QUANTUM MECHANICS

Classical Mechanics: Mechanics of particles (finite position, finite momenta)

F = ma

= + = +2

1 22

( , , ) ( , , )2pE mv V x y z V x ym

z

Quantum Mechanics: mechanics of waves.

A simple harmonic wave that moves with a constant velocity v can be written as

π νλ

Ψ = − 1( , ) cos 2 ,xx t t (1)

where λ is the wave length andν is the frequency. The velocity of the wave is the product of length of each cycle,λ, and the number of cycles in unit time, ν, so that

λν= .v (2)

The fundamental relationship in quantum mechanics is

ν= ,E h (3)

where h is the Plank’s constant = 6.636 × 10–34 J s. This is a revolutionary equation because it relates the energy to the frequency rather than the amplitude, as in the case of a classical wave.

A harmonic wave can also be described by the sine function:

π νλ

Ψ = − 2( , ) sin 2 .xx t t (4)

More generally, one could combine the two relationships (1) and (4), and write

ω−Ψ = (( , ) ,i kx tx t e ) (5)

where k = 2π/λ, ω = 2πν. The wave velocity v = k/ω.

The de Broglie Relationship

For a particle of zero rest mass (photon), the momentum is given by

ν

λ= = = .E h hpc c

(6)

Extending this relationship to material systems, Louis de Broglie in 1924, postulated that

λ =hmv

(7)

for particles with non-zero rest mass. In other words, particles have wave properties.

1

This relationship was verified rather quickly by electron diffraction experiments. The de Broglie wavelength establishes a useful (but soft) boundary between classical and quantum systems. Particles those whose de Broglie wavelengths are larger than or comparable to their dimensions require quantum mechanics to describe their behavior. The momentum “operator” Returning to Eq. (6), we see that

π

= = .2 /hpk

k (8)

Note that we have introduced a new constant, = h/2π (pronounced “h-bar”) which is, in a sense, more fundamental to quantum mechanics than Planck’s constant h. If we wished to “extract” the momentum k from the wave in Eq. (5), one way to do that is to differentiate the wave function with respect to x, then multiply it by /i, as in

ω ω−∂Ψ= =

∂( ) ( .i kx t i kx t

xk e p ei x

− ) (9)

For this reason, we will now call the mathematical operator ( /i)∂/∂x, the momentum operator in one dimension.

The Kinetic Energy “operator” The kinetic energy operator (in one dimension) can be “derived” from the momentum operator by recognizing that kinetic energy is given by p2/(2m) = k2 2/(2m). Therefore,

∂ ∂Ψ − ∂ Ψ = = ∂ ∂ ∂

2 2 2 2

21 .

2 2k

m i x i x m x mΨ

2 (10)

In other words, the kinetic energy operator in one dimension is

− ∂

∂

2 2

2 .2m x

The Schrödinger Equation

Let us now write the second fundamental equation of classical mechanics in quantum mechanical language, by recognizing that in quantum mechanics, E = ω [and using tactics similar to those employed in Eq. (9) to get ω∂Ψ ∂ = Ψ = Ψ( / )i t ], E

− ∂ Ψ ∂Ψ

+ Ψ =∂ ∂

2 2

2( , ) ( ) ( , ) .

2x t V x x t i

m x t (11)

This is the fundamental equation of non-relativistic quantum mechanics, namely, the Schrödinger equation. In three-dimensional Cartesian space, this equation becomes

− ∂

∇ + Ψ = ∂

22 ( , , , )( , , ) ( , , , ) ,

2x y z tV x y z x y z t i

m tΨ

(12)

2

where, of course,

∂ ∂ ∂

∇ = + +∂ ∂ ∂

2 2 22

2 2 .x y z2

(13)

The operator in square brackets on the left hand side of Eq. (12) is called the Hamiltonian operator for the system being studied. It is clear that it is the definition of the potential energy that distinguishes one system from another.

The time-independent Schrödinger Equation If the potential energy function is independent of time, it is possible to show that we may write

(14) ψΨ = /( , ) ( ) ,iEtx t x e

where ψ(x) satisfies the time-independent Schrödinger equation,

ψ ψ ψ−

+ =2 2

2 ( ) ( ) ( ).2d V x x E x

m dx (15)

In these lectures, we will be concerned with this equation and its solutions for a few systems that can serve as useful models for real systems.

The solutions of Eqs. (11), (12), or (15) are called wave functions.

Physical Interpretation of the wave function The wave function of Eq. (5) is complex, i.e., it has a real and an imaginary part, because

cos sin .ie iθ θ θ= +

Let a complex number c = a + ib. The complex conjugate of c is denoted as c* and is defined as c* = a – ib. For any complex number or function, the complex conjugate is obtained simpy replacing all occurrences of i with –i, where i is 1− . The “absolute-value-squared” of a complex number (or function) is the product of the number (or function) and its complex conjugate, denoted by

*2 .c c= c

There is no universally accepted consensus as to the physical meaning of the wave function itself. The commonly accepted interpretation for or ( , )x tΨ 2( , )x tΨ is that of Max Born, who

postulated that 2( , )x tΨ is the probability of finding the system at a specific point in space, x, at

a given time, t. Considering the time-independent wave functions, the quantity 2( )x dxψ is the probability that the one-dimensional system under study is in the line segment (x, x+dx). Likewise, 2( , , )x y z dxdydzψ is the probability of finding a three-dimensional system in a volume element defined by [(x, x + dx), (y, y + dy), (z, z + dz)].

It is important to note that this is not a physical interpretation of the wave function itself. The wave function itself has positive and negative values in different parts of space. The function

2Ψ is a probability distribution function that is non-negative everywhere in space.

3

Normalization

Since the probability for the system (let us think of an electron) existing somewhere in the Universe must be 1, we require that all wave functions satisfy the normalization condition

* 1dxψ ψ∞

−∞

=∫ (16)

where the integration is from one end to the other of a one-dimensional Universe.

Measurement: Eigenvalues and Expectation Values It is a postulate of quantum mechanics that every measurable (observable) property of a quantum system is represented by an operator that acts on the wave function. Let us denote these operators by the generic symbol , where the “hat” is to indicate that we are dealing with an operator, not a variable. Depending on the nature of the operator and the wave function, two kinds of outcomes are possible:

A

1) ,A aψ ψ= where a is a constant. Example: ( ) ( )ikx ikxd e k ei dx

= . Such wave

functions are called eigenfunctions of the operator under consideration, and the constant that results is called the eigenvalue. The Hamiltonian operator, designated as H [see Eq. (12)] satisfies the relationship

.H Eψ ψ=

Quantum mechanics, therefore, is primarily concerned with finding, and studying the properties of, eigenfunctions of the Hamiltonian operator.

2) ,Bψ ϕ≠ where ϕ is a different function. Example: sin( ) cos( ).d kkx kxi dx i

= ×

Clearly, sin(kx) is not an eigenfunction of the momentum operator.

The physical act of making a measurement on a quantum system is a hotly debated subject. The mathematical apparatus for quantum measurements consists of evaluating integrals very much like those used for calculating average values from probability distribution functions.

Consider the Maxwell-Boltzmann molecular speed distribution function for a gas at a specific temperature:

2

3/22 /(( ) 4 ,

2Bmv k T

B

mF v v ek T

ππ

− =

)

where kB = 1.381 × 10–23 J K–1 is Boltzmann’s constant. The average speed for an ensemble of molecules can be calculated from the distribution function as

0

( ) ,v vF v dv∞

= ∫

and similar integrals for other properties.

4

Now consider the fact that 2( )xψ is a probability distribution function. The average value of a property then, can be considered as the result of evaluating the integral

2

.A A dψ x∞

−∞

= ∫

However, quantum operators act on ψ, not 2( )xψ . Therefore, the integral we must actually evaluate is

* ,A Aψ ψdx∞

−∞

= ∫ (17)

for a one-dimensional system. The quantity A is called the expectation value of A . The uncertainty in measuring the property A is the standard deviation of A, which is calculated using standard statistical relationships as

22

2* * .

A A A

AA dx A dxψ ψ ψ ψ∞ ∞

−∞ −∞

∆ = −

= −

∫ ∫

(18)

When 22 ,A A= ∆A = 0, and we can measure the property exactly, at least in principle. This

is indeed the case if ψ happens to be an eigenfunction of A : ,aAψ ψ= and we get

* * 2 *

2* * 2

1.

1.

AA dx a A dx a dx a

A dx a dx a

ψ ψ ψ ψ ψ ψ

ψ ψ ψ ψ

∞ ∞ ∞

−∞ −∞ −∞

∞ ∞

−∞ −∞

2= = =

= = ×

∫ ∫ ∫

∫ ∫

×

The Heisenberg Uncertainty Relations As explained in the handout “Introduction to Quantum Mechanics,” Heisenberg’s Uncertainty Principle places a fundamental limit on the accuracy with which certain pairs of properties can be simultaneously measured. One such pair is the position and the momentum of a particle. In one dimension, this principle is expressed as

.2xx p∆ ∆ ≥ (19)

Since the right hand side is a constant, this relationship implies that nature places a fundamental limit on the accuracy with which the positions and momenta of quantum particles can be simultaneously known, regardless of the sophistication of the measurement method.

5

Requirements of an acceptable wave function

1. An acceptable wave function ψ(x) must be continuous, must have continuous first derivatives, and the second derivative must be non-zero somewhere in space.

2. An acceptable wave function ψ(x) must be square integrable, i.e.,

2 .dxψ∞

−∞

< ∞∫

An additional requirement will be introduced later for a wave function describing more than one particle.

A few “toy” quantum systems Let us consider the artificial case where the potential energy is constant over a certain interval in space and abruptly assume other, but constant, values elsewhere. As a specific example, let us say V(x) = V0 for –∞ ≤ x < 0, V(x) = 0 for 0 ≤ x ≤ a, and V(x) = V0 for a < x ≤ ∞. A sketch of this potential is shown in the figure.

E

a 0

V0

x

1 2 2 3

The “system” naturally divides into three regions, as shown. Let us write the Schrödinger equation in each region for the total energy E indicated by the dashed line.

( )

( )

21

0 12 2

22

22 2

23

0 32 2

2 0; 0

2 0; 0

2 0;

d m E V xdx

d mE x adx

d m E V a xdx

ψ ψ

ψ ψ

ψ ψ

+ − = − ∞ ≤

− = ≤ ≤

+ − = < ≤ ∞

<

(20)

6

It is easy to show (show this!) that functions of the form jik xj jN eψ ±= (j = 1,2,3) are solutions

in each region denoted by the index j if we define

1 3 0

2

2 ( ) / , and

2 / .

k k m E V

k mE

= = −

= (21)

Now, it is clear that E < V0. Therefore, k k1 3 ,iκ= = where 02 ( ) / .mV Eκ = − With this substitution, we get

2

1 1

2 2

3 3

( ) ,

( ) ,

( ) .

x

ik x

x

x N e

x N e

x N e

κ

κ

ψ

ψ

ψ

±

±

∼

∼

∼

(22)

The square-integrability requirement on the wave function requires that we should eliminate from consideration functions that exponentially grow as x → ±∞. In region 2, we may wish to choose the trigonometric form to give us additional flexibility. Therefore, we choose

1 1

2 2 2

3 3

( ) ; 0,

( ) cos( ) sin( );0 ,

( ) ; .

x

x

x N e x

x A k x B k x x a

x N e a x

κ

κ

ψ

ψ

ψ −

= −∞ ≤ <

= + ≤

= < ≤ ∞

≤ (23)

In other words, we get exponentially decaying functions in regions 1 and 3, and an oscillatory function in region 2. To determine the unknown constants, we will need to apply boundary conditions, which are given to us by the requirement that the wave function and its derivative be continuous. In other words, we require

1 21 2

0 0

2 32 3

(0) (0), and ,

( ) ( ), and .

x x

x a x a

d ddx dxd da adx dx

ψ ψψ ψ

ψ ψψ ψ

= =

= =

= =

= =

This will allow us to determine all constants in terms of one undetermined constant. If necessary, the final undetermined constant is determined by the normalization requirement of Eq. (16). Often, we are interested in the ratios of the constants to each other and not their absolute magnitudes. The handout “Linear Motion” on the course web page provides a detailed analysis of four cases of constant potentials including the one just discussed.

One of the great “surprises” of quantum mechanics is that the wave function extends into regions 1 and 3. Classically, the system should not enter these regions because there the potential energy is higher than the total energy, i.e., the kinetic energy is negative! Since 2( )xψ at a given x is related to the probability of finding the particle there, quantum mechanics predicts that the particle can indeed enter these regions, although the probability exponentially dies off as one goes deeper into the classically forbidden regions. This tendency of quantum systems to penetrate the classically disallowed regions is called quantum mechanical tunneling.

7

The infinite well

Let us consider the special case where V0 = ∞. Then ψ1 = 0 everywhere in region 1, and ψ3 = 0 everywhere in region 3. Therefore, due to the requirement that the wave function be continuous, we require that ψ2(0) = 0 = ψ2(a). From Eq. (23), we therefore set 2 2 2(0) cos( 0) sin( 0) 0.A k B kψ = + =

Obviously, this condition can be satisfied only if A = 0. Therefore, we get ψ2(x) = B sin(k2x). Now, the second boundary condition, ψ2(a) = 0, is satisfied only if k2 = nπ/a, where n = 1,2,3,... Thus, we get

2( ) sin ; 1,2,3,...nx B x naπψ = =

(24)

Since 2 2 /mE= ,k we get

2 2 2 2 2 2 22

2 2 ; 1,2, 3,...2 2 8nk n n hEm ma ma

π= = = =n (25)

Only the energies given by Eq. (25) are associated with wave functions that satisfy the boundary conditions. In general, a quantum system bound to a region of space by confining potentials can only have discrete energy levels, given by expressions similar to Eq. (25), which are associated with wave functions which satisfy the boundary conditions, similar to those given by Eq. (24).

The constant B in Eq. (24) is determined by the normalization condition

2

02 2 2

1 2 30

2 2

0

1, which gives

1, or

sin 1,

a

aa

dx

dx dx dx

nB x dxa

ψ

ψ ψ ψ

π

∞

−∞∞

−∞

=

+ + =

=

∫

∫ ∫ ∫

∫

which leads to the result B = (2/a)1/2.

The particle in the one-dimensional infinite well can be used as a model for various systems of interest in the context of nanotechnology. Some examples are, an electron on a superconducting nanowire terminated in two insulators, or an electron or an ion trapped in a carbon nanotube. A system of chemical interest that appears to be well-described by this model, at least at as a first approximation, is the iodine-starch complex which has a deep blue color.

The particle in the infinite well will only absorb light whose frequency matches the differences in the energy levels, i.e.,

( )2

2 22 ,

8f ihh E n nma

ν = ∆ = − (26)

where “f” and “i” denote initial and final states. The absorption (or emission) of radiation due to transitions between discrete energy levels of bound quantum systems is the general principle behind all forms of spectroscopy.

8

The Harmonic Oscillator

The harmonic oscillator is the model for a “perfect” spring connecting two masses A and B which is a useful model for examining the quantum properties of a vibrating diatomic molecule. A perfect spring is one for which the force required to change its length increases linearly with length: F = khx; x = (r – re), where r is the actual length of the spring, and re is its “equilibrium” length. The potential energy for such a system is

1 22

0 0

( ) ( ) .x x

hV x F y dy k ydy k x= = =∫ ∫ h (1)

Since we are only interested in the relative motion of the two masses, we express the Schrödinger equation in terms of displacement coordinates (displacement relative to the equilibrium length re). The result of this transformation is that the two masses mA and mB are replaced with a single “reduced” mass, µ = mAmB/(mA + mB). Therefore, we get

2 2

22

1( ) ( ) ( )2 2 hdH x k x x E xdx

ψψ ψµ

−= + = ψ (2)

where E the total energy of the system.

The classical picture that we obtain for the behavior of the harmonic oscillator is as follows. During the vibrations of the spring, the spring extends outwards or compresses inwards from its equilibrium length until E = V(x), which occurs when The values of x for which this occurs are called the classical turning points, because the spring reverses itself at these points and starts to travel in the opposite direction. Let us denote these points by ±a = (2E/kh)1/2. Classically, the spring cannot access the regions x < –a or x > a. Therefore, we call these the classically forbidden regions.

Quantum mechanicajust discussed for theexpect the solution towe expect wave-likeproblem therefore, ob

Imposing these condithose corresponding

lly, however, these regions are very similar to the x < 0 and x > a regions finite potential well problem. In these regions, E < V(x) and, therefore, we exhibit exponentially decaying behavior whereas in the interval –a ≤ x ≤ a, oscillatory behavior. Acceptable solutions of the harmonic oscillator ey the boundary conditions

( ) 0xLim xψ→±∞

= .

tions leads to the result that of all the possible solutions to Eq. (2), only to certain energies E are selected.

The quantized energy levels corresponding to the acceptable solutions of the harmonic oscillator problem are given by

( )12

, where 0,1,2,...v hE v h vν= + = (3)

Note that v is used here as a quantum number and has nothing to do with velocity. The quantity νh is called the harmonic frequency of the oscillator, and is related to the spring constant (commonly called force constant) and the reduced mass as follows:

1 .2

hh

kνπ µ

= (4)

It is very important to note that the lowest possible energy of the quantum oscillator is not zero, but rather ½hνh, which is called the zero point energy of the oscillator.

The solutions of the Schrödinger equations for this case is written as

2 /2( ) ( ) x

v v vx C H x e αψ α −= (5)

where α = 2πµνh/ , Cv is the normalization constant, and Hv(y) are a class of orthogonal polynomials called Hermite polynomials. The figure below shows a few of the wave functions of the harmonic oscillator potential corresponding to the values of v shown.

The wave functions are extremely sensitive to the energy. If we try to plot the wave functions corresponding to energies even slightly different from that predicted by Eq. (3), we get functions that will have to be rejected for failing to meet one or more of the requirements for an acceptable wave function. This is illustrated in the figure below. For v = 0, the correct energy is, of course, 0.500hνh. The wave functions we get for 0.499hνh and 0.501 hνh clearly violate the square-integrability criterion.

Afvea

s mentioned earlier, the harmonic oscillator is a reasonably good model for the potential energy

unction of a diatomic molecule at values of the interatomic distance r close to the equilibrium alue re, in other words, in the region close to x = 0. A sketch of a typical molecular potential nergy function along with the harmonic oscillator potential (with the value of kh adjusted to get good fit near x = 0) is shown below.

Quantum Mechanical Treatment of Atoms The Schrödinger equation in spherical polar coordinates

When dealing with the Schrödinger equation for atoms, Cartesian coordinates are not very convenient for applying boundary conditions. Therefore, we switch to spherical polar coordinates, which are related to Cartesian coordinates (x,y,z) as follows:

X

Z

Yφ

θ r

(x,y,z)

x = r sin θ cos φ

y = r sin θ sin φ

z = r cos θ The volume element dxdydz in these new coordinat es is given by

dxdydz = r2dr sin θ dθ dφ. The ranges of the new coordinates are as follows:

0 ≤ r ≤ ∞, 0 ≤ θ ≤ π, 0 ≤ φ ≤ 2π. The (three-dimensional) kinetic energy operator in these coordinates has the following form:

2 2 2

2 22 2 2 21 1 1sin .

2 2 sin sinr

m m r rr r rθ

θ θθ θ − − ∂ ∂ ∂ ∂ ∂ ∇ = + + ∂ ∂ ∂ ∂ ∂

2φ (1)

Let us consider a system of two masses mA and mB connected by a rigid rod of length r, rotating in three dimensional space. Assume that the potential energy is exactly zero everywhere. As in the case of the harmonic oscillator, we can convert this to a system of a single mass µ rotating at a distance r about a point in space; this is the “particle on a sphere” problem. The Schrödinger equation is obtained directly from Eq. (1) by remembering that r is a constant and, therefore, the solutions will have no r-dependence:

2 2

2 2 2 21 1sin ( , ) ( , ).

2 sin sinE

r rθ ψ θ φ ψ θ φ

µ θ θθ θ φ − ∂ ∂ ∂ + = ∂ ∂ ∂

(2)

In order to ensure the continuity of the wave function and its single-valuedness, we impose the following boundary conditions: (0, ) ( , ); ( , 0) ( ,2 ).ψ φ ψ π φ ψ θ ψ θ π= ± = The solutions that solve Eq. (2) and satisfy the boundary conditions are called sherical harmonics, and are commonly denoted as Y ( , ).m

l θ φ They can be expressed as a product of two functions as follows:

,1( , ) (cos ) ,2

m ml l m lY N P eimφθ φ θ

π= (3)

where l = 0, 1, 2, …. ∞, and m = 0, ±1, ±2, …, ±l.

1

The functions (cos )mlP θ are called associated Legendre polynomials. The normalization

constants Nl,m can be found by the usual procedure:

2

2 2,

0 0

1 sin ( ,ml m lN d d Y ) .

π π

θ θ φ θ φ= ∫ ∫

The energy values resulting from these spherical harmonics are given by

2

, 2( 1) ; 0,1,2,..., .2l mE l l lrµ

= + = ∞ (4)

Note that the energy does not depend on the m quantum number. Since, for a given value of l, there are (2l+1) allowed values for m (0, ±1, ±2, …, ±l), each energy level of the system has 2l+1 states associated with it.

Angular momentum

The angular momentum of a particle is a vector defined as the cross-product of the position vector r and the linear momentum vectorp :

.

x y z

i j kL r p x y z

p p p= × = (5)

The cross-product of two vectors is a new vector oriented perpendicular to the plane containing the other two vectors. The Cartesian components of the angular momentum vector are obtained by expanding the determinant:

(6) ( ) ( ) (

.

z y x z y x

x y z

L i yp zp j zp xp k xp yp

iL jL kL

= − + − + −

= + +

)

The quantum mechanical expression for the square of the angular momentum L2 = L is L•Obtained by substituting the momentum operators for the momentum components in Eq. (6), and converting the expression to spherical polar coordinates:

22 2

2 21 sin .sin sin

Lr r

θθ θθ θ

∂ ∂ ∂ = − + ∂ ∂ ∂ 2 2

1φ (7)

So, it turns out that the Schrödinger equation for the particle on a sphere problem can be expressed as

2 2

2 ( , ) ( 1) ( , ).2 2

ml

L Y l l Yr r2

mlθ φ

µ µ= + θ φ

ml

This also shows that the spherical harmonics are eigenfunctions of L : 2

2 2( , ) ( 1) ( , ).mlL Y l l Yθ φ = + θ φ (8)

The quantum mechanical operator corresponding to Lz has a particularly simple form in spherical polar coordinates:

2

.zLi φ

∂=

∂

It is easy to verify that the spherical harmonics are also eigenfunctions of : zL

,1( , ) (cos ) ( , ).2

m m imz l l m lL Y N P e m Y

iφ m

lθ φ θφ π∂ = = ∂

θ φ

Therefore, it is possible to measure the properties associated with L and exactly and simultaneously. The property associated with

2zL

zL is, of course, the Z-component of the angular momentum vector. The length of the vector is related to the

2L operator as follows: the length

of the angular momentum vector for a particular (l,m) state is 2 ( 1)L l l= • = = + .L L L A geometrical picture of the (l,m) states can be obtained from these results as follows: for any given (l,m), the quantity ( 1)l l + is the length of the vector, and the quantity m gives the value of the Z-component of the vector. As a consequence of Hesenberg’s Uncertainty Principle, we cannot know all three components of the quantum angular momentum vector simultaneously and precisely.

The Central Force Problem: The Hydrogen Atom The potential energy function in the case of the Hydrogen-like atom is determined by the laws of electrostatics. For two charges Ze (the nucleus with Z protons) and –e (the electron) separated by a distance r, the potential energy is given by

2

0 0

( )( ) 1 1( ) ,4 4Ze e ZeV r

r rπε πε− −

= = (9)

where ε0 is the permittivity of vacuum, and the 4πε0 factor is needed to enforce SI units. Strictly speaking, in the Schrödinger equation for this two particle problem (nucleus and the electron), we should use the reduced mass µ = mHme/(mH + me) but, since the mass of the nucleus is much larger than that of the electron, µ ≈ me. Therefore, we write the Schrödinger equation as

22 2

22 2

0

1 1 ( , , ) ( , , ).2 42e e

L Zer rm r r rr m r

ψ E rψ θ φ ψ θ φπε

− ∂ ∂ + − = ∂ ∂ (10)

Since the only part that depends on the angles θ and φ is the operator, the solution has to be of the form

2L

( , , ) ( ) ( , ),mlr DR r Yψ θ φ θ φ=

where D is the normalization constant, and R(r) is a radial function to be determined by solving the equation. Substituting this form for ψ(r,θ,φ) into the Schrödinger equation above and simplifying, we get

3

2 2 2

22 2

0

1 ( 1) 1 ( ) ( ).2 42

ll l

e e

R l l Zer Rm r rr m r πε

− ∂ + + − = ∂ r ER r (11)

The subscript on the solutions R(r) is to explicitly show that they will depend on the value of the angular momentum quantum number l, since the quantity in the square brackets in Eq. (11) depends on the value of l. We also require the solutions to satisfy the boundary condition that

so as to ensure square-integrability. Such solutions belong to a class of

polynomials known as associated Laguerre polynomials. Some of the details of solving Eq. (11) and applying the boundary conditions are given in a separate handout (Central Force Problem). As explained in that handout, the application of the boundary conditions results in a new quantum number n, and since the form or R(r) depends on n as well as l, it is now denoted as R

lim ( ) 0rR r

→∞=

nl(r). The complete solutions of the hydrogen atom problem are thus

, ,

2 2

20 0

( , , ) ( ) ( , ),

1 .8

mn l m nl nl lr D R r Y

Z eEa n

ψ θ φ θ φ

πε

=

= − (12)

The quantity a0 appearing in Eq. (12) is called the Bohr radius, and is defined as

2

00 2

4 ,e

am eπε

=

and has a value of 0.529177 × 10–10 m or 0.0529 nm.

The ranges of the quantum numbers for which the solutions satisfy the boundary conditions are:

1,2,3,...,

0,1,2,...,( 1)

0, 1, 2,...,

n

l n

m l

= ∞

= −

= ± ± ±

Note that the energy levels depend only on n. Here, n is called the principal quantum number, l is the angular momentum quantum number, and m is called the magnetic quantum number. The solutions for a given set of (n,l,m) quantum numbers are called atomic orbitals. The explicit forms of several atomic orbitals are tabulated in the handout “Central Force Problem. Plots of a few radial functions Rnl(r) and the three-dimensional isosurface plots of a few atomic orbitals are provided as an additional handout. Physical Significance of the Orbital Quantum Numbers It is customary to associate letter symbols to various values of l as in

Value of l 0 1 2 3 Letter symbol s p d f

so that we would refer to the orbitals with n = 1, l = 0 as the 1s, that with n = 3, l = 1 as the 3p and so on. The probability of finding the electron in the ground state of a hydrogen atom in a small element of space defined by [(r, r + dr);(θ, θ + dθ), and (φ + dφ)] is given by

4

1s(r, , )2r2dr sin d d = 1s(r)

2r2dr sin d d In the second equality above, we explicitly take into account the fact that the 1s wave function does not depend on the angles. Let us now integrate this function over the angles to obtain the probability as a function of the distance from the nucleus:

2

2 22 21 1

0 0

( ) sin ( ) 4 ( ) .nl s sP r dr r dr d d r r r drπ π

θ θ φ ψ π ψ= =∫ ∫

This quantity is called the radial probability distribution function and gives the probability of finding the electron in a spherical shell of radius r and thickness dr. The maximum in a plot of the radial distribution function vs. r is the most probable distance of the electron. For the ground state of the hydrogen atom, that value happens to be exactly equal to the Bohr radius, a0. The average distance of the 1s electron from the nucleus is given by the expectation value

…r1s   = 4 ¶0

∞

1s& (r)r 1s(r)r2dr

= ¶0

∞

rP1s(r)dr

= 32 a0.

A more general result is

…rnl   = ¶0

∞

rPnl(r)dr

= n2a0Z 1 + 1

21 −

l(l + 1)n2

.

which leads to the conclusion that the excited state electrons, on average, are further away from the nucleus.

Spin quantum numbers

Experiments with high magnetic fields revealed the existence of another type of magnetic moment in the electron. Goudsmit and Uhlenbeck proposed in 1925 that an electron has an intrinsic angular momentum S (called the “spin” angular momentum) and an accompanying magnetic moment µs. They further proposed that

(a) the length of the vector S is |S| = [s(s+1)]1/2®, where s ≡ ½ [therefore, |S| = (√3/2)®], and

(b) the Z-component Sz of this “spin” angular momentum is ms®; ms = ±½.

Therefore, the complete specification of the state of an electron in an atom requires that four quantum numbers, n, l, m and ms, be specified. The wave function labeled by these four quantum numbers corresponds to a specific spin state of the electron and is called a spin orbital.

The spin part of a spin orbital corresponding to ms = ½ is denoted by α, and that for ms = –½ is denoted as β. Therefore, a 1s electron could be in a 1s α or a 1s β spin orbital.

5

Pauli Principle

Wolfgang Pauli discovered an important and fundamental property of multi-particle quantum systems consisting of particles like the electrons, which have half-integer intrinsic spins (1/2, 3/2, 5/2, etc..). Such particles are called Fermions, named after the Italian-American scientist Enrico Fermi. The Pauli Principle may be stated as follows:

“The total wave function of a system of interacting Fermions must be antisymmetric with respect to the exchange of any two particles.”

To illustrate this idea, let us examine the Helium atom, which has two electrons. The Schrödinger equation for this system is

2

1 1 1 2 2 2 1 1 1 2 2 2 1 1 1 2 2 20 12

1( , , ) ( , , ) ( , , , , , ) ( , , , , , ),4eh r h r r r E r rr

θ φ θ φ ψ θ φ θ φ ψ θ φ θπε

+ + =

φ

)

(13)

where the operators h are given by the left hand side of Eq. (10), with Z = 2. The third term on the left hand side represents the repulsive interaction between the two electrons, and r12 is the separation between them. Because of this term, the Schrödinger equation cannot be solved except numerically. However, an approximate solution can be written down by neglecting the electron-electron repulsion:

1 1 1 2 2 2 1 1 1 1 1 2 2 2( , , , , , ) ( , , ) ( , ,

1 (1)1 (2),s sr r r r

s s

ψ θ φ θ φ ψ θ φ ψ θ φ=

= (14)

where we have adopted a short-hand notation in the second line. Now, including the spin part of the electrons, we have several possibilities:

1s(1)α(1) 1s(1)α(2),

1s(1)β(1) 1s(1)β(2),

1s(1)α(1) 1s(1)β(2), or

1s(1)β(1) 1s(1)α(2).

Of these, the last two assume that we can distinguish between electrons 1 and 2. In order to remove this artificial distinguishability, we form combinations of these two wave functions to get

[ ][ ]

1212

1 (1) (1)1 (2) (2)

1 (1) (1)1 (2) (2)

1 (1) (1)1 (2) (2) 1 (1) (1)1 (2) (2)

1 (1) (1)1 (2) (2) 1 (1) (1)1 (2) (2)

s s

s s

s s s s

s s s s

α α

β β

α β β α

α β β α

+

−

Of these, only the last expression is antisymmetric with respect to electron exchange. The other three possibilities are eliminated by Pauli’s principle. For this reason, the Pauli principle is commonly refered to as the Pauli Exculsion Principle. A consequence of the Pauli principle is that no two electrons in an atom can have the same values for all four quantum numbers (n, l, m and ms). This means that only two electrons can occupy an atomic orbital (n,l,m).

6

When we have to consider systems of three (Lithium atom) or four (Beryllium) or more electrons, forming wave functions that are antisymmetric with respect to the exchange of any two electrons may seem like a rather complicated task. However, Slater proposed an ingenious method to accomplish this. His method consists of writing the wave function in the form of a determinant, with each row corresponding to one electron. When two rows (or two columns) of a determinant are exchanged, the determinant changes sign. The Slater determinant for the 3 electrons in the ground state of a Lithium atom is given by

1 (1) (1) 1 (1) (1) 2 (1) (1)1(1,2,3) 1 (2) (2) 1 (2) (2) 2 (2) (2) .3!

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

s s s

s s s

s s s

α β α

ψ α β

α β α

= α (15)

Note that we could have also written this wave function with the 2sβ orbital for the third column. Approximate methods for solving the Schrödinger equation

As already noted, it is impossible to analytically solve the Schrödinger equation for most systems of chemical interest. Therefore, methods for obtaining approximate solutions are very valuable. Most of the quantum mechanical calculations for atoms and molecules are done in what are known as atomic units (au):

Energy: 1 Hartree = Eh = 2

0 04eaπε

= 4.36 × 10–18 J = 27.2114 eV.

Length: 1 Bohr = a0 = 5.29177 × 10–11 m

Mass: 1 me = 9.109 × 10–31 kg

Charge: 1 e = 1.60219 × 10–19 C

Time: 1 aut = 2.42 × 10–17 s

In these units, ® = 1 Hartree aut. Therefore, the ground state energy of the hydrogen atom in atomic units is given by

E1 = − Z2e2

8 0a0% 1

12 = − 12 Hartree.

This means that if we supplied ½ Hartree energy to the electron, it would become a “free particle,” leaving behind a proton. Therefore, the ionization potential for hydrogen is ½ Hartree.

If we calculate the energy of the 2s electron in the lithium atom, using Z = 3 and n = 2 above, we get

E2 = − 9

2 %14 = −1.125 Hartree

The experimental value is –0.198 Hartree, an error of 468%! Clearly, neglecting the presence of the 2 electrons in the 1s orbital is a very bad assumption. We can deduce that one of the effects of the 1s electrons is to alter the effective nuclear charge experienced by the 2s electron.

7

Therefore, let us write

E2 = −

Zeff2

2 % 122 = −0.198 Hartree,

which gives us the effective nuclear charge Zeff = 1.26. The wave function corresponding to the 2s electron may (approximately, since we are neglecting the presence of the other two!) be written as

2s(r) = 1

4 2

Zeffa0

3/2(2 − Zeffr

a0 )e−Zeffr/a0 .

One way to think about this is that Zeff is an adjustable parameter in the approximate wave function (or trial function). Finding the “best” value of this parameter will yield the “best” energy for the system.

The Variation Method is a systematic way of finding the best value of the parameters in a trial function.

The Variation Method

The basic idea in the variational method is to

(a) calculate the energy functional using the trial function ϕ,

(b) minimize the energy functioanl with respect to the adjustable parameters,

(c) get the optimum value of the parameters,

(d) calculate the “best” energy using the optimum value of the parameters.

The energy functional is simply

*

,*

H dW

d

ϕ ϕ τ

ϕ ϕ τE= ≥∫

∫ (16)

where dτ is a generic volume element, and the denominator will be unity if ϕ is normalized. The exact ground state energy of the system is given by E. The variational energy W, therefore, provides an upper limit to the exact ground state energy of the system being studied.

An example of the application of this strategy:

The trial function ( ) ( ) 03/2 /01/ / effZ r a

effZ a eϕ π −= when substituted into Eq. (16) where the Hamitonian operator is for the Helium atom, gives

Step (a): W = e2

a0Zeff

2 − 278 Zeff .

Step (b): ØWØZeff

= 0 = e2a0

2Zeff − 27

8

Step (c): Therefore, Zeff = 27/16 = 1.688.

Step (d): W = –2.85 Hartree (compare to E = –2.9031 Hartree.)

8

MOLECULAR STRUCTURE The simplest molecule consists of two atoms, called a diatomic molecule. Consider the diatomic molecule AB with a total of 2 electrons. The Hamiltonian operator for this molecule will consist of five sets of terms:

• the kinetic energy terms for each of the nuclei,

e1

e2

A B

• the kinetic energy terms for the electrons,

• the potential energy of attraction between the electrons and nuclei,

• the potential energy of repulsion between the two electrons, and

• the potential energy of repulsion between the two nuclei.

In atomic units, this is given by

( )2 2

2 21 2

1 1 2 2 12

1 1 1 .2 2

A B A B A B A B

A B A B A B AB

Z Z Z Z Z ZHm m r r r r r r ∇ ∇

= − + − ∇ + ∇ − + + + + +

(1)

For the more general case of M nuclei and N electrons, we write (we use Greek letters for nuclei and Roman letters for electrons):

2

2

1 1 1 1 1 1

1 1 1 .2 2

M N M N N M

ii iji i i j i

Z ZZm r r r

α βα α

α αα α α β= = = = = > = >

∇− − ∇ − + +∑ ∑ ∑∑ ∑∑ ∑ ∑

αβα (2)

No one has succeeded in solving the Schrödinger equation analytically with this Hamiltonian!

A very important step towards its solution is the Born-Oppenheimer approximation, which is based on recognizing that the the electrons in a molecule move many orders of magnitude faster than the nuclei. Therefore, as a first step, it is justifiable to solve the electronic part of the Schrödinger equation for the instantaneous, fixed, positions of the nuclei.

Invoking the Born-Oppenheimer approximation for the case of Eq. (1), therefore, means that we wish to solve the Schrödinger equation for the electrons for a fixed value of rAB:

( )2 21 2

1 1 2 2 12

1 1 ( )2

A B A Bel el AB el

A B A B

Z Z Z Z E rr r r r r

.ψ ψ − ∇ + ∇ − + + + + =

(3)

This electronic energy depends on the value of rAB because (r1A, r1B) and (r2A, r2B) are related to each other through rAB. Now, we can return to Eq. (1) and write it as

2 21 ( )

2A B A B

el AB nuc nucA B AB

Z ZE r Em m r

ψ ψ ∇ ∇− + + + =

(4)

1

As we have done in the case of other two-body systems, we may re-write this as

21 ( ) ( ) ( )2 AB AB ABU r r E rµ ψ ψµ

−∇ + =

, (5)

where

( ) ( ) , an

.

ABAB el AB

AB

A B

A B

ZU r E rr

m mm m

µ

= +

=+

d (6)

The combination of the electronic energy and the internuclear repulsion energy, therefore, acts as the potential energy for the movement of the atomic nuclei.

Typically, the electronic energy decreases as the distance rAB decreases. On the other hand, the nuclear repulsion energy obviously increases as rAB decreases. If the atoms A and B form a stable molecule AB, the sum of these two terms goes through a minimum as shown in the figure below. The distance at which the energy is a minimum is the prefered internuclear distance, or the equilibrium bond length between A and B, re.

Thus, the concepts of bond length and bond angle, for molecules with more than two atoms, arise from the Born-Oppenheimer approximation.

We cannot think of molecular structure without the concepts of bond length and bond angle.

Without the notion of molecular structure, there is no chemistry!

Eel + ZAZB/rAB

ZAZB/rAB

Eel

AB molecule energy

0 2 4 6 8 10

rAB

-2.5

-1.5

-0.5

0.5

1.5

2.5

Ener

gy

2

Let us discuss Eq. (5) a bit more. The low-energy movements of the nuclei subject to the potential functions like U(rAB) are called the vibrations of the molecule. In the figure below, we compare a model for the molecular potential energy to the more familiar harmonic oscillator potential. It is clear that the harmonic oscillator model is a poor, but qualitatively correct, approximation to the molecular potential energy function at low energies, near the equilibrium geometry of the molecule.

The high energy movements of the nuclei subject to potentials like U(rAB) lead to bond dissociation. Movements of nuclei of different molecules relative to one another may lead to reactions. The Born-Oppenheimer potential energy, therefore, plays a very central role in our fundamental understanding of the behavior of molecules.

The accurate solution of the electronic Schrödinger equation to obtain the electronic energy Eel and the solutions ψel for a given set of nuclear coordinates is the central concern of the field of research called quantum chemistry.

Our focus for the present is on the qualitative nature of the electronic wave functions ψel corresponding to the different atomic orbitals on atoms A and B. These molecular electronic wave functions are called molecular orbitals. The formation of molecular orbitals may be said to be a result of the overlap (or linear combination) of atomic orbitals on the different nuclei. This mechanism, the “linear combination of atomic orbitals to form molecular orbitals” is refered to as the LCAO-MO method. The variational method discussed earlier plays a central role in the application of this method, as we shall see below.

3

The LCAO-MO Method: The H2+ molecular ion

erB

A B

rA

The H2+ molecular ion is the simplest of all

molecules, consisting of exactly three particles. Since only one electron is present, the electron-electron repulsion term is absent from the Hamiltonian.

Within the Born-Oppenheimer approximation, the Schrödinger equation for the electronic wave function ψel is

− 12 =

2el −

1rA + 1

rB el = Eel el.

The single electron could belong to nucleus A, in which case ψel = 1sA = 1 Are

π− or it could belong to nucleus B, in which case ψel = 1sB = 1 Bre

π− .

It is, however, unphysical to assign the electron to only one of the two nuclei. Therefore, a good variational trial function for this problem is

= c11sA + c21sB, where the coefficients c1 and c2 are the variational parameters. It is important to keep in mind that two different origins (one at each nucleus) and coordinate systems are necessary to define these atomic orbitals. In order to solve the problem, we should transform to a common set of coordinates. The suitable coordinates are called “confocal-elliptic coordinates.” We will gloss over the details in our treatment below.

We adopt the following notation for the various integrals needed:

S = ¶(1sA)&(1sB)d

HAA = ¶(1sA)&H(1sA)d

HAB = ¶(1sA)&H(1sB)d

HBB = ¶(1sB)&H(1sB)d

In addition, also recall that since the atomic orbitals are normalized,

¶(1sA)&(1sA)d = ¶(1sB)&(1sB)d = 1.

4

Now, the variational functional is given by

W =¶[c11sA + c21sB]H[c11sA + c21sB]d

¶[c11sA + c21sB]2d

=c1

2HAA + 2c1c2HAB + c22HBB

c12 + c2

2 + 2c1c2S .

We now minimize W with respect to the two parameters c1 and c2, i.e.,

ØWØc1

= 0, and ØWØc2

= 0.

This gives rise to two equations which ultimately gives rise to the system of equations

c1(HAA − W) + c2(HAB − WS) = 0,c1(HAB − WS) + c2(HBB − W) = 0.

A trivial solution is obviously c1 = c2 = 0. It is a theorem of linear algebra that such a system of equations can have non-trivial solutions if and only if

(HAA − W) (HAB − WS)(HAB − WS) (HAA − W) = 0,

where we have recognized that since the nuclei A and B are identical, HAA = HBB. This determinant is called the secular determinant. Expanding the determinant gives two possible solutions for W, the variational energy, as

W! =HAA "HAB

1 " S . Substituting W+ into the system of equations above gives rise to the result c1 = c2, and substituting W− gives c1 = −c2. From the normalization condition

¶ !&

!d = 1, we get

! =

1sA ! 1sB

2(1 ! S).

Thus, the linear combination of two atomic orbitals gives rise to two molecular orbitals. Let us now examine the nature of these orbitals.

5

A few molecular orbitals of F2 (HF/3-21G*)

Compare these orbital shapes to the schematic sketches on the last page of “Molecular Structure -2” handout.

HOMO – 6 (σg2s) E = –1.74694 au Compare to σg1s. This orbital is actually formed from 2s atomic orbitals.

HOMO – 5 (σu2s) E = –1.46344 au

A few molecular orbitals of F2 (HF/3-21G*)

HOMO – 4 (πu2p) E = –0.78671 au

HOMO – 3 (πu2p) E = –0.78671 au

A few molecular orbitals of F2 (HF/3-21G*)

HOMO – 2 (σg2p) E = –0.71849 au

HOMO – 1 (πg2p) E = –0.63893

A few molecular orbitals of F2 (HF/3-21G*)

HOMO (πg2p) E = –0.63893 au

LUMO (σu2p) E = +0.13789 au

A few molecular orbitals of H2O (HF/3-21G*)

HOMO – 3 E = –1.321 au

HOMO – 2 E = –0.68563

A few molecular orbitals of H2O (HF/3-21G*)

HOMO – 1 E = –0.52980

HOMO E = –0.4772

A few molecular orbitals of H2O (HF/3-21G*)

LUMO E = +0.2611

LUMO + 1 E = +0.3604

DENSITY FUNCTIONAL THEORY Prepared by Pedro Derosa

Correlated Methods The difference between the energy predicted by the Hartree-Fock method and the “exact one” is known as correlation energy. The exact mathematical formula is not known. Correlated methods differ from each other in the way they deal with correlation. Different approaches can be followed:

1. Better wave functions: The trial wave function is set to a linear combination of Slater determinants. The parameters used for the linear combination are adjusted for the energy to be a minimum. This approach improves the description of the electron interactions reducing the correlation energy, thus approaching the energy to the exact one. This is known as the Configuration Interaction Method. The determinants used in the linear combination are the ground state and some of the excited states predicted by Hartree-Fock. The number and the order of the excited states included in the linear combination determines the level of the approximation CIS, CID, CIT indicate that singly, doubly and triply excited states are used in the combination. CISD means that singly and doubly excited states are included

2. Perturbational approach: The Hartree-Fock Hamiltonian (uncorrelated) is corrected by adding a perturbation to it. This works as a series expansion. The more terms are added the more precise the method become. M∅ller and Plesset introduced a commonly used perturbation method. They are known as MPn where n is the order of perturbation; MP2 is a perturbation up to a second order, MP4 up to fourth order.

3. Improved functionals: Other methods are based on finding a more appropriate mathematical description for the kinetic energy and exchange operator and/or adding and extra term for correlation. Different methods differ on the mathematical form used for this potential. Density Functional Theory.

Computational cost Many of the methods described above would converge to the exact energy if enough terms are added; unfortunately adding terms increases the computational cost of the calculation. Formally, Hartree-Fock scale with the fourth power of the system size, which mean that if the number of electrons in the system is doubled, the calculation time increases 16 times (24). MP2 scales with the power of 5, MP4 with the power of 7. CISDT scales with the power of 8. All of this implies that highly accurate system can only be used for very small molecules.

1

Density Functional Theory In 1964, Hohemberg and Khon, demonstrated a fundamental theorem giving birth to Density Functional Theory (DFT). “The external potential )r(extV is (to within a constant) a unique functional of the

electron density )r(ρ ; since, in turn )r(extV fixes H we see that the full many particle

ground state is a unique functional of )r(ρ ” A second Hohemberg and Khon Theorem ensures that DFT is a variational method which implies that we can play around with the electron density and we will always be above the true ground state unless we have the true wave function. The DFT energy can be expressed as

( ) ( ) ( ) ( oNeoeeooo EETE ρρρρ ++= )

)

)

(1) Where what the theorem called “external potential” is the nuclear-electron interaction. Out of the three terms in equation (1) only the third term depends on the molecular structure, the other two are general terms, given an electron density they assume a unique value, so (1) can be rewritten as

( ) ( ) ( oNeoHKoo EFE ρρρ += (2) Where is a generic functional of the density of the ground state, it would equally applied to any systems from a Hydrogen atom to a protein. Unfortunately the mathematical form of that function, consisting on a kinetic energy term plus the electron-electron interaction, is unknown except for the coulomb part of the electron-electron interaction.

( oHKF ρ

Out of all the terms involved in (1), the functional form is known for: Nuclear-electron interaction:

( ) ( ) rdrRrZE

M

A A

ANe ∑∫

= −=

1

ρρ (3)

Coulomb interaction:

( ) ( ) ( )∫∫ −

= 'rdrd'rr'rrJ ρρρ

21 (4)

Different methods within DFT differ on the description of the kinetic and the exchange-correlation term.

2

Electron density The electron density is obtained by integrating the square of the wave function over N-1 electron coordinates V

( ) ( ) NN xdxddsx,x,xNr LLL 21

2

21∫ ∫ Ψ=ρ (1)

ds1 is the spin coordinate of one of the electrons. This is the probability of finding one of the electrons in volume element 1rd with arbitrary spin and for any position of the other N-1 electrons. The electron density has the following properties ( )( ) Nrdr

r

=

=∞→

∫ 1

0

ρ

ρ (2)

The first condition is consistent with electrons attached to a molecule, the second indicates the total number of electrons is N The advantage of this approach is evident comparing to the wave function approach. A wave function for an N-electron system has 3N coordinates. The electron density only depends on the three coordinate regardless the number of electrons. The Khon-Sham Molecular Orbitals For all practical purposes there is no wave function in DFT; however DFT would be of limited practical application in chemistry if we were not able to obtain molecular orbital from it. Following the approach used for Hartree-Fock we should be able to setup a system of non-interacting electrons in an effective potential. The kohn-sham one-electron Hamiltonian looks like

( )rVf SKS +∇−= 2

21 (3)

For non-interacting electrons the kinetic energy is exactly the first term on equation (3).

( ) ( ) ( ) ∑∫=

−+=M

A A

AXCS r

ZxVxdrxrV

1 112

12

2ρ (4)

In equation (4) only VXC is unknown. The one-electron wave functions obtained from (4) are restricted to give the “exact density”

3

4

•

•

It is difficult to realize from the limited description offered here but, unlike Hartree-Fock no approximations have being taken so far, in Hartree-Fock the wave function was described as a single Slater determinant which warranty that the solution will not be exact. In DFT instead, if we knew the exact functionals the approach is exact, the approximations come in when the exact form of the functionals, in particular VXC needs to be determined. References This class was prepared using

A Chemist’s Guide to Density Functional Theory, Wolfram Koch and Max Holthausen. Wiley-VCH (2001) Introduction to Computational Chemistry, Frank Jensen, Wiley & Sons (2002)