomic molecule and finds the minimum energy pathway for the reaction:
A2 + B → A + AB
% Reaction coordinates
r_A2 = linspace(1, 3, 100); % Bond length of A2
r_AB = linspace(0.5, 2, 100); % Bond length of AB
% Lennard-Jones potential for A2 and AB
sigma = 1; epsilon = 0.2;
V_A2 = 4 * epsilon * ((sigma ./ r_A2).^12 - (sigma ./ r_A2).^6);
V_AB = 4 * epsilon * ((sigma ./ r_AB).^12 - (sigma ./ r_AB).^6);
% Potential energy surface
[E_grid, R_grid] = meshgrid(r_A2, r_AB);
V_total = V_A2(:)' + V_AB(:);
contourf(R_grid, E_grid, reshape(V_total, size(R_grid)), 20);
colorbar;
xlabel('Bond length A2 (Å)');
ylabel('Bond length AB (Å)');
title('Potential Energy Surface for Reaction Pathway');
Example 3: Monte Carlo Simulation of Molecular Interactions
Monte Carlo methods are used to simulate molecular interactions by sampling configurations according to a Boltzmann distribution.
% Monte Carlo parameters
N_particles = 100;
L = 10; kT = 1;
positions = rand(N_particles, 3) * L; % Initial positions
% Lennard-Jones potential
sigma = 1; epsilon = 1;
potential = @(r) 4 * epsilon * ((sigma ./ r).^12 - (sigma ./ r).^6);
% Monte Carlo loop
n_steps = 1000; energy = zeros(n_steps, 1);
for step = 1:n_steps
i = randi(N_particles); % Randomly select a particle
displacement = (rand(1,3) - 0.5) * 0.1;
new_pos = positions(i,:) + displacement;
% Calculate energy change
r = sqrt(sum((positions - new_pos).^2, 2));
dE = sum(potential(r)) - sum(potential(sqrt(sum((positions - positions(i,:)).^2, 2))));
% Metropolis acceptance criterion
if dE < 0 || rand < exp(-dE / kT)
positions(i,:) = new_pos;
end
energy(step) = sum(potential(r));
end
plot(1:n_steps, energy);
xlabel('Step'); ylabel('Energy');
title('Monte Carlo Simulation of Molecular Interactions');
Example 4: Vibrational Spectrum of a Triatomic Molecule
MATLAB can be used to calculate vibrational frequencies of molecules by diagonalizing the Hessian matrix of the potential energy function.
% Masses of atoms (in atomic units)
m1 = 1; m2 = 16; m3 = 1;
% Approximate Hessian matrix (harmonic approximation)
H = [2 -1 0; -1 2 -1; 0 -1 2];
% Vibrational frequencies
k = 1; % Force constant
omega = sqrt(eig(H) * k ./ [m1; m2; m3]);
disp('Vibrational frequencies (cm^-1):');
disp(omega / (2 * pi * 3e10)); % Convert to cm^-1
Example 5: Solving the Hartree-Fock Equations for a Simple Atom
The Hartree-Fock method approximates the wavefunction of an atom or molecule using a set of orthogonal orbitals. For simplicity, we consider the helium atom:
Hψ = Eψ, where H is the Fock operator.
% Define radial grid
r = linspace(1e-3, 20, 500); dr = r(2) - r(1);
% Initial guess for wavefunction
psi = exp(-r); psi = psi / sqrt(sum(psi.^2) * dr);
% Define potential and Hamiltonian
V = -2 ./ r; % Nuclear attraction
H = -0.5 * diag(-2 * eye(length(r)) + diag(ones(length(r)-1,1),1) + diag(ones(length(r)-1,1),-1)) / dr^2 + diag(V);
% Iterative Hartree-Fock solution
for iter = 1:100
rho = psi.^2; % Electron density
V_H = cumtrapz(r, rho ./ r) * dr; % Hartree potential
H_eff = H + diag(V_H); % Effective Hamiltonian
[psi_new, E] = eigs(H_eff, 1, 'smallestabs');
psi_new = psi_new / sqrt(sum(psi_new.^2) * dr);
if norm(psi_new - psi) < 1e-6
break;
end
psi = psi_new;
end
plot(r, psi.^2);
xlabel('r'); ylabel('|\psi|^2');
title('Electron Density for Helium Atom');
Example 6: Molecular Orbital Energy Diagram for H2
Computing the molecular orbital energy diagram involves diagonalizing the Fock matrix for the H2 molecule.
% Overlap and Hamiltonian matrices
S = [1, 0.5; 0.5, 1]; % Overlap matrix
H = [-1, -0.5; -0.5, -1]; % Hamiltonian matrix
% Generalized eigenvalue problem
[C, E] = eig(H, S);
disp('Molecular Orbital Energies:');
disp(diag(E));
disp('Molecular Orbital Coefficients:');
disp(C);
% Plotting MO diagram
figure; hold on;
plot([1, 2], [E(1,1), E(1,1)], '-k', 'LineWidth', 2); % Bonding
plot([1, 2], [E(2,2), E(2,2)], '-r', 'LineWidth', 2); % Antibonding
text(1.1, E(1,1), 'Bonding Orbital');
text(1.1, E(2,2), 'Antibonding Orbital');
xlabel('Orbital Index'); ylabel('Energy');
title('Molecular Orbital Energy Diagram for H2');
Example 7: Vibrational Spectroscopy for a Diatomic Molecule
Compute vibrational transitions for a diatomic molecule using the Morse potential:
V(r) = De(1 – exp(-a(r – re)))²
\[ V(r) = D_e \left( 1 – \exp(-a(r – r_e)) \right)^2 \]
% Parameters
D_e = 4.6; a = 1.2; r_e = 1.0;
r = linspace(0.5, 2, 500); dr = r(2) - r(1);
% Morse potential
V = D_e * (1 - exp(-a * (r - r_e))).^2;
% Discretized Hamiltonian
T = -0.5 * diag(-2 * eye(length(r)) + diag(ones(length(r)-1,1),1) + diag(ones(length(r)-1,1),-1)) / dr^2;
H = T + diag(V);
% Solve Schrodinger equation
[psi, E] = eigs(H, 5, 'smallestabs');
disp('Vibrational Energy Levels:');
disp(diag(E));
% Plot wavefunctions
for i = 1:5
plot(r, psi(:,i).^2 + E(i,i)); hold on;
end
plot(r, V, '--k');
xlabel('r'); ylabel('Energy');
title('Vibrational Energy Levels for Morse Potential');
Example 8: Time-Dependent Quantum Dynamics
Simulate the time evolution of a wave packet in a one-dimensional potential:
% Parameters
x = linspace(-10, 10, 500); dx = x(2) - x(1);
V = 0.5 * x.^2; % Harmonic potential
T = -0.5 * diag(-2 * eye(length(x)) + diag(ones(length(x)-1,1),1) + diag(ones(length(x)-1,1),-1)) / dx^2;
H = T + diag(V); % Hamiltonian
% Initial wave packet
psi_0 = exp(-x.^2); psi_0 = psi_0 / sqrt(sum(psi_0.^2) * dx);
% Time evolution
dt = 0.01; t_max = 10;
for t = 0:dt:t_max
psi_t = expm(-1i * H * dt) * psi_0;
plot(x, abs(psi_t).^2);
xlabel('x'); ylabel('|\psi|^2');
title(['Wave Packet at t = ', num2str(t)]);
drawnow;
end
Useful MATLAB Functions for Computational Chemistry
Practice Problems
Test Yourself
1. Simulate the reaction kinetics for a second-order reaction and plot the concentration vs. time graph.
2. Solve the Schrödinger equation for a particle in a finite potential well.
3. Optimize the geometry of a triatomic molecule using MATLAB’s optimization functions.
4. Generate a 3D plot of the Lennard-Jones potential for two particles as a function of their distance.