Atomic Model Theory: Shapes of Electron Orbitals

MTH 426: Mathematical Modeling

Atomic Model Theory

Amar Sha 13822

Anshul Singh 11547

Pushpendra Bhati 11547

Siddharth Vishwanath 10712

Siddhartha Sanghi 10714

March 2014

Contents

List of Figures ii

1 Introduction 1

1.1 The Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 The Goal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 System Characterization 3

3 Atomic Models 5

3.1 Thomson’s Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3.1.1 Pitfalls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3.2 Rutherford’s Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3.2.1 Mathematical Formulation . . . . . . . . . . . . . . . . . . . . . . 7

3.2.2 Pitfalls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.3 Bohr’s Atomic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.3.1 Mathematical Formulation . . . . . . . . . . . . . . . . . . . . . . 11

3.3.2 Pitfalls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4 Schrodingers wave equation 14

4.1 System Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4.2 Mathematical Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4.2.1 Schrodingers Equation in Three Dimensions . . . . . . . . . . . . . 18

4.2.1.1 Radial Solution . . . . . . . . . . . . . . . . . . . . . . . 20

4.2.1.2 Azimuthal Solution . . . . . . . . . . . . . . . . . . . . . 20

4.2.1.3 Polar Solution . . . . . . . . . . . . . . . . . . . . . . . . 20

4.3 Shape of orbitals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.3.1 1s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.3.2 2px . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.3.3 3dxy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.3.4 3dz2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

5 References 26

i

List of Figures

3.1 Thomson’s Plum Pudding Model . . . . . . . . . . . . . . . . . . . . . . . 5

3.2 Rutherford’s Gold Foil Experiment . . . . . . . . . . . . . . . . . . . . . . 6

3.3 Rutherford’s Planetary Model of the Atom . . . . . . . . . . . . . . . . . 7

3.4 Determination of radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.5 Bohr’s Atomic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.6 Energy levels for Hydrogen atom: Bohr’s Atomic Model . . . . . . . . . . 13

4.1 Orbital Shape: 1s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.2 Probability Density 1s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.3 Orbital Shape: 2px . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.4 Orbital Shape: 3dxy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.5 Orbital Shape: 3dz2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

ii

Chapter 1

Introduction

Quantum mechanics is a branch of physics which deals with physical phenomena at

nanoscopic scales. The earliest versions of quantum mechanics were formulated in the

first decade of the 20th century. About this time, the atomic theory and the corpuscular

theory of light (as updated by Einstein) first came to be widely accepted as scientific

fact.

These theories, however, were analogous to the theories of classical mechanics that deals

with continua on macroscopic levels in the Newtonian or Eulerian framework. Physicists

later discovered, both experimentally and with intuition, that the analogy from the

macroscopic level to the nanoscopic level were not consistent.

Many of the erstwhile widely accepted theories started to disintegrate on the basis of

some experimental and mathematical inconsistencies. It is during this time that math-

ematics became a very fundamental tool in developing new physical theories pertinent

to the atom.

Mathematics was already being used extensively by physicists in the early 20th century,

but with the increasing demand for answers in Quantum Mechanics, mathematics for

the first time played a very crucial role in the development of new theories purely based

on mathematical theory, models and formulations.

Mathematical formulations are those mathematical formalisms that permit a rigorous

description of quantum mechanics. Many of the structures used in Quantum Mechan-

ics come from Functional Analysis - a research area within pure mathematics that was

influenced in part by the needs of quantum mechanics. These formalisms are character-

ized by the use of infinite dimensional Hilbert spaces and operators. In brief, values of

physical observables such as energy and momentum were no longer considered as values

1

Atomic Model Theory 2

of functions on phase space, but as eigenvalues in a Hilbert space. With these historical

milestones in perspective, we would like to use the opportunity in the mathematical mod-

elling course to revisit these models and examine them in the mathematical-modelling

framework.

1.1 The Problem

Physical experiments like Einsteins photoelectric effect, Thompsons cathode ray exper-

iment, Rutherfords gold foil experiment and the like, strongly advocated the existence

of a fundamental particle the atom. As more experiments validated these theories,

scientists were fascinated by the nature of these nanoscopic particles. Thus, emerged

the discipline of Quantum Mechanics. And, as the research progressed in this field, they

discovered that the energy that we derive from these nanoscopic particles is attributed

primarily to a negatively charged miniscule that is in perpetual motion around the nu-

cleus of the atom the electron. The most fundamental question that the scientists were

puzzled by was the location/position of the electron in the atom. Many theories and

models have tried to describe this property of the electron. Most of them unsuccessful,

and a few of them revolutionary. Our objective is to try and describe the location of

the electron.

1.2 The Goal

To use the principles of mathematical modelling to formulate a model that can describe

the location/position of the electron with respect to the atom.

Chapter 2

System Characterization

Our System is atom, as we are dealing with the atomic structure. Initially, we can begin

with taking certain objects – negative charge, positive charge. Our variables will be –

location of negative charge, location of positive charge, as these only will be responsible

to specify the atomic structure. There are few Parameters, which are intrinsic to the

atom : mass of negative charge , mass of positive charge , volume of electron, volume of

positive charge, charge of negative, positive charge.

• Closed system as there is nothing out of system that we require to for our model

of structure of atom.

• System is Static as the variables and model doesnt depend on time.

Deterministic as degree of uncertainty in our system is small final outcome of structure

is taken as certain and deterministic Further, Rutherfords experiment led to conclude

that there is central positive nucleus surrounded by negative orbiting electrons. There

are neutral particles with positively charged in nucleus to give atom stability.

Modifications

Neutral particle known as neutron, and location of positive charge (proton) is fixed

at center of atom and R is find out by using the law of conservation of energy - in

the moment.. So, variable is location of negative charge, which is revolving round

the nucleus. The potential energy of different electrons is also a variable in order to

understand what all is going on.

Further since, electron moving around the nucleus should emit an electromagnetic wave

and Electron should than move not by the circle but helical and finally collide with the

3


nucleus. But atom is stable. Therefore, it resulted that Electrons orbit the nucleus in

fixed orbits without losing energy; We, now use angular momentum and electrostatic

energy to calculate orbital energy of electrons. If we talk about the system character-

ization which includes Schrdinger equation then we will be having these parameters:

this models find these three parameters principal (n), angular (l), and magnetic (m)

quantum numbers, Plank constant, mass of the electron, charge of the electron, total

energy of the electron at any time. This will be our main variable which we need to find

using wave function of the electron : Probability density of an particular electron. The

System can be dynamic or static in nature which can be defined two ways: It will be

dynamic in nature as we the particle location (wave function) will change with time or

one can also make it time independent by taking time with other quantity to make the

quantity dimensionless then it becomes a static.

It will be a stochastic function for various reasons like particles do not have exactly

determined properties, and when they are measured, the result is randomly drawn from a

probability distribution. It predicts the probability distributions are, but fundamentally

cannot predict the exact result of each measurement.

Later developments, the application of the new quantum theory to electromagnetism

resulted in quantum field theory. Quantum field theory has driven the development

of more sophisticated formulations of quantum mechanics. System characterization is

given in later chapter.

Chapter 3

Atomic Models

3.1 Thomson’s Model

In this model, the atom is made up of negative electrons that surrounded by a soup

of positive charge to balance the electrons’ negative charges, like negatively charged

”raisins” surrounded by positively charged ”pudding”. Here the positive charge dis-

tribution was assumed to be spherical in shape with a radius of the known order of

magnitude of the radius of an atom , 10−10m.

Figure 3.1: Thomson’s Plum Pudding Model

3.1.1 Pitfalls

1. This model didnt explain the neutrality of an atom nor did it explain why atoms

undergo reactions. It was merely a visualization made by Thomson.

2. This model could not explain the large angle scattering of alpha particles by thin

metal foils.

5


System Atom

Objects electron , positive charge

Variables location of electron , location of positive charge

Parameters mass of electron , mass of positive charge , volume of elec-tron, volume of positive charge, charge of electron, positivecharge

Environment Closed as there is nothing out of system that we require tofor our model of structure of atom

Static vs Dynamic Static as the variables and model doesnt depend on time

Stochastic vs Deterministic Deterministic as degree of uncertainty in our system is smalland final outcome of structure is taken as certain and deter-ministic

Table 3.1: System Characterization of Thomson’s Model

3.2 Rutherford’s Model

Ernest Rutherford publishes his atomic theory describing the atom as having a central

positive nucleus surrounded by negative orbiting electrons. This model suggested that

most of the mass of the atom was contained in the small nucleus, and that the rest of

the atom was mostly empty space.

Figure 3.2: Rutherford’s Gold Foil Experiment

Observations out of experiment:

• Most of the alpha particles pass straight through the gold foil.

• Some of the alpha particles get deflected by very small amounts.

• A very few get deflected greatly.

• Even fewer get bounced of the foil and back to the left.

Inferences:


Figure 3.3: Rutherford’s Planetary Model of the Atom

• The atom is 99.99percent empty space.

• The nucleus contains a positive charge and most of the mass of the atom.

• The nucleus is approximately 100,000 times smaller than the atom.

He thus developed the planetary model of the atom which put all the protons in the

nucleus and the electrons orbited around the nucleus like planets around the sun. This

model was alternatively called the nuclear atom.

System Atom

Objects electron , positive charge , neutron




Static vs Dynamic System is static as the variables and model doesnt dependon time (if the time frame is taken larger than the orbitaltime period)


Table 3.2: System Characterization of Rutherford’s Model

3.2.1 Mathematical Formulation

Between an alpha particle and an atomic nucleus there subsists an interaction - the

repulsing - according to Coulomb force:

F =1

4πε0

2.Z.e2

r2(3.1)


Where,

2e : Charge on the Alpha Particle

Ze : Charge on the Atomic Nucleus

r : Distance between Nucleus and Alpha Particles

ε0 : Permittivity of free space

Also, dN-the number of alpha particles scattered in a time unit inside the solid angle

dQ is equal:

dN

dQ= n.

2e.Ze

4πε04.E

1

sin4 G2

(3.2)

Where,

n : alpha particle flux density

G : angle of scattering alpha

E : particles’ energy

On the base of the number of alpha particles scattered by the G angle, the Z ,number of

elementary, positive charges in the nucleus can be calculated. He found that the number

is equal to the atomic number.

From the charge of the nucleus we can calculate the upper limit of its radius (by the

assumption that the nucleus is a ball). the alpha particles and the nucleus radius is

smaller than the minimum limit r0 between their centers in the moment of collision.

To evaluate r0

Lets consider the central collision - scattering by the angle G = 180 degree. By the law

of conservation of energy - in the moment, when the distance between the alpha parti-

cles and the nucleus is minimal, the kinetic energy of that alpha particle is completely

changed to the energy of the interaction:

m.v2

2=

2e.Ze

4πε0r0(3.3)

Where,

m : alpha particle mass


Figure 3.4: Determination of radius

v : alpha particle velocity before the collision

r0 =2e.Ze

4πε0m.v2(3.4)

Result of Model

From the last equation for the gold of this experiment the r0 value was about 3.1 x 10−14

meter (the velocity of alpha particles was equal 1.9 x 107 meter per second), atom

consisted of a nucleus (of size 10−15 to 10−14 meter).

3.2.2 Pitfalls

The model created by Rutherford had still some serious discordance. According to the

classic science, electron moving around the nucleus should emit an electromagnetic wave.

That kind of emission is connected with the escape of some energy from the electron-ion

circuit. Electron should than move not by the circle but helical and finally collide with

the nucleus. But atom is stable. In effect other scientist shown that electrons have

to emit energy whilst electrons move onto curved orbitals so electrons should be loose

energy it might have to fall onto nucleus.

3.3 Bohr’s Atomic Model

Neils Bohr (18851962) refined Rutherford’s model in 1913 by proposing another model

of the atom where electrons are arranged in circular paths about the nucleus.

In this model , he says that electrons:

• orbit the nucleus without losing energy.


• could move only in fixed orbits of specific energies.

• electrons with low energy would orbit closer to the nucleus while electrons with

high energy orbit further from the nucleus.

Figure 3.5: Bohr’s Atomic Model

As long as an electron continues to have that energy, it would remain in that path or

energy level.

From experiments, He deduced then, that when energy was added, electrons were pushed

to new and higher energy levels that were discrete like rungs on a ladder. He concluded

that electrons could not exist at just any level within the atom-only at the defined energy

levels which, unlike a ladder, were not evenly spaced around the nucleus. He termed

these packets of energy ”quanta”.

System Atom

Objects electron , positive charge , neutron




Static vs Dynamic System is static as the variables and model doesnt dependon time (if the time frame is taken larger than the orbitaltime period)


Table 3.3: System Characterization of Bohr’s Model


3.3.1 Mathematical Formulation

In an hydrogen atom, the centripetal force is being supplied by the coulomb force between

it and the proton in the hydrogen nucleus.

Fcentripital = Felesctrostatic

mv2

rn= k.−e.(Ze)r2n

(3.5)

mv2 = k.Ze2

rn(3.6)

Z represents the atomic number (the number of protons), that electrons and protons

have the same magnitude charge, e, and that a negative Felectrostatic merely means that

the electrostatic force is attractive. Also note that the values of vn and rn are unknowns

in this equation.

As a means of evaluating these two unknowns, Bohr first hypothesized that the

electron’s angular momentum was quantized.

L = n.h

2π(3.7)

mr2vnrn

= n.h

2π(3.8)

mvnrn = n.h

2π(3.9)

Upon solving the angular momentum equation for vn, substituting it into the centripetal

force equation yields the following expression for rn :

rn = n2.h2

4π2kmZe2(3.10)

Bohr’s second hypothesis in his model was that an electron only loses or releases

energy (and therefore a photon) when it goes through de-excitation or drops from a


higher energy state to a lower energy state. In order to determine the energy lost by the

electron, an expression for an electron’s total energy has to be developed. The electric

potential energy for an electron would equal to:

EPE = qVabs

EPE = −k(Ze2)

rn(3.11)

By extending the centripetal force relationship, an expression can also be derived for

the electron’s kinetic energy using (3.5):

KE = k.Ze2

2rn(3.12)

Thus, the total energy, En, of an electron equals

En = −k.Ze2

2rn(3.13)

In this equation, notice that the total energy is negative. This is interpreting as meaning

that the electron is trapped in an energy well about the nucleus; that is, it would take

the addition of energy to ionize or free the electron.

Substituting in the value for r1 into this total energy expression yields a ground state

energy of 2.18x10−18 Joules or −13.6 eV for a hydrogen atom.

rn = n2r1 (3.14)

3.3.2 Pitfalls

• It violates the Heisenberg Uncertainty Principle because it considers electrons to

have both a known radius and orbit.

• The Bohr Model provides an incorrect value for the ground state orbital angular

momentum.

• It makes poor predictions regarding the spectra of larger atoms.

• It does not predict the relative intensities of spectral lines.


Figure 3.6: Energy levels for Hydrogen atom: Bohr’s Atomic Model

• The Bohr Model does not explain fine structure and hyperfine structure in spectral

lines.

• It does not explain the Zeeman Effect.

Chapter 4

Schrodingers wave equation

With the study of quantum mechanics, a theory developed at the beginning of the twen-

tieth century. The core equation of this theory, the analogue of Newton’s second law, is

called Schrdinger’s equation.

”In classical mechanics we describe a state of a physical system using position and mo-

mentum where we can find the position of a particle at a certain time if we know the

initial condition by using the classical mechanics laws, the kind of question we then ask

is: if we know the initial conditions of a system, that is, we know the system at time

t0, what is the dynamical evolution of this system? And we use Newtons second law

for that. In quantum mechanics we ask the same question, but the answer is tricky be-

cause position and momentum are no longer the right variables to describe [the system].”

The system consists of electrons, protons and neutrons. In this model we try to find out

maximum likelihood of finding the electron in a particular orbital.

4.1 System Characterization

• Objects: Goal is to find a wave equation in terms of the wave function which

predicts analytically and precisely the probability of events or outcome so objects

will be: position-space wave function, atom in which we have to find the probability

of electrons location, electrons mass, total, kinetic and potential potential energies.

• Open/Close system: It will be a close system as there are no other factors which

influencing the system.

14


• Stochastic/ Deterministic: In quantum mechanics, particles do not have ex-

actly determined properties, and when they are measured, the result is randomly

drawn from a probability distribution. The Schrdinger equation predicts what the

probability distributions are, but fundamentally cannot predict the exact result of

each measurement. So, it will be stochastic in nature.

• Static/ dynamic: It will be dynamic in nature as we the particle location (wave

function) will change with time. One can also make it time independent by taking

time with other quantity to make the quantity dimensionless. Potential energy

will also change with time.

• Variables: Probability density of the particle with location, location of the par-

ticle, potential energy, Kinetic energy.

• Parameters: total energy of the particle at any point of time, mass of the particle,

the angular frequency as it depends on total energy, this models find these three

parameters principal (n), angular (l), and magnetic (m) quantum numbers.

Schrodinger combined the equations for the behavior of waves with the de Broglie equa-

tion to generate a mathematical model for the distribution of electrons in an atom. The

advantage of this model is that it consists of mathematical equations known as wave

functions that satisfy the requirements placed on the behavior of electrons.

The Schrodinger model assumes that the electron is a wave and tries to describe the

regions in space, or orbitals, where electrons are most likely to be found. Instead of

trying to tell us where the electron is at any time, the Schrodinger model describes the

probability that an electron can be found in a given region of space at a given time.

This model no longer tells us where the electron is; it only tells us where it might be.

The three coordinates that come from Schrodinger’s wave equations are the principal

(n), angular (l), and magnetic (m) quantum numbers. These quantum numbers describe

the size, shape, and orientation in space of the orbitals on an atom.

Following Sommerfelds considerations let us then consider a particle moving in one spa-

tial dimension subject to a roller coaster-like potential. How do we expect the wave

function to behave? We would expect the wavelength to be shortest where the potential

is lowest, in the minima, because thats where the particle is going the fastest. Our

task then is to construct a wave equation which leads naturally to the relation follow-

ing from (classical) energy conservation, E = p2/2m + V (x). In contrast to the free

particle, the relevant wave function here will no longer be a simple plane wave, since

the wavelength (determined through the momentum via the de Broglie relation) varies

with the potential. However, at a given position x, the momentum is determined by the


local wavelength. The appropriate wave equation is the one-dimensional Schrodinger

equation.


4.2 Mathematical Formulation

The Schrodinger equation plays the role of Newton’s laws and conservation of energy in

classical mechanics - i.e., it predicts the future behavior of a dynamic system.

Kinetic energy+Potential energy=Total energy

Classical Conservation of Total Energy: mv2 + kx2

Quantum conservation of Total Energy : p2/2m+ kx2

In making transition to a wave equation where physical variables take the form of oper-

ators:

P :~i

∂

∂x

H : − ~2

2m

∂2

∂x2

the kinetic and potential energies are transformed into the Hamiltonian which acts upon

the wave function to generate the evolution of the wave function in time and space. The

Schrodinger equation gives the quantized energies of the system and gives the form of

the wave function so that other properties (n, l, and m) may be calculated.

In three dimensions, the time-independent Schrodinger equation takes the following

form,

− ~2

2m[∂2ψ

∂x2+∂2ψ

∂y2+∂2ψ

∂z2] + U(x, y, z)ψ(x, y, z) = Eψ(x, y, z) (4.1)

for cartesian coordinates. This can also be written in a more compact form by making

use of the Laplacian operator.

ψ(x) is the wave function of the quantum particle.

ψ(x, y, z) is the wave function in three dimensional space.

ψ(x, y, z) ∈ C i.e it can take complex values and ψ(x, y, z) is single valued.

By definition,

ψ∗(x, y, z)ψ(x, y, z) = Probability of finding the quantum particle at time (x,y,z )

ψ(x, y, z) = 1 (4.2)


Constraints on ψ(x, y, z):

1. It must be a solution to Scrdinger Wave Equation.

2. It must be normalizable, i.e.

lim(x,y,z)→∞

ψ(x, y, z) = 0

3. It must be a continuous function, i.e.

ψ : R3 → R

4. It must be an orthogonal function, i.e.

∫∞−∞

∫∞−∞

∫∞−∞ ψ

∗(x, y, z)ψ(x, y, z)dxdydz = 1

Probability of finding the particle in the entire space is equal to 1.

4.2.1 Schrodingers Equation in Three Dimensions

− ~2

2m∆2ψ(x, y, z) + U(x, y, z)ψ(x, y, z) = Eψ(x, y, z) (4.3)

∆2: Laplacian Operator,

∆2 = [∂2ψ

∂x2+∂2ψ

∂y2+∂2ψ

∂z2]

using spherical transformation,

∆2ψ = r∂2

∂r2rψ(r, θ, φ) +

1

sinθ

∂

∂θsinθ

∂

∂θψ(r, θ, φ) +

1

r2sin2θ

∂2

∂φ2ψ(r, θ, φ) (4.4)

Where,

x= r sinθ cosφ

y= r sinθ sinφ

z= r cosθ

Thus, in spherical coordinates the Schrodinger Wave Equation becomes:

− ~2

2m

1

r2sin2θ[sinθ

∂

∂r(r2

∂φ

∂r) +

∂

∂θ(sinθ

∂φ

∂θ) +

1

sinθ

∂2ψ

∂φ2] + U(r)ψ(r, θ, φ) = Eψ(r, θ, φ)

(4.5)


Where ψ(r, θ, φ) is the wave function.

To solve the second order partial differential equation given by (4.5), we use the variable

separation technique:

ψ(r, θ, φ) = R(r)Y (θ, φ)

Where,

R(r) is a function of only r

Y (θ, φ) is a function of only (θ, φ)

Hence,

∂R(r)

∂θ=∂R(r)

∂φ= 0 (4.6)

∂Y (θ, φ)

∂r= 0 (4.7)

Substituting this into (4.5) and dividing by R(r)Y (θ,φ)2m , we get:

[~2

R(r)

∂

∂r(r2

∂R

∂r) + 2mr2(U(r)− E)] +

~2

Y (θφ)[

1

sinθ

∂

∂θ(sinθ

∂Y

∂θ) +

1

sin2θ

∂2Y

∂φ2] = 0

(4.8)

⇒ ~2

R(r)

∂

∂r(r2

∂R

∂r) + 2mr2(U(r)− E) = constant (4.9)

⇒ ~2

Y (θφ)[

1

sinθ

∂

∂θ(sinθ

∂Y

∂θ) +

1

sin2θ

∂2Y

∂φ2] = −constant (4.10)

such that 4.9 + 4.10 = 0

Y (θφ) can be separated into two further parts: Y(θ, φ) = P (θ).F (φ)

where, ∂P (θ)∂φ = ∂F (φ)

∂θ = 0

Equation (4.10) becomes:

1

sin2θ

~2

F (φ)

∂2F

∂φ2+

~2

sinθP (θ)

∂

∂θ(sinθ

∂P (θ)

∂θ) = −constant (4.11)


Here, the constant= ~2l(l + 1)

where l= 0,1,2,3...,n-1

and n ∈ Z+ is the principal quantum number.

4.2.1.1 Radial Solution

1

R(r)

∂

∂r(r2

∂R

∂r) + 2mr2(U(r)− E) = l(l + 1) (4.12)

Solution:

R(r) = e− r

na0 rlLn,l (4.13)

Ln,l is the Laguerre polinomial which exists only when n,l are integers and n ≥ l.

4.2.1.2 Azimuthal Solution

1

F (φ)

∂2F

∂φ2= k (4.14)

F (φ) = F (2π + φ)⇒ Complex Solution

Solution:

F (φ) = Aekφ = F (2π + φ) = Ae(2πk+φ) (4.15)

⇒ F (φ) = Aeimlφ (4.16)

where k = −m2l

4.2.1.3 Polar Solution

~2

sinθP (θ)

∂

∂θ(sinθ

∂P (θ)

∂θ)−m2

l ~2 = −l(l + 1)~2 (4.17)


Solution:

P (θ) = P lml(cosθ) (4.18)

where P lml(x) is the Legendre polynomial whose solution exists only when —m+l— ≤ l

So,

ψ(r, θ, φ) = Rn,l(r)Pml,l(θ)Fml(φ)

where:

Rn,l(r) = e− r

na0 rlLn,l

Fml(φ) = Aeimlφ

Pml,l(θ) = P lml(cosθ)

constraints:

1. n ∈ N

2. n ≥ l

3. l ≥ |ml|

n: principal quantum number

l: azimuthal quantum number

ml : magneticquantumnumber


4.3 Shape of orbitals

4.3.1 1s

n=1, l=0, m=0

ψ1,0,0 = Rn,0(r)P0,0(θ)F0(φ)

ψ1,0,0 = ψ1S = ( 1π )1/2( 1

a0)3/2e−r/a0

Figure 4.1: Orbital Shape: 1s

Figure 4.2: Probability Density 1s


4.3.2 2px

n=2, l=1, m=0

ψ1,1,0 = ψ2px = ( 1

4√2πa

3/20

( ra0

)e−r/2a0cosθ

Figure 4.3: Orbital Shape: 2px

4.3.3 3dxy

n=3, l=2, m=0

ψ3,2,0 = ψ3dxy = 181√2π

( 1a0

)3/2( ra0

)2e−r/3a0sin2θsin(2φ)

Figure 4.4: Orbital Shape: 3dxy


4.3.4 3dz2

n=3, l=2, m=1

ψ3,2,1 = ψ3dz2= 1

81√2π

( 1a0

)3/2( ra0

)2e−r/3a0(3cosθ − 1)

Figure 4.5: Orbital Shape: 3dz2

Chapter 5

References

1. Wolfram Burgard, University of Freiburg, Advanced Artificial Intelligence

2. R. Atkins, Physical Chemistry, 2009

3. Mike Goodrich, Brigham Young University, Artificial Intelligence, 2005

4. MathWorks, Introduction to MATLAB, http://www.mathworks.in/help/index.html

5. HyperPhysics Blog, http://hyperphysics.phy-astr.gsu.edu/

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Atomic Model Theory: Shapes of Electron Orbitals

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