Useful math for spectroscopic applications

Two-dimensional coordinate system

A two-dimensional coordinate system shown below illustrates the relationship between polar and cartesian coordinates. The projection of vector OA along x and y is given r sinq and r cosq, respectively.

If the radius of the circle is r then we know that x² + y² = r² (the Pythagorean theorem) or r = Ö(x² + y²). The definition of of sinq and cosq are given below.

Given that x = r cosq and y = r sinq we find that

The bottom relation is extremely useful for manipulating trigonometric expressions. For example, we can write

and

The latter expression is useful for evaluating Legendre polynomials, angular momentum operators, and in many other contexts.

The above representation of the blue arrow is a designation of a vector in two dimensions. If we call this vector A then we can represent it as

The components A_x and A_y are the projections of the vector along the x- and y-axes, respectively. The i and j are unit vectors along x and y, respectively. A vector has a magnitude and a direction. This is often easier to specify in polar coordinates. The magnitude is r and the direction is given by the angle q.

Vectors are extremely important for representing dipole moments, angular momenta, and electric field polarization which are some of the applications that will arise in the study of molecular spectroscopy.

We can also define a polar coordinate system with real and imaginary axes. It looks very similar to the one above.

In this representation the position of the vector can be specified by defining the function eⁱq = cosq + i sinq. In this expression i is the square root of -1, i = Ö-1. The way to view this definition is to realize that eⁱq contains information along both the real and imaginary axes and hence both vector components. Note further that e^-iq = cosq - i sinq.

We will find that use of eⁱwt to represent a sinusoidal function is useful for the description of electromagnetic radiation. Here w is the angular frequency w = 2pn where n is the frequency Hz (or s^-1) and t is the time.

The sine and cosine functions are 90 degrees out of phase with one another and thus when we write e^-iwt = coswt + i sinwt we are able to consider both in-phase and out-of-phase components at the same time. The fact that we can write a sum of sine and cosine functions as an exponential is also an aid to integration. Instead of taking the integral of coswt we can take the integral of eⁱwt and then take the real part.

To illustrate this we take the integral of cos(x).

Alternatively, we can take the real part (Re) of the integral of e^ix

We have implicitly used a substitution to perform the exponential integral and then used the fact that 1/i = -i. The real and imaginary parts are both physically real parts of a response function. They differ only in phase.

Three-dimensional coordinate system

The projections of a vector in a three dimensional coordinate system are indicated in the Figure below.

The extension from the two coordinate system follows the convention that the q angle represents the azimuthal angle or tilt angle from the z axis. The projections of the vector on the three-dimensional cartesian coordinate system are given by

x = r sinq cosf

y = r sinq sinf

z = r cosq

These relations will become important for deriving the averages of molecular properties over all orientations of the molecular in a solution or gas phase. The definition of a vector in three dimensions is an extension of the definition in two dimensions.

A vector may also be considered a tensor of rank one. The dot product or inner product of two vectors

is given by

where q is the angle between the two vectors. The result is a scalar. A scalar is a number (i.e. it has a magnitude but there is directional information contained in a scalar).

The cross-product is defined by

The magnitude of the cross-product is |A||B|sinq and the direction is perpendicular to the directions of A and B as given by the right hand rule. Imagine using your fingers to push A into B. Your thumb will point in the direction of A x B.

The del or grad (gradient) operator returns a vector quantity when operating on a scalar. Since the operator is the first derivative, the gradient is the slope of a three-dimensional function given as the components along x, y, and z.

This operator should not be confused with the divergence which is the dot product of del and a function. Since it is a dot product the divergence is a scalar.

The Laplacian (also called del squared) is defined as

The curl is the cross product of del with a vector.

Matrices

A matrix is an array of numbers. An n by m (or n x m) matrix is represented below.

Two matrices can be added or subtracted only if they have the same dimensions n and m. The result A + B = C is just the result of adding (or subtracting) each of the individual elements a_ij + b_ij = c_ij.

A square matrix has n = m.

A diagonal matrix has elements only along the diagonal. The unit matrix I is a diagonal matrix with only ones along the diagonal. The elements of a unit matrix can be represented by the Kroenecker delta, d_ij = 1 for i = j and d_ij = 0 for i ¹ j.

A column vector is a matrix of dimension n x 1.

A row vector is a matrix of dimension 1 x n.

In order to multiply two matrices, AB = C, the number of columns on the left (A) matrix must equal the number of rows on the right (B) matrix. Each element in C can be obtained by multiplying the column elements of A by the row elements of B

The product of a (left) row and a right (column) matrix is a scalar.

However, the product of a column with a row (reverse order is a matrix).

The transpose of a matrix, A^T, is obtained by interchanging rows and columns. If A^T = B then b_ij = a_ji. The inverse of a square matrix is defined such that CC^-1 = I.

The determinant of a square matrix of dimension n is the result of summing n! terms. It is represented as detA or |A|. The determinant of a 2 x 2 matrix is:

For larger dimensions the determinant is resolved by breaking down into smaller dimensions until ultimately the result is a sum of 2 x 2 determinants that can be obtained from the above formula.

For example, a 3 x 3 is

Note that the sign of the terms alternate as one moves across the rows of the matrix. This property means that the magnitude of the determinant changes sign if the order of any two rows are reversed. The determinant vanishes if any two rows or columns are identical.

Determinants are important in the solution of the eigenvalue problem. This is discussed further in section A.2 of appendix A.