Computational SpectroscopyIII. Spectroscopic Hamiltonians(a) Introduction to MATLAB(b) Matrix operations(c) Programing(d) Diagonalizing Hamiltonian matricesUpdated: March 25, 2008(a) What is MATLAB? MATLAB is the computational environment that we will use to solve spectroscopicHamiltonians to find the resulting energy levels and wavefunctions. MAT LAB stands for Matrix Laboratory The basic object in MATLAB is the matrix. Matrices are constructed, manipulated, matrix problems solved, and the results displayed. Any problem that can be expressed in terms of matrices, that is in terms of linear algebra, canbe easily solved in MATLAB. A matrix is an m × n array of numbers A vector is a 1 × n matrix A scalar is a 1 × 1 matrix, that is just a number. Complex numbers are no problem in MATLAB. MATLAB calculations can be run in two ways:1. Type in commands one at a time in the command window2. Create a program, an “M-file”, that contains a series of such commands. MATLAB comands read like ordinary algebra Many high-level matrix operations are available. Each command packs a big punch; you get a lot of work done without much programming.Warnings about MATLAB The hardest part is learning the user interface, and the various kinds ofwindows that it gives you. You do need to work through the tutorials to learn this. Be careful where you save your stuff. Make a folder in your “My Documents” folder and keep all your MatLab stuff there. Don’t leave any files in the MALAB directory or elsewhere on the computer. The current directory is shown at the top of your main MATLAB window. Your workspace contains all of the data and matrices that you have created. The default name for your workspace is “matlab.mat”, but you can use other nameswith the “.mat” extension on the file name. Your programs are saved separately as “M-files”, such as “myprog.m”. These are just text files that can be opened and edited with any text editor. You can save your matrices as text files:save ABC.txt ABC -asciiThis saves the matrix ABC in the text file ABC.txt in the current directory.ABC.txt can then be opened with any text editor, with Excel, or other program. Variable and matrix names are case-sensitiveHow will MATLABhelp us compute spectra? To compute molecular energy levels, we need to solve the Schroedinger equationHΨ = EΨ . (1) We will be concerned with the nuclear motion, therefore the Hamiltonian isH = T + Eelec(R), (2)where T is the nuclear kinetic energy operator and the effective potential energy is Eelec(R), theelectronic energy calculated as a function of the nuclear coordinates R. Eelec(R) is what we could getfrom Gaussian or from another electronic structure program. To solve the Schroedinger equation (1), we expand the wavefunction as a linear combination of a setof n basis functions, The wavefunction Ψ then becomes a vector in the n-dimensional space of the basis functions, the Hamiltonian H becomes an n×n matrix, E is a constant, and HΨ = EΨ becomes linear algebra eigenvalue problem, that is solved in MATLAB with thesingle command “eig”. Hilbert space is the infinite-dimensional vector space of wavefunctions. For practical reasons,we truncate to n dimensions. Most of our work will be in setting up the matrix H. We will also use matrix algebra to make the computer do most of that work. Spectral transition frequencies are obtained by subtracting to get the difference in energy betweenenergy levels connected by the selection rules.! " = ci#ii=1n$What we didn’t get fromGaussian or Spartan Didn’t solve the quantum mechanics of the nuclear motion. No rotational energy levels or rotational (microwave) spectra. No vibrational energy levels for fundamentals or for overtone and combination bands. Got only classical harmonic frequencies No treatment of large-amplitude motion such as Torsion, inversion Conformational change Motion along a reaction coordinate No coupling of nuclear degrees of freedom Vibration-vibration (IVR = intramolecular vibrational redistribution) Rotation-vibration (Coriolis coupling and centrifugal distortion) Quantum chaos Restricted ergodicity (Ergodicity is the core assumption of molecular mechanics calculations) No coupling of electronic and nuclear degrees of freedom Radiationless transitions, dynamics at conical intersections(b) Matrix Operations Live demonstration. A = [1,2,3;4,5,6;7,8,9] Let A, B be matrices and c and d be constants. The elements of A are Aij, i labels the rows, i=1,m, andj labels the columns, j=1,n scalar multiplication: cA is c*A Scalar addition: A+c is A+c Matrix multiplication: C=AB or Cik=ΣjAijBjk C=A*B The number of columns in A must be equal to the number of rows in B. Indentity matrix: I I = eye(n,n) Always a square matrix Matrix Inversion: I/A = inv(A)= I/A (I must have same dimension as A) A must be square and “non-singular” to have an inverse Matrix division: B/A = B*inv(A)= B/A A matrix raised to a power: A2=AA; A3=AAA; etc. A^3 A must be square Exponentiation of a matrix: exp(A) = 1 + A + A2/2! +A3/3! + A4/4! + …= expm(A) Many other functions of matrices are defined. Search “funm” in MATLAB help.! A " A =A11A12A13A21A22A23# $ % & ' ( ! I =1 0 00 1 00 0 1" # $ $ $ % & ' ' 'Fun with Complex Numbers z =c+d*i with i = sqrt(-1) and c, d real i and j are both predefined in MATLAB as the imaginary unit. Complex conjugation: z* is conj(z) Exponentiation of imaginary number uses Euler’s formula: exp(i*pi) gives -1.000 + 0.000i Functions and matrix operations work fine with complex numbers (wheremathemetically allowed). Complex numbers are user-friendly!!! exp iz( )= cos z + isin zMore fun with matrices Matrix transpose: At is A.’ Matrix transpose conjugate: (At)* is A’. This is the more useful form. Select a sub-matrix: A(1:2,:) selects the first two rows and all of the columns of A. A(2,3) gives the 3rd element from the 2nd row. Concatenate 2 matrices side-by-side [A,B] or one on top of the other [A;B]. Inner product: A*B Examples are matrix multiplication and “dot” product If x is a column vector containing the variables xi and p is a column vectorthen A*x=p is a set of 3 linear equations in the 3 unkowns x1, x2, and x3.Get