Mathematical Review for Theoretical Chemistry

Randal Hallford

Department of Chemistry

Midwestern State University

Introduction: This review of the basic mathematical techniques that are generally necessary to

understand the material in an upper level undergraduate chemistry courses is not intended

to be complete, but is a review of the minimum tools that will be required to solve

problems in theoretical chemistry. Chemistry is a fascinating subject as it deals with the

nature of the physical world, but there is a requirement for a modest level of skill in using

mathematics. Consider this a brief guide to chemistry mathematics, and allow this to help

you build some facility in applied mathematics. For some this will be a review, but very

commonly using the calculus in this applied way generates significant proficiency for

those who have not been exposed to higher calculus.

Calculus:

Essential underlying operations in the calculus are differentiation and integration.

Consider functions of one variable such as y = f(x), and assume the functions are well

behaved -meaning the derivatives exist and the functions are single valued for a given

value of x.

1. Derivatives

The derivative of the function f(x) at the point x0 is defined by:

( ) ( ) ( ) ( )0

0 000 0 0

0

( ) ( )' lim limx x h

f x h f xf x f xdf x f xdx x x h→ →

− −−= = =

−

This definition gives the instantaneous slope of a line tangent to a curve.

the derivative can also be written as f’(x) (f prime of x). The above equation allows derivation of the general formula for polynomials:

1( ) '( ) ( )n ndf x f x nx for f x xdx

−= = =

In general, if given a function in the denominator, write the expression as a negative exponent. This will simplify finding the derivative.

Example,

( ) ( ) (1 23 2 3 2 23 2

1 5 4 5 4 3 105 4

d d )x x x x xdx x x dx

− −⎛ ⎞ = − + = − − + −⎜ ⎟− +⎝ ⎠x

where the function 1/f(x) is treated as f(x)-1 and therefore n = -1 and nf(x)n-1= (-1)f(x)-2. This example utilizes the chan rule. The chain rule applies when one function is a function inside another, e.g. g(f(x)).

( )( )( ) ( )( ) ( )' 'd g f x g f x f xdx

=

First, take the derivative with respect to g(x) treating the whole of f(x) as the variable, then take the derivative with respect to f(x):

( )2 2

2x xd e xdx

eα αα− −= −

This is the derivative of a Gaussian function with respect to x. Note that here g(f(x)) = ef(x) and f(x) = -ax2.

The derivative of an exponential is itself an exponential (d/dx)ex = ex.

2

Derivatives To Memorize

1

cos sin

sin cos

1log

n n

x x

ax x

d x nxdxd e edxd e aedxd x xdxd x xdxd xdx x

−=

=

=

= −

=

=

Special Derivative Relationships

Chain Rule: [ ( ( ))]d F u x dF dudx du dx

=

Ratio Rule: ( )2

/du dvv ud u v dx dx

dx v

⎛ ⎞ ⎛−⎜ ⎟ ⎜⎝ ⎠ ⎝=

⎞⎟⎠

Product rule ( )d uv du dvv udx dx dx

= +

Differential Equations Equations which involve derivatives of functions giving rise to a solution (or family of solutions) for a function are referred to as differential equations. A set of boundry conditions must exist, either as a function of the physical system, or be soluble for them. A differential statement has a physical relationship to rate.

3

Ordinary Differential Equations

A differential statement with derivatives with respect to only one variable, or expressions with separable terms with derivatives of only one variable are referred to as ordinary differentials.

Newton’s second law in vector form is F ma= . The acceleration is written as dvdt

,

where is the velocity vector, or as v2

2

d rdt

, where r is the displacement vector. Thus, a

differential can be written for any mechanical system where motion is the result of a force. The force of a mechanical system is given by the first derivative of the potential energy function

( ) V x

Fx

∂= −

∂

where V(x) is the potential energy function. The acceleration is the second derivative with respect to time

2

2

d xadt

=

and since the form of the potential is known, the derivative gives the force

2

2( ) d xF x mdt

=

An important classical and quantum mechanical model is the harmonic oscillator. This models the oscillations of spring systems and the vibrations of molecules. The potential function is given by

2 212

V mx ω=

where ω is the oscillator frequency. The force is determined from the differential form to give 2( )F x mxω= − The negative sign indicates that the force acts in opposition to the motion along x and is referred to as the restoring force. The results are set equal to the F(x) expression above

2

22

d xm mxdt

ω= −

and setting equal to zero

2

22 0d x x

dtω+ =

gives a second order (highest exponential is 2) homogeneous (separable and equals zero, not some function of t) linear (exponent on x is one) differential equation.

4

2. Properties of Derivatives

The Total differential of a function that depends upon more than one other variable can

be written as:

dyyfdx

xfdZ

yxfZ

xy⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+⎟⎠⎞

⎜⎝⎛∂∂

=

= ),(

hold const. hold const.

This is convenient for depicting relationships between variables in several dimensions.

The partial derivative symbol, , replaces the standard d when it necessary to hold one or

more variables constant while taking the derivative of a function with respect to another

variable. Subscripts on the parenthetical terms note the variable(s) held constant.

∂

The total differential for the function depicted in the

Graph is: dxdxdydy =

For functions that represent physical systems responding to more than one variable at a

time, the total differential accounts for the variables, and describes the change in the

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system. For the x y z graph below, a path is shown that can be described by the total

differential for z:

dxxzdy

yzdz

yx

⎟⎠⎞

⎜⎝⎛∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

=

this implies that a general form of differential can be written:

∑ ⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

=i

ii

dqqzdz

where the qi are the variables in the system.

Example: From physics it is known that for a quantity of an ideal gas, the pressure

responds to temperature and volume changes:

mV

RTP = ; Find the total differential for dP

Write a partial term for each variable, holding the others constant:

dTTPdV

VPdP

mVTm⎟⎠⎞

⎜⎝⎛∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

=

2−−=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

mTm

RTVVP ; 1−=⎟

⎠⎞

⎜⎝⎛∂∂

mv

RVTP

m

6

the function then is: 2 mm m

R RTdP dT dVV V

= −

Independent and Dependant Variables

For multivariate total differentials, functions may contain both dependant and

independent variables.

f (x,y,w)=0 and g (x,z,w)=0 contain four variables.

If then let 222 zyxw ++= 22 yxz +=

So that x, y are independent: yx

w⎟⎠⎞

⎜⎝⎛∂∂ . The differential then is: ( )( )22222 yxyx

x+++

∂∂ =

2x + 4x3 + 4xy2

and if x, z are independent: zx

w⎟⎠⎞

⎜⎝⎛∂∂ . The differential is: ( )( )2 2 2x z x z

x∂

+ − +∂

= 0

This has a physical interpretation:

↔ 0 =zx

w⎟⎠⎞

⎜⎝⎛∂∂ w does not change in x-y plane but does in the xz ↕ =

yxw⎟⎠⎞

⎜⎝⎛∂∂

Eigenfunctions Equations that have an operator multiplied by some function can return a value and the function again. This process produces an eigen equation, where the value is an eigenvalue.

ax

axde aedx

=

The function, , is multiplied by the operator axe ddx

and returns the function multiplied by

a value, a. Not all math functions can behave in an eigenvalue equation. One example is the log function. Cycle (cyclic)Rule in Differential Equations Total differentials in thermodynamics may be independent or related as a result of associated state variables. Determination of which case is true, and how many exist, is

7

accomplished via the cyclic rule. This also indicates the function is real, i.e., is on the surface of a gaph:

( )

:

if the changes in variables are restricted so that z is unchanged,then dz = 0 and

0

divide through by :

0

m

y x

y x

z

y z x

z zdz dx dyx y

z zdx dyx y

y

z x zx y y

⎛ ⎞∂ ∂⎛ ⎞= + ⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠

⎛ ⎞∂ ∂⎛ ⎞= + ⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠∂

⎛ ⎞ ⎛ ⎞∂ ∂ ∂⎛ ⎞= +⎜ ⎟ ⎜ ⎟⎜ ⎟∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠

ultiply through by yields:

1

x

x yz

yz

y x zz y x

∂⎛ ⎞⎜ ⎟∂⎝ ⎠

⎛ ⎞∂ ∂ ∂⎛ ⎞ ⎛ ⎞ = −⎜ ⎟⎜ ⎟ ⎜ ⎟∂ ∂ ∂⎝ ⎠ ⎝ ⎠⎝ ⎠

This result indicates that the partial differential statements are related. A different representation of the cyclic rue allows the determination of missing partial terms.

For the expression :, ,V n T n P n

P P VT V T∂ ∂ ∂⎛ ⎞ ⎛ ⎞ ⎛ ⎞= −⎜ ⎟ ⎜ ⎟ ⎜ ⎟∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ,

find the partial term

,T n

VP∂⎛ ⎞

⎜ ⎟∂⎝ ⎠.

This representation is written as a ratio:

,

,

,

P n

V n

T n

VTPVTP

∂⎛ ⎞⎜ ⎟∂∂ ⎝ ⎠⎛ ⎞ = −⎜ ⎟ ∂∂ ⎛ ⎞⎝ ⎠⎜ ⎟∂⎝ ⎠

Moving the partial terms to the other side leaves :, ,V n P n T n

T V VP T P∂ ∂ ∂⎛ ⎞ ⎛ ⎞ ⎛ ⎞− =⎜ ⎟ ⎜ ⎟ ⎜ ⎟∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ,

Exactness Tests If some differential form is exact, the coefficients must be the derivative of the function.

For some set of functions du = M(x,y) dx + N(x,y) dy then:

8

( , ) ; ( , )y x

u uM x y N x yx y

⎛ ⎞∂ ∂⎛ ⎞= = ⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠

The Euler reciprocity relation 2 2u u

x y y∂ ∂

=∂ ∂ ∂ ∂x

guarantees that:

y x

N Mx y

⎛ ⎞∂ ∂⎛ ⎞ = ⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠

for an exact differential. Example:

Show that 2 2

22

9 32 x xdu xy dx x dyy y

⎛ ⎞ ⎛= + + −⎜ ⎟ ⎜⎝ ⎠ ⎝

⎞⎟⎠

Is exact.

2 2

2

3 22

2 2

9 92 2

3 92

x

y

x xxy xy y y

x xx xx y y

⎡ ⎤⎛ ⎞∂+ = −⎢ ⎥⎜ ⎟∂ ⎝ ⎠⎣ ⎦

⎡ ⎤⎛ ⎞∂+ = −⎢ ⎥⎜ ⎟∂ ⎝ ⎠⎣ ⎦

Since both derivatives are the same, the differential is exact. The Chain Rule If z = z(u,x,y) but x can be expressed as a function of u,v and y, then

,, ,u vu v u v

z z xy x y

⎛ ⎞ ⎛ ⎞∂ ∂ ∂⎛ ⎞=⎜ ⎟ ⎜ ⎟⎜ ⎟∂ ∂ ∂⎝ ⎠⎝ ⎠ ⎝ ⎠

If a function A(B,C) is written where B and C are also functions of the variables C and D so that B(D,E) and C(D,E) then:

C E C D C

A A D A EB D B E B∂ ∂ ∂ ∂ ∂⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞= +⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

.

Notice that D and E in the first and second term can also be a result of division of a total derivative for dA by dB at constant C. Reciprocity

If , ,V n V n

P nR TthenT V P∂ ∂⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠

VnR

More details on the use of derivatives are in the definitions and theorems section.

3. Integration

The integral is the sum of small areas under a curve, as the areas approach an infinitely

small dimension:

9

0lim ( )

n

ih iI f x h

→= ∑

Where f(x) is the height and h is the width of a small area centered on the point xi. In the

limit of a very small h, this converts to the integral:

( )I f x d= ∫ x

Integration is the antiderivative – that is, it is the inverse operation of a derivative. If a

derivative is known, the integral can be deduced. x xe dx e c= +∫ is an indefinite

integral, where c is the integration constant. The integrating constant is a value

determined by the physical application of the integrand to a specific problem. In cases

where the problem exists within definite limits, the definite integral can be written as

cos sin | sin sinb b

aaI x dx x b a= = = −∫ . The difference of the function values at a and b

result in the loss of the integrating constant (it is the same for both function values and

subtracts out) and the answer is simply the numerical difference of the function at a and

b.

Integration for physical processes: summation of small steps

The expansion of an ideal gas against an external force produces a graph of given

area. However, there is no functional dependence between the two variables as they can

change independently of each other.

The work done by a gas under these conditions does depend on both the external

pressure on the gas and the volume change by the gas:

)( 12 VVPw ext −−=

which is just the area under the curve. By measuring the area, we can determine the work

without the need for consideration of all of the other parameters in the experiment.

In less straightforward cases, where several variables are changing, measuring the

area is problematic without the calculus, and the evaluation of integrals describing the

complex area changes are a necessary step in the solution of physical problems. When a

problem involves several variables that are related to a change in one variable during a

process, an analytical solution is not available unless the function relating the variables

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over the path is known. Allowing only one variable to change and integrating the path for

this variable can be accomplished by removing a constant variable from the integral in

some cases. For instance, causing the external pressure to remain constant and removing

it from the integral provides an analytical solution:

∫=2

1

V

V extdvPA becomes which is –P∫= 2

1

V

Vext dVPA ext (V2 – V1).

If the system is reversible, then the external pressure and the internal pressure

become equal, and the temperature variable is introduced. The resulting integral looks

like:

dVVTPAV

V gas ),(2

1∫=

and a direct evaluation is not possible.

Integrals to Memorize

( )

1

1

1

ln

1

1sin( ) cos( )

c

nn

n n

axax

axax dx cn

a dx a x cxa adx cx n x

ee dx ca

ax dx ax ca

+

−

= ++

= +

= − +−

= +

= − +

∫

∫

∫

∫

∫1os( ) sin( ) ax dx ax ca

= +∫

4. Methods of Integration

When more complicated integrals result from analysis of a problem, integral tables can

provide templates for the solution of the integrals. Many times the integral you have

determined is necessary to solve the problem does not seem similar to any in the tables.

This requires manipulation to re-arrange the integral to “standard form”.

11

1. Break integrals into small steps: 0

0

( ) ( ) ( )

( ) ( ) ( )a b

a b

f x dx f x dx f x dx

f x dx f x dx f x dx

∞ ∞

−∞ −∞

∞

−∞

= +

+ +

∫ ∫ ∫

∫ ∫ ∫

=

2. Change a variable (dummy variable)

The results of an integration is independent of the identity of the variable over which

integration is carried out, and can be treated as a dummy variable, u, such that it’s

derivative allows substitution in the integrand.

( ) ( )b b

x a x u

f x dx f u du= =

=∫ ∫

Change of variables

If u=kx, then x=u/k:

1

1( ) ( )b kb

x a u ka

dxdx du dudu k

uF x dx F duk k= =

= =

=∫ ∫

3. Reversing Limits

Changing the limits changes the sign of the integral.

4. Integration by Parts

An important method of integration stems from the identity:

d(uv) = udv + vdu

and leads to the integral:

udv uv vdu= −∫ ∫

This is the formula presented in most calculus books.

12

A useful example to consider is the real part of a Fourier transform. In general:

( ) ( )0

1 i tF e f t dωωπ

∞ −= ∫ t

this can be solved for the exponential function f(t) = e-t/τ to yield two terms, one real and one imaginary. Solving for the real and imaginary parts separately using the identity:

( ) ( )cos sini te t iω tω ω− = −

Results in

( ) ( ) ( ) ( ) ( )0 0

1 cos siniF t f t dt tω ω ωπ π

∞ ∞f t dt−

= −∫ ∫

From this form it is clear that the real and imaginary parts are separated prior to carrying out integration.

The real part is ( ) /

0

1 cos tt e dtτωπ

∞ −∫

Apply the integration by parts at this point.

Let u = cos(ωt) and dv = e-t/τdt. Then v = τe-t/τ and du = ωsin(ωt)dt. Using the formula for integration by parts (and ignoring the factor of 1/π) :

/ /

00 0

cos( ) cos( ) sin( )t tt e dt t e t t e dtτ τω τ ω ω ω∞ ∞

− − ∞= −∫ ∫ /t τ−

The first term on the right hand side evaluates to τ. The same integration by parts is performed for the second integral on the right hand side.

/ /

00 0

sin( ) sin( ) cos( )t tt e dt t e t e dtτ τω τ ω ωτ ω∞ ∞

− − ∞= +∫ ∫ /t τ−

The first term on the right hand side evaluates to zero. Substituting the second integral into the first :

13

/ 2 2

0 0

cos( ) cos( )t tt e dt t e dtτ τω τ ω τ ω∞ ∞

− −= −∫ ∫ /

and solving for the integral term we find:

/

2 20

cos( )1

tt e dtω

τ τωω τ

− =+∫

Another example:

Exponential Function e-ax

The area under the curve is:

0

1axe dxa

∞− =∫

To obtain this answer use the substitution u = -ax and du = -adx. Therefore dx = -du / a.

The integral can be written:

( )0

0

1 1 1 0 1|u ue du ea a a

∞∞

= − = − − =∫1a

In general exponential integrals multiplied by a polynomial can be solved using integration by parts. For example, the integral

14

axxe dx−∫

can be solved by substitution u = x and dv = e-ax dx. Then

du = dx and v = (-1/a)e-ax to obtain the integral. In general

udv uv vdu= −∫ ∫

so that in the present case:

0 20 0

1 1ax ax axxe dx xe e dxa a

∞ ∞− − ∞ −= +∫ ∫

1a

=

Notice that the first (uv) term is zero and the second term contains the integral of e-ax that was solved above. By this technique any polynomial can be solved yielding the general formula

10

!n axn

nx e dxa

∞−

+=∫

5.. Even-oddness

If an integrand is even over the limits [f(x)=f(-x)] then:

0

( ) 2 ( )a a

a

f x dx f x dx−

=∫ ∫

And if odd [f(x)=-f(-x)]:

( ) 0a

a

f x dx−

=∫

5. Multiple Integration Multiple dimensional spaces require multiple integrations and are written over sets of coordinates or variables. The multiple integral is evaluated step-wise, holding all but one variable constant. A two dimensional integral looks like:

( , )b d

a cf x y dxdy∫ ∫

15

If the function f(x,y) is in the form of f(x,y) = g(x)h(y) then the function variables are separable, and the integral becomes the product of two integrals.

( , ) ( ) ( )b d b d

a c a c

f x y dxdy g x dx h y dy=∫ ∫ ∫ ∫

Frequently, this is not possible and evaluation of multiple integrations becomes more complex.

Changing variables in Multiple Integration Commonly, limits on multiple integrals must be converted form cartesian coordinates to polar coordinates. Recall that x = r cos θ and y = r sin θ with ranges of 0≤ r ≤∞ and 0 ≤ θ ≤ 2π. The function must be expressed in terms of r and θ, and the area or volume element is changed from dxdy to J drdθ, where J is the Jacobian determinant of the coordinate transform:

x xrJy yr

θ

θ

∂ ∂∂ ∂=∂ ∂∂ ∂

The two dimensional determinant is given bya b

ad bcc d

= −

After solving the four partial derivatives and inserting them, the solution gives ( )2 2

2 2

cos sin

since cos sin 1 then

J r

J r

θ θ

θ θ

= +

+ = =

The volume (or area for a two dimensional system) change is dxdy rdrdθ. The units remain the same. In the three dimensional case, the variable change is x = r sinθ cosφ; y = r sinθ sinφ; z = r cosθ with ranges of 0≤ r ≤∞; 0 ≤ θ ≤ π; 0 ≤ φ ≤ 2π. The determinant can be taken from a calculus text. The change in the volume element is dxdydz r2 dr sinθdθdφ.

6. Vector Algebra

Vectors are noted in type as a bold letter, a, and in written form as a . Since vectors are directional, that is they are a combination of magnitude (scalar) and direction, there is a notation set of subscripts and operators that give direction to each component of a vector scalar.

x y za a i a j a k= + + and unit vectors multiply:

0

0

0

i i i j k

j j j k

k k k i j

× = × =

i× = × =

× = × =

The notation (“i hat”) is the unit vector in the x direction. i

16

Sum of Vectors

( ) ( ) ( )x x y y z za b a b i a b j a b k+ = + + + + + Represents the sum of vectors a and b Multiplication by a constant distributes the constant throughout as a scalar multiplier.

x y za a i a j aα α α α= + + k The scalar value of a vector can be found as the norm of the vector.

( )1

2 2 2 2x y za a a a= + +

The unit vector for a vector is found as the ratio of the vector to the norm:

1 unit value; this is parallel to aa aa

= =

The scalar product of vectors (dot product) is found from the product of the norms of the vectors and the cosine of the angle between them:

cosa b a b θ⋅ = And the magnitude of the vector is

a a= a . For an orthogonal set of vectors cos θ is zero, therefore the orthogonal vectors have a zero dot product The cross product of vectors (vector product) requires a unit vector, n , that is orthogonal to both vectors.

sin 0a b a b nθ θ π× = ≺ ≺ Complex algebra includes expressions having a real and imaginary part, typically denoted by z= x + iy. The complex portion contains i , a symbol representing √-1. The modulus of a complex number is

( ) ( )2 2 2 2Z real z imaginary z x y= + = +

17

And satisfies :

( )2 is the complex conjugate of ZZ ZZ Z∗ ∗=

(Z = x + iy then Z* = x – iy) Complex cartesian coordinates can be expressed as polar coordinates such that Z = r(cosθ + i sinθ), where θ is the argument of the expression. Values can be obtained for r, sinθ, and cosθ :

2 2

2 2 2 2; cos ; sinx yr x y

x y xθ θ= + = =

+ + y

7. Series Expansions Many approximations to physical processes for which no exact solution exists can be found by substituting terms from a series expansion into an empirical or analytical function. Several series which should be kept in mind are listed here.

2 3

0

2 1 3 5 7

0

2 2 4 6

0

1 ...! 1! 2! 3!

( 1)sin ...(2 1)! 3! 5! 7!

( 1)cos 1 ...(2 )! 2! 4! 6!

nx

n

n n

n

n n

n

x x x xen

x x x xx xn

x x x xxn

∞

=

+∞

=

∞

=

= = + + +

−= = − +

+

−= = − + −

∑

∑

∑

−

+

An application to quantum wave mechanics is related to the sin and cosine functions. Using a series of expansions provides a simplification and a very fundamental relationship used in a wide variety of applications.

18

( ) ( )

2 3

0

2 3

2

2 3 4 5

1 ...! 1! 2! 3!

let z = , where is real and z is, therefore, imaginary.

1 ...1! 2! 3!

if 1

11! 2! 3! 4! 5!

separate terms based on alternating real

nz

n

i

i

i

z z z zen

e

i iie

ii i ie

θ

θ

θ

θ

θ θθ

θ θ θ θ θ

∞

=

= = + + + +

= + + + +

= −

= + − − + + −

∑

2 4 6 3 5 7

and imiaginary parts

1 ... ...2! 4! 6! 1! 3! 5! 7!

cos sin:

cos sin Euler's Formula

i

i

i i i ie i

ithereforee i

θ

θ

θ θ θ θ θ θ θ

θ θ

θ θ

⎛ ⎞ ⎛= − + − + + − + −⎜ ⎟ ⎜⎝ ⎠ ⎝

= +

⎞⎟⎠

In quantum mechanics it is necessary to constrain systems of wavefunctions to real positive values. Utilizing complex conjugates ensures this is true. If Z = r ie θ , then Z* = r ie θ− = r2, a real value sinc r = |Z|. Then ZZ* = |Z|2. For non-zero values of r we have:

1 1 122

ii re ZZ r e so that Z

r Z

θθ

− ∗− − − −= = =

If a wavefunction requires a shift by some angle then Euler’s formula can be used to ensure only integer values exist for the shift:

( ) ( )

( 2 ) 2

2

2 ( ):

:1

using Euler's formula: cos(2 ) sin(2 ) 1cos 2 1 sin 2 0cos 1 if =0, 2 , 4 ... sin =0 if =0, , 2 ,....

so that 0, 1, 2

im

im im im im

im

i m

f f where f ethen

e e e ecancelling e

em i m

m and m

m

θ

θ θ π θ π

θ

π

φ φ π φ

π ππ πφ φ π π φ φ π π

+

= + =

= =

== + =

= == ± ± ± ±= ± ± , 3,... integer values onlym± =

19

Delta functions

A delta function is an infinitely narrow, infinitely high function whose area is normalized to one. This is a little difficult to understand unless we give a model for a delta function. One model is the square function. Imagine a function whose value is 1/a over the range -a/2 < x < a/2 and 0 outside these values. This function is plotted below for three values of a.

The area under each rectangle is the same because the base length is a and the height is 1/a. Since (1/a)(a) = 1 the area is always one. Imagine we take this function in the limit that a goes to zero. The function becomes narrower and taller, but always has an area of one. In the limit that a goes to zero this is a delta function. Since the delta function δ(x - x0). Note that since the delta function is centered about 0 in this instance the value of x0 is 0. In this case our delta function is δ(x).

In spectroscopy, a delta function is often used to represent a matching condition. If we imagine the energy levels of a molecule as being extremely narrow then the energy of an incident photon must match exactly to the energy difference in order for a transition to occur. The Einstein relation states that E2 - E1 = ΔE = hν. One way to write this for infinitely narrow level widths is to use a delta function δ(ΔE - hν).

Properties

1. The delta function is the eigenfunction of the position operator. For a free particle we can operate with the position operator x (hat). The eigenvalue equation is:

20

( ) ( )ˆ o o ox x x x x xδ δ− = −

The eigenvalue is x0 the actual position of the particle. The delta function (δ) specifies that of all the possible x values only x0 is non-zero.

2. The integral properties of a delta function are as follows:

a. the integral over δ(x - x0) is equal to the function evaluated at x0.

0 0( ) ( ) ( )f x x x dx f xδ

∞

−∞

− =∫

b. The area under the delta function is one.

0( )x x dxδ∞

−∞

1− =∫

3. The value of a delta function is zero everywhere except where the argument is zero.

0 0( ) 0

( ) 0 0

x x for x

x for x

xδ

δ

− = ≠

= ≠

4. A change of argument by a factor results in multiplication by the inverse of the factor.

1( ) ( )kx xk

δ δ=

To see this consider the above rectangular function. The delta function is 1/a over the limits -a/2 < x < a/2. Thus, the height is 1/a and the base is a. If we multiply the height by k then it becomes k/a. This means that we should multiply the base by k. In other words since

( ) 1x dxδ∞

−∞

=∫

21

we require that

( ) 1k kx dxδ∞

−∞

=∫

Thus, ( ) ( )k kx xδ δ= In spectroscopy we make use of this (non-intuitive) property of the delta function to make the equation:

( ) ( )21 21E Eδ ω ω δ+ = + A final point is that any of the lineshape functions are a representation of a delta function. A Gaussian, Lorentzian or sin(x)/x function are all delta functions in the limit that their width goes to zero. In fact

( ) 12

ikxx e dxδπ

∞

−∞

= ∫

is also a representation for a delta function.

8. Some important mathematical definitions and theorems ∇ : Vector Operator

used like ddx

, has no meaning unless a function or scalar is present

ˆˆ ˆi j kx y z∂ ∂ ∂

∇ = + +∂ ∂ ∂

Gradient ( ): Directional derivative vector ∇

ˆˆ ˆgrad i j kx y zφ φ φφ φ ∂ ∂ ∂

∇ = = + +∂ ∂ ∂

the vector is perpendicular to the surface of Φ (= constant at some point, x,y,z) Divergence ( φ∇i ): Flux scalar Dot product

yx z

x y zφφ φφ∂∂ ∂

∇ = + +∂ ∂ ∂

i

22

If φ represents the velocity (change of position per unit time, change in density, etc.) then div is the change per unit volume (the flux). Useful for E-M theory, liquid mechanics, and etc. Curl ( φ∇× ): Rotation (angular velocity) Vector Fundamental to Stoke’s theorem

y yx xz z

x y z

i j k

i j ky z z x x y x y z

φ φφ φφ φφ

φ φ φ

∂ ∂⎛ ⎞ ⎛ ⎞∂ ∂∂ ∂ ∂ ∂ ∂⎛ ⎞∇× = − + − + − =⎜ ⎟ ⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠⎝ ⎠ ⎝ ⎠

div grad ( 2φ φ∇ = ∇⋅∇ ) Laplacian

2φ φ∇ =∇⋅∇ =22 2

2 2yx z

2x y zφφ φ∂∂ ∂

+ +∂ ∂ ∂

Important aspects of Laplacian mathematics in the physical sciences are: ∇ =

∇ =∂∂

∇ =∂∂

2

22

2

2

22

01

1

φ

φ φ

φ φ

Laplace's equation

The wave equation

diffusion equation (of heat conduction)

a t

a t

Green’s Theorem (line integrals)

The double integral of the function ∂∂FHGIKJx Q x y( , ) where an area enclosed by the function

Q(x,y) and P(x,y) has continuous first partial differentials, is equal to the line integral.

∂∂

−∂∂

FHG

IKJ = +

∂zzz Q

xPy

dxdy Pdx QdyAA

The line integral is a counterclockwise rotation of the total area.

Stokes’ Theorem (open surface relationship to the line integral)

23

Vdr ndsurfacecurve bounding

= ∇×∇ •zzz b g σσσ

Butterfly net: the rim is the curve bounding the surface made of the net. Surfaces treated with this theorem must be two sided in order to know the sense of the normal vector ( not a Moebius surface). Matrices Definition of a matrix

A matrix is a two-dimensional array. The dimensions of a matrix are given by its number of rows n and columns m. We designate a matrix by a bold letter. The matrix A is A. The dimensions of a are n x m (n by m). The elements of a are aij. The index i is the row index and the index j is the column index. We represent A as:

11 12 13

21 22 23

31 32 33

...

...

...

a a aA a a a

a a a

⎛ ⎞⎜ ⎟= ⎜ ⎟⎜ ⎟⎝ ⎠

Special cases

A square matrix has n = m.

A diagonal matrix D has values only along the diagonal d11, d22, d33, etc. and zero elsewhere.

11

22

33

0 0...0 00 0 ...

dD d

d

⎛ ⎞⎜ ⎟= ⎜ ⎟⎜ ⎟⎝ ⎠

...

The unit matrix or identity matrix I has ones along the diagonal.

1 0 0...0 1 0...0 0 1...

I⎛ ⎞⎜ ⎟= ⎜ ⎟⎜ ⎟⎝ ⎠

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

24

1

2

3...

rr r

r

⎛ ⎞⎜ ⎟= ⎜ ⎟⎜ ⎟⎝ ⎠

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

( )1 2 3...r r r r= Matrix addition and subtraction

Two matrices can be added or subtracted only if they have the same dimesions. The result of adding matrices A and B is obtained by adding their corresponding elements. A + B = C is obtained by adding aij + bij = cij for all i and j.

11 12 13 11 12 13 11 12 13

21 22 23 21 22 23 21 22 23

31 32 33 31 32 33 31 32 33

... ... ...

... ... ...

... ... ...

A B Ca a a b b b c c ca a a b b b c c ca a a b b b c c c

+ =

⎛ ⎞ ⎛ ⎞ ⎛⎜ ⎟ ⎜ ⎟ ⎜+ =⎜ ⎟ ⎜ ⎟ ⎜⎜ ⎟ ⎜ ⎟ ⎜⎝ ⎠ ⎝ ⎠ ⎝

⎞⎟⎟⎟⎠

b

Matrix multiplication

In order to multiply two matrices AB = C the number of columns in the A matrix must be the same as the number of rows in the B matrix. If A is a k x l matrix and B is a l x m matrix, then C has dimensions k x m. The prescription for matrix multiplication is:

1

l

ij ir rjr

c a=

= ∑

The first row elements of the A matrix are multiplied by the first column elements of the B matrix to give the first element of the C matrix.

11 12 13 11 12 13 11 12 13

21 22 23 21 22 23 21 22 23

31 32 33 31 32 3331 32 33

... ... ...

... ... ...

... ......

AB C

a a a b b b c c c

a a a b b b c c ca a a c c cb b b

=

⎛ ⎞⎛ ⎞ ⎛ ⎞⎜ ⎟⎜ ⎟ ⎜ ⎟

=⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠⎝ ⎠

Note that the above equation indicates the sum of the products of the elements c11 = a11b11 + a12b21 + a13b31 + … until the row and column are finished.

The c21 element is obtained in a similar fashion:

25

11 12 1311 12 13 11 12 13

21 22 23 21 22 23 21 22 23

31 32 33 31 32 3331 32 33

...... ...... ... ...

... ......

b b ba a a c c ca a a b b b c c c

b b b c c ca a a

⎛ ⎞⎛ ⎞ ⎛ ⎞⎜ ⎟⎜ ⎟ ⎜ ⎟

+ =⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠ ⎝ ⎠

This process is continued until all of the elements of the C matrix have been obtained.

The order of operation is important. This can be simply illustrated with the following important example.

The product of a row matrix and a column matrix is a scalar.

( )1

1 2 3 2 1 1 2 2 3 3

3

... ...

...

ba a a b a b a b a b

b

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟ = + + +⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

The product of a n x 1 column matrix and a 1 x n row matrix is a n x n matrix.

( )1 1 1 1 2

2 1 2 3 2 1 2 2 2 3

3 3 1 3 2

...... ...

......

b b a b ab a a a b a b a b ab b a b a

⎛ ⎞⎜ ⎟ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎜ ⎟ = ⎜ ⎟⎜ ⎟ ⎜ ⎟

⎝ ⎠⎜ ⎟⎜ ⎟⎝ ⎠

1 3

3 3

b a

b a

⎞⎟⎟⎟⎠

Matrix Transpose The transpose of a matrix is obtained by interchanging the rows and columns of a matrix.

11 12 11 21

21 22 12 22

31 13

... ...

... , ...... ...

T

a a a aA a a A a a

a a

⎛ ⎞ ⎛⎜ ⎟ ⎜= =⎜ ⎟ ⎜⎜ ⎟ ⎜⎝ ⎠ ⎝

Matrix Inverse

The inverse of a matrix is defined such that AA-1 = A-1 A = I where I is the identity matrix.

The determinant of a matrix

The determinant of a matrix is a number obtained by summing n! products of a n x n matrix. This is best illustrated by starting with a 2 x 2 matrix. The determinant of a 2 x 2 matrix is:

26

11 2

11 22 12 2121 22

a aDet a a a a

a a⎛ ⎞

= −⎜ ⎟⎝ ⎠

For larger dimension matrices, the determinant can be obtained by reduction to lower-order determinants (minors). This is illustrated for a 3 x 3 matrix.

11 12 13

22 23 21 23 22 2321 22 23 11 12 13

32 33 31 33 32 3331 32 33

a a aa a a a a a

Det a a a a Det a Det a Deta a a a a a

a a a

⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎜ ⎟ = − +⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎜ ⎟

⎝ ⎠The 2 x 2 determinant is obtained as above for each of the minors.

The determinant of A is obtained by summing across any row as follows:

( )1

| 1n

i j ijij

i

A a+

=

= −∑ A

Matrix solution of linear equations

Matrices are important tools for the solution of linear equations. The general form of a set of simultaneous linear equations is

Ax = b. This illustrated below:

11 12 13 1 1

21 22 23 2 2

31 32 33 3 3

...

...

...... ...

a a a x ba a a x ba a a x b

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎛ ⎞⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟=⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟

⎝ ⎠⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

Here the xi are the unknowns and aij are the linear coefficients. To find the ith coefficient xi we replace that column of the A matrix by the b vector.

11 1 13

2 21 2 23

31 3 33

...

...

...

a b aA a b a

a b a

⎛ ⎞⎜ ⎟= ⎜ ⎟⎜ ⎟⎝ ⎠

The unknown is given by the ratio of the determinants,

( )( )

ii

Det Ax

Det A=

27

Braket notation In quantum spectroscopy it is commonly necessary to solve integrals of the type

*m nA dτΨ Ψ∫

where the Ψm and Ψn are wavefunctions, A (hat) is an operator, and the volume element dτ represents the integral over all space. The significance of this expression is that it gives the average of the physical quantity corresponding to operator A for the wavefunctions in question. There are several equivalent ways of writing this integral. The above integral can be rewritten using the following abbreviations:

*m n m nA d A m AτΨ Ψ = Ψ Ψ =∫ n

Keep in mind that all of these expression mean integration over all space. These expressions are referred to as matrix elements. This is because often there are many different wavefunctions for different electronic, vibrational, or rotational states of a molecule and the operator may represent the transitions between them. This is often the case in spectroscopy. For example, if the operator A is the dipole operator then the above matrix element represents transition dipoles between states m and n. In the same manner the overlap of two states can be expressed as

* | |m n m nd mτΨ Ψ = Ψ Ψ =∫ n Another way to view the above terms is that they represent the projection of one state onto another. The shorthand notation with angle brackets is a result of P.A.M. Dirac . This is known as braket notation. One important aspect of the shorthand notation is that the bra ⟨m| implies that the complex conjugate of Ψm* is represented. The ket |n⟩ represents Ψn. Note that since:

**

m n m nd dτ τΨ Ψ = Ψ Ψ∫ ∫

then :

*m n n m=

There is no particular significance to the choice of symbol for the wavefunction. Variously Ψ, ψ, φ, χ, or the ket |n⟩ is used to represent a wavefunction. The average properties and probabilities are required to be real. In other words Ψ*Ψ must be real. Wavefunctions that satisfy this condition are called Hermitian. In order to solve matrices (groups of vectors or spatial functions, etc.) that contain imaginary and real parts, performing operations requires them to be Hermitian. A square matrix is Hermitian if the complex matrix transpose is equal to the original matrix:

( )* TA A=

28