Lecture 8: Introduction to Quantum Chemical Simulation graduate course taught at MIT in Fall 2014 by Heather Kulik. This course covers: wavefunction theory, density functional theory, force fields and molecular dynamics and sampling.
This presentation shows a technique of how to solve for the approximate ground state energy using Schrodinger Equation in which the solution for wave function is not on hand
These slides are especially made to understand the postulates of quantum mechanics or chemistry better. easily simplified and at one place you will find each of relevant details about the 5 postulates. so go through it & trust me it will help you a lot if you are chemistry or a science student.
well done
This is an introduction to modern quantum mechanics – albeit for those already familiar with vector calculus and modern physics – based on my personal understanding of the subject that emphasizes the concepts from first principles. Nothing of this is new or even developed first hand but the content (or maybe its clarity) is original in the fact that it displays an abridged yet concise and straightforward mathematical development that provides for a solid foundation in the tools and techniques to better understand and have a good appreciation for the physics involved in quantum theory and in an atom!
This presentation is the introduction to Density Functional Theory, an essential computational approach used by Physicist and Quantum Chemist to study Solid State matter.
This presentation shows a technique of how to solve for the approximate ground state energy using Schrodinger Equation in which the solution for wave function is not on hand
These slides are especially made to understand the postulates of quantum mechanics or chemistry better. easily simplified and at one place you will find each of relevant details about the 5 postulates. so go through it & trust me it will help you a lot if you are chemistry or a science student.
well done
This is an introduction to modern quantum mechanics – albeit for those already familiar with vector calculus and modern physics – based on my personal understanding of the subject that emphasizes the concepts from first principles. Nothing of this is new or even developed first hand but the content (or maybe its clarity) is original in the fact that it displays an abridged yet concise and straightforward mathematical development that provides for a solid foundation in the tools and techniques to better understand and have a good appreciation for the physics involved in quantum theory and in an atom!
This presentation is the introduction to Density Functional Theory, an essential computational approach used by Physicist and Quantum Chemist to study Solid State matter.
UCSD NANO 266 Quantum Mechanical Modelling of Materials and Nanostructures is a graduate class that provides students with a highly practical introduction to the application of first principles quantum mechanical simulations to model, understand and predict the properties of materials and nano-structures. The syllabus includes: a brief introduction to quantum mechanics and the Hartree-Fock and density functional theory (DFT) formulations; practical simulation considerations such as convergence, selection of the appropriate functional and parameters; interpretation of the results from simulations, including the limits of accuracy of each method. Several lab sessions provide students with hands-on experience in the conduct of simulations. A key aspect of the course is in the use of programming to facilitate calculations and analysis.
The postulates of quantum mechanics have been successfully used for deriving exact solutions to Schrodinger equation for problems like A particle in 1 Dimensional box Harmonic oscillator Rigid rotator Hydrogen atom • However for a multielectron system, the SWE cannot be solved exactly due to inter-electronic repulsion terms.
The SWE is solved by method of seperation of variables.
• However, the inter-electronic repulsion term cannot be solved because the variables cannot be seperated and the SWE cannot be solved. • Approximate methods have helped to generate solutions for such and even more complex real quantum systems. • Approximate methods have been developed for solving Schrodinger equation to find wave function and energy of the complex system under consideration. • Two widely used approximate methods are, 1. Perturbation theory 2. Variation method
Perturbation theory is an approximate method that describes a complex quantum system in terms of a simpler system for which the exact solution is known. • Perturbation theory has been categorized into, i. Time independent perturbation theory, proposed by Erwin Schrodinger, where the perturbation Hamiltonian is static. ii. Time dependent perturbation theory, proposed by Paul Dirac, which studies the effect of time dependent perturbation on a time independent Hamiltonian H0.
PERTURBATION THEOREM
FIRST ORDER PERTURBATION THEORY
FIRST ORDER ENERGY CORRECTION
FIRST ORDER WAVE FUNCTION CORRECTION
APPLICATIONS OF PERTURBATION METHOD
SIGNIFICANCE OF PERTURBATION METHOD
Lecture slides from a class introducing quantum mechanics to non-majors, giving an overview of black-body radiation, the photoelectric effect, and the Bohr model. Used as part of a course titled "A Brief history of Timekeeping," as a lead-in to talking about atomic clocks
lecture slide on:
Gibbs free energy and Nernst Equation, Faradaic Processes and Factors Affecting Rates of Electrode Reactions, Potentials and Thermodynamics of Cells, Kinetics of Electrode Reactions, Kinetic controlled reactions,Essentials of Electrode Reactions,BUTLER-VOLMER MODEL FOR THE ONE-STEP, ONE-ELECTRON PROCESS,Current-overpotential curves for the system, Mass Transfer by Migration And Diffusion,MASS-TRANSFER-CONTROLLED REACTIONS,
(If visualization is slow, please try downloading the file.)
Part 2 of a tutorial given in the Brazilian Physical Society meeting, ENFMC. Abstract: Density-functional theory (DFT) was developed 50 years ago, connecting fundamental quantum methods from early days of quantum mechanics to our days of computer-powered science. Today DFT is the most widely used method in electronic structure calculations. It helps moving forward materials sciences from a single atom to nanoclusters and biomolecules, connecting solid-state, quantum chemistry, atomic and molecular physics, biophysics and beyond. In this tutorial, I will try to clarify this pathway under a historical view, presenting the DFT pillars and its building blocks, namely, the Hohenberg-Kohn theorem, the Kohn-Sham scheme, the local density approximation (LDA) and generalized gradient approximation (GGA). I would like to open the black box misconception of the method, and present a more pedagogical and solid perspective on DFT.
UCSD NANO 266 Quantum Mechanical Modelling of Materials and Nanostructures is a graduate class that provides students with a highly practical introduction to the application of first principles quantum mechanical simulations to model, understand and predict the properties of materials and nano-structures. The syllabus includes: a brief introduction to quantum mechanics and the Hartree-Fock and density functional theory (DFT) formulations; practical simulation considerations such as convergence, selection of the appropriate functional and parameters; interpretation of the results from simulations, including the limits of accuracy of each method. Several lab sessions provide students with hands-on experience in the conduct of simulations. A key aspect of the course is in the use of programming to facilitate calculations and analysis.
The postulates of quantum mechanics have been successfully used for deriving exact solutions to Schrodinger equation for problems like A particle in 1 Dimensional box Harmonic oscillator Rigid rotator Hydrogen atom • However for a multielectron system, the SWE cannot be solved exactly due to inter-electronic repulsion terms.
The SWE is solved by method of seperation of variables.
• However, the inter-electronic repulsion term cannot be solved because the variables cannot be seperated and the SWE cannot be solved. • Approximate methods have helped to generate solutions for such and even more complex real quantum systems. • Approximate methods have been developed for solving Schrodinger equation to find wave function and energy of the complex system under consideration. • Two widely used approximate methods are, 1. Perturbation theory 2. Variation method
Perturbation theory is an approximate method that describes a complex quantum system in terms of a simpler system for which the exact solution is known. • Perturbation theory has been categorized into, i. Time independent perturbation theory, proposed by Erwin Schrodinger, where the perturbation Hamiltonian is static. ii. Time dependent perturbation theory, proposed by Paul Dirac, which studies the effect of time dependent perturbation on a time independent Hamiltonian H0.
PERTURBATION THEOREM
FIRST ORDER PERTURBATION THEORY
FIRST ORDER ENERGY CORRECTION
FIRST ORDER WAVE FUNCTION CORRECTION
APPLICATIONS OF PERTURBATION METHOD
SIGNIFICANCE OF PERTURBATION METHOD
Lecture slides from a class introducing quantum mechanics to non-majors, giving an overview of black-body radiation, the photoelectric effect, and the Bohr model. Used as part of a course titled "A Brief history of Timekeeping," as a lead-in to talking about atomic clocks
lecture slide on:
Gibbs free energy and Nernst Equation, Faradaic Processes and Factors Affecting Rates of Electrode Reactions, Potentials and Thermodynamics of Cells, Kinetics of Electrode Reactions, Kinetic controlled reactions,Essentials of Electrode Reactions,BUTLER-VOLMER MODEL FOR THE ONE-STEP, ONE-ELECTRON PROCESS,Current-overpotential curves for the system, Mass Transfer by Migration And Diffusion,MASS-TRANSFER-CONTROLLED REACTIONS,
(If visualization is slow, please try downloading the file.)
Part 2 of a tutorial given in the Brazilian Physical Society meeting, ENFMC. Abstract: Density-functional theory (DFT) was developed 50 years ago, connecting fundamental quantum methods from early days of quantum mechanics to our days of computer-powered science. Today DFT is the most widely used method in electronic structure calculations. It helps moving forward materials sciences from a single atom to nanoclusters and biomolecules, connecting solid-state, quantum chemistry, atomic and molecular physics, biophysics and beyond. In this tutorial, I will try to clarify this pathway under a historical view, presenting the DFT pillars and its building blocks, namely, the Hohenberg-Kohn theorem, the Kohn-Sham scheme, the local density approximation (LDA) and generalized gradient approximation (GGA). I would like to open the black box misconception of the method, and present a more pedagogical and solid perspective on DFT.
Lecture 5: Introduction to Quantum Chemical Simulation graduate course taught at MIT in Fall 2014 by Heather Kulik. This course covers: wavefunction theory, density functional theory, force fields and molecular dynamics and sampling.
Lecture 3: Introduction to Quantum Chemical Simulation graduate course taught at MIT in Fall 2014 by Heather Kulik. This course covers: wavefunction theory, density functional theory, force fields and molecular dynamics and sampling.
Lecture 4: Introduction to Quantum Chemical Simulation graduate course taught at MIT in Fall 2014 by Heather Kulik. This course covers: wavefunction theory, density functional theory, force fields and molecular dynamics and sampling.
PART V - Continuation of PART III - QM and PART IV - QFT.
I intended to finish with the Hydrogen Atom description and the atomic orbital framework but I deemed the content void of a few important features: the Harmonic Oscillator and an introduction to Electromagnetic Interactions which leads directly to a formulation of the Quantization of the Radiation Field. I could not finish without wrapping it up with a development of Transition Probabilities and Einstein Coefficients which opens up the proof of the Planck distribution law, the photoelectric effect and Higher order electromagnetic interactions. I believe this is the key contribution: making it more understandable up to, but not including, quantum electrodynamics!
Lecture 2: Introduction to Quantum Chemical Simulation graduate course taught at MIT in Fall 2014 by Heather Kulik. This course covers: wavefunction theory, density functional theory, force fields and molecular dynamics and sampling.
Describe the Schroedinger wavefunctions and energies of electrons in an atom leading to the 3 quantum numbers. These can be also observed in the line spectra of atoms.
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This pdf is about the Schizophrenia.
For more details visit on YouTube; @SELF-EXPLANATORY;
https://www.youtube.com/channel/UCAiarMZDNhe1A3Rnpr_WkzA/videos
Thanks...!
Nutraceutical market, scope and growth: Herbal drug technologyLokesh Patil
As consumer awareness of health and wellness rises, the nutraceutical market—which includes goods like functional meals, drinks, and dietary supplements that provide health advantages beyond basic nutrition—is growing significantly. As healthcare expenses rise, the population ages, and people want natural and preventative health solutions more and more, this industry is increasing quickly. Further driving market expansion are product formulation innovations and the use of cutting-edge technology for customized nutrition. With its worldwide reach, the nutraceutical industry is expected to keep growing and provide significant chances for research and investment in a number of categories, including vitamins, minerals, probiotics, and herbal supplements.
Richard's entangled aventures in wonderlandRichard Gill
Since the loophole-free Bell experiments of 2020 and the Nobel prizes in physics of 2022, critics of Bell's work have retreated to the fortress of super-determinism. Now, super-determinism is a derogatory word - it just means "determinism". Palmer, Hance and Hossenfelder argue that quantum mechanics and determinism are not incompatible, using a sophisticated mathematical construction based on a subtle thinning of allowed states and measurements in quantum mechanics, such that what is left appears to make Bell's argument fail, without altering the empirical predictions of quantum mechanics. I think however that it is a smoke screen, and the slogan "lost in math" comes to my mind. I will discuss some other recent disproofs of Bell's theorem using the language of causality based on causal graphs. Causal thinking is also central to law and justice. I will mention surprising connections to my work on serial killer nurse cases, in particular the Dutch case of Lucia de Berk and the current UK case of Lucy Letby.
Multi-source connectivity as the driver of solar wind variability in the heli...Sérgio Sacani
The ambient solar wind that flls the heliosphere originates from multiple
sources in the solar corona and is highly structured. It is often described
as high-speed, relatively homogeneous, plasma streams from coronal
holes and slow-speed, highly variable, streams whose source regions are
under debate. A key goal of ESA/NASA’s Solar Orbiter mission is to identify
solar wind sources and understand what drives the complexity seen in the
heliosphere. By combining magnetic feld modelling and spectroscopic
techniques with high-resolution observations and measurements, we show
that the solar wind variability detected in situ by Solar Orbiter in March
2022 is driven by spatio-temporal changes in the magnetic connectivity to
multiple sources in the solar atmosphere. The magnetic feld footpoints
connected to the spacecraft moved from the boundaries of a coronal hole
to one active region (12961) and then across to another region (12957). This
is refected in the in situ measurements, which show the transition from fast
to highly Alfvénic then to slow solar wind that is disrupted by the arrival of
a coronal mass ejection. Our results describe solar wind variability at 0.5 au
but are applicable to near-Earth observatories.
THE IMPORTANCE OF MARTIAN ATMOSPHERE SAMPLE RETURN.Sérgio Sacani
The return of a sample of near-surface atmosphere from Mars would facilitate answers to several first-order science questions surrounding the formation and evolution of the planet. One of the important aspects of terrestrial planet formation in general is the role that primary atmospheres played in influencing the chemistry and structure of the planets and their antecedents. Studies of the martian atmosphere can be used to investigate the role of a primary atmosphere in its history. Atmosphere samples would also inform our understanding of the near-surface chemistry of the planet, and ultimately the prospects for life. High-precision isotopic analyses of constituent gases are needed to address these questions, requiring that the analyses are made on returned samples rather than in situ.
Observation of Io’s Resurfacing via Plume Deposition Using Ground-based Adapt...Sérgio Sacani
Since volcanic activity was first discovered on Io from Voyager images in 1979, changes
on Io’s surface have been monitored from both spacecraft and ground-based telescopes.
Here, we present the highest spatial resolution images of Io ever obtained from a groundbased telescope. These images, acquired by the SHARK-VIS instrument on the Large
Binocular Telescope, show evidence of a major resurfacing event on Io’s trailing hemisphere. When compared to the most recent spacecraft images, the SHARK-VIS images
show that a plume deposit from a powerful eruption at Pillan Patera has covered part
of the long-lived Pele plume deposit. Although this type of resurfacing event may be common on Io, few have been detected due to the rarity of spacecraft visits and the previously low spatial resolution available from Earth-based telescopes. The SHARK-VIS instrument ushers in a new era of high resolution imaging of Io’s surface using adaptive
optics at visible wavelengths.
2. MIT
10.637
Lecture 8
A quantum refresher
• This course teaches some concepts that rely
on a basic understanding of quantum
mechanics.
• For a deeper background into quantum
mechanics, consider the courses: 5.73
(Graduate quantum mechanics), 5.61
(undergraduate quantum mechanics).
• Useful textbooks:
– Quantum Chemistry by Ira N. Levine
– Physical Chemistry by McQuarrie & Simon
3. MIT
10.637
Lecture 8
Glimpse of a quantum world
Black body radiation: when objects are heated, maximum wavelength of radiation
shifts to shorter wavelengths.
Classical theory
Expt.
Classical: Rayleigh-Jeans law
Planck (1900): Energies of
oscillations in the black body are
discrete or quantized!
Planck distribution law:
Planck’s constant
4. MIT
10.637
Lecture 8
Glimpse of a quantum world
Hydrogen atom line spectrum: Rydberg (1888) discrete lines of
hydrogen spectrum fit formula:
where:
Bohr (1911) model:
1) stationary electron orbit,
2) integral # of wavelengths
where:
5. MIT
10.637
Lecture 8
Physical laws in a quantum
world
Wave-particle duality in the double-slit experiment:
particle waveelectrons
de Broglie (1924) matter can have wave-like properties:
Particle Mass (kg) Speed (m/s) l (pm)
accelerated electron 9.1x10-31 5.9x106 120
fullerene (C60) 1.2x10-24 220 2.5
golf ball 0.045 30 4.9x10-22
6. MIT
10.637
Lecture 8
Uncertainty for quantum
particles
Heisenberg’s uncertainty principle:
Position and momentum cannot be precisely known exactly and simultaneously.
This is a consequence of the fact that position and momentum operators do not
commute –
i.e. no wavefunction can be simultaneously a position and momentum
eigenstate, making it not possible to exactly know the position and
momentum at the same time.
This is typically expressed as:
Can also be expressed in terms of standard deviations over a large number of
measured values:
7. MIT
10.637
Lecture 8
Postulates of quantum
mechanics
The wavefunction defines the state of a QM system completely:
The probability that a particle lies in an interval :
The wavefunction is normalized:
1)
Properties of the wavefunction: it only has one value at a given point in
space and time, it is finite and continuous at all points, as are its first and
second derivatives with respect to distance.
8. MIT
10.637
Lecture 8
Postulates of quantum
mechanics
2) Every classical observable has a QM linear, Hermitian operator:
Property Symbol Operator
Position multiply by
Momentum
Kinetic
energy
Potential
energy
multiply by
Total energy
Angular
momentum
9. MIT
10.637
Lecture 8
Operators in other coordinates
Kinetic energy in spherical coordinates:
Angular momentum operators in spherical coordinates:
Combined:
10. MIT
10.637
Lecture 8
Postulates of quantum
mechanics
3) Observables associated with an operator are eigenvalues of the wavefunction.
4) The average value of an observable is defined as:
The time-independent Schrödinger equation is a special case:
11. MIT
10.637
Lecture 8
Postulates of quantum
mechanics
5) The wavefunction is solution to the time-dependent Schrödinger equation:
Typically, we can separate the spatial and temporal:
Most wavefunctions of interest are stationary-state solutions:
In this course we focus only on solving the time-independent equation:
12. MIT
10.637
Lecture 8
A note about notation
complex
conjugate
Dirac or “bra-ket” notation is a useful representation for QM equations:
bra ket
Useful relations Schrödinger equation
TDSE:
TISE:
13. MIT
10.637
Lecture 8
Properties of operators
Here are some rules for operator algebra:
1) If then
2)
3)
4) Operators corresponding to physical quantities are linear:
5) and Hermitian:
14. MIT
10.637
Lecture 8
Commuting operators
Two operators commute if their order of application does not change the result
(i.e. they share common eigenfunctions):
Using another notation, we can say this as:
Examples of commuting operators:
Total angular momentum and components commute.
But the components don’t commute with each other.
15. MIT
10.637
Lecture 8
A free particle
The simplest Schrodinger equation to solve is for the free particle in one
dimension.
The Hamiltonian is simply the kinetic energy operator:
The Schrodinger equation: Rearranged:
Solutions to this simple differential equation:
where
16. MIT
10.637
Lecture 8
Particle in an infinite well
Particle in a 1D infinite well with boundary conditions:
For 0 ≤ x ≤ a, TISE (same as the free particle):0 x a
8
8
with:
Wavefunction must go to zero outside of the box because the potential is
infinite there. This sets boundary conditions on the wavefunction:
18. MIT
10.637
Lecture 8
Particle in an infinite well
Second boundary condition sets constraint:
Energy levels are quantized! Higher solutions have nodes.
Can rearrange to get energy, even the lowest solution has
some kinetic energy (zero point energy):
From normalization, we obtain the wavefunction:
n = 1, 2, …
19. MIT
10.637
Lecture 8
Particle in a 3D box
0 x a
0zc
This is an extension of the infinite well problem,
where now the Hamiltonian is:
It can be shown that separation of variables can be carried out. That is the
wavefunction can be written as a product of functions that each depend only on one
coordinate (this is relevant for later):
The solution looks just like a product of the 1D solutions:
Energy levels can be
degenerate (multiple
states) for cubic boxes.
20. MIT
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Lecture 8
Moving on…
• We won’t solve every Hamiltonian.
• What you should understand so far is that:
– Solving the Schrodinger equation gives us
eigenvalues (energies) and eigenfunctions
(wavefunctions).
– Solutions to this quantum mechanical master
equation are often quantized.
– The lowest energy solution will have non-zero
kinetic energy: zero point energy, Heisenberg
uncertainty principle.
– There may be degeneracy in the energy levels.
– We are just solving differential equations.
21. MIT
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Lecture 8
Hydrogen atom
+
We use a reference frame where the nucleus is fixed:
proton
Coulomb
potential The potential is simply the Coulomb interaction
between two point charges:
The total Hamiltonian:
The geometry motivates use of spherical coordinates. Recall the Laplacian:
22. MIT
10.637
Lecture 8
Hydrogen atom
It’s a little too painful to do the full derivation here.
In the spirit of separation of variables, we will treat the three
variables (r, q, f) separately, but they depend on each other
through parameters. We solve the Schrodinger equation
sequentially and replace an operator with the operator’s
eigenvalue.
A general strategy:
1. Start with the f – get eigenvalues and eigenstates of second
derivatives in f.
2. Solve the q part – finding eigenvalues and eigenvectors of
angular momentum operator.
3. Replace operator with eigenvalues so Hamiltonian depends
only on r.
4. Solve the Schrodinger equation in r.
1+2 are the same as solving the quantum rigid rotor.
23. MIT
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Lecture 8
Rigid rotor
moment of inertia for
linear molecule.
Schrodinger equation in spherical (I is constant):
Group together constants:
Rearrange Schrodinger equation (multiply by sin2q):
terms only contain q terms only contain f
24. MIT
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Lecture 8
Rigid rotor
The wavefunction must be a product of the independent functions:
And the two sides of the equations should be equal to a constant (arbitrarily set to m2
for now) and the derivatives are no longer partial:
(1)
(2)
Solution to (2) is simple, same form as for free electron:
25. MIT
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Lecture 8
Rigid rotor
These are associated Legendre polynomials of cosq
Solution to (1) is more challenging and outside of the scope of this review. Solution
introduces a second constant l and m is also referred to as ml:
26. MIT
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Lecture 8
Rigid rotor
Combined solution with normalization are called spherical harmonics:
These are also referred to by the variable
The energy levels are:
where:
27. MIT
10.637
Lecture 8
Hydrogen atom
Now, we return to the hydrogen atom, noting that we are working with separation
of variables because the potential is only dependent on r and we can separate the
components of the Laplacian:
Write out a simplified version of the Hamiltonian on this separated wavefunction
and with operators only acting on relevant components of the wavefunction:
28. MIT
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Lecture 8
Hydrogen atom
We can separate out the one component dependent on f and q from our previous
analysis of the rigid rotor:
So we insert the eigenvalues into Hamiltonian and divide by :
This differential equation can be solved (not shown here) and the resulting
solutions are related to the associated Laguerre polynomials.
31. MIT
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Lecture 8
Hydrogen atom
Quantum numbers from gen. chem. directly from solving the hydrogen atom
Hamiltonian:
Real parts of the spherical harmonics in hydrogen atom wavefunction:
Linear combinations of spherical harmonics to produce only real orbitals.
Radial part: Spherical harmonics (rigid rotor solutions):
Principle quantum number
Determines # nodes (n-1)
Azimuthal QN – angular
momenta (s=0,p=1,d=2…)
Magnetic QN –
distinguishes between
wfns of same AM
34. MIT
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Lecture 8
Multi-electron atoms
Once we have two or more electrons, the Schrodinger
equation cannot be solved exactly: fundamental
challenge for quantum chemistry!
Helium atom hamiltonian:
fixed nucleus at origin
1
2r12
r2r1
He
electron
kinetic energy
electron-nuclear
coulombic attraction
electron
repulsion
Not exactly solvable because of electron repulsion term – the electrons are
correlated. If electrons don’t interact, energy is just sum of each electron’s energy.
35. MIT
10.637
Lecture 8
Quantum mechanics for
molecules
1) Born-Oppenheimer approximation: Large difference in mass (de broglie
wavelength) of nuclei and electrons means we can view the nuclei as fixed.
2) Hamiltonian for multi-electron molecule:
nuc
KE
el
KE
el-nuc
attraction
el-el
repulsion
nuc-nuc
repulsion
electronic
energy
electronic
wavefunction
a parameter
36. MIT
10.637
Lecture 8
Where do we go from here?
• Perturbation theory allows us to make a
zeroth order Hamiltonian and then can add in
corrections perturbatively – e.g. Helium using
Hydrogen as initial guess.
• Can use variation theorem to guess the form
of the wavefunction and then iterativey
improve it until the energy is lowered.
• Next classes: quantum chemistry allows us to
approximately solve the Schrodinger
equation.
Editor's Notes
Possible extensions: perturbation theory or variation theorem for He from H. Also more pictures of hydrogen atom orbitals – e.g. scan from a book or get a better source.
Changed the slide – note I had the wrong expression before.
Should be able to show why the integral falls out…it’s essentially the integral of the derivative of the function, which is just the function itself.
More later about vibrational modes and when it doesn’t work.