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I BASICS OF
CLASSICAL MECHANICS
1 Principles of classical dynamics
1.1 Newtonian dynamics
1.2 Space and time
1.3 Mass
1.4 Energy
1.5 Electric charge
1.6 Reference system of coordinates
1.7 Newtonian time
1.8 Linear motion
1.9 Angular motion
1.10 Descriptions between inertial
reference frames
2 Foundations of Newtonian dynamics
2.1 First Newton’s law
2.2 Second Newton’s law
2.3 Third Newton’s law
2.4 Reduced mass of a two-particle
system
2.5 Time reversibility
2.6 Angular momentum and torque
2.7 Impulse, work and power
2.8 Kinetic and potential energies
2.9 Energy conservation
3 Many-particle systems
3.1 Reference frame of a
many-particle system
3.2 Angular momentum and torque of a
many-particle system
3.3 Mechanical energies of a many-particle
system
3.4 Transformation of the energy
components
3.5 Energy balance equation
3.6 Statistical and time averages of
physical observables
3.7 Ergodic hypothesis
3.8 Breaking the ergodic hypothesis
3.9 Velocity distribution function
3.10 Temperature of a system of
particles
3.11 Temperature scaling as a
thermostat
3.12 Temperature fluctuations
3.13 Pressure and volume
3.14 The virial and the equation of
state
4 Mechanical descriptors
4.1 Caloric curve
4.2 Interatomic distance
fluctuations
4.3 Root mean square deviation of
positions
4.4 Orientational order parameter
4.5 Pair correlation distribution
function
4.6 Correlation functions
4.7 Properties of correlation
functions
4.8 Vibrational spectra from
autocorrelation functions
5 Rigid body
5.1 Angular momentum of a rotating
system of particles
5.2 External torques acting on a
rotating body
5.3 Total energy of a rotating rigid
body
6 Analytical Mechanics
6.1 Action function
6.2 Principle of stationary action
6.3 Classifying molecular systems
6.4 Lagrange’s equations of motion
6.5 Newtonian equations of motion
from Lagrange theory
6.6 Non-uniqueness of the Lagrangian
6.7 Invariance of the Lagrange
equations of motion
6.8 Motion with constraints
6.9 Hamilton’s function
6.10 Preservation of the Hamiltonian
in time
6.11 Conserved observables and
symmetries
6.12 Space homogeneity
6.13 Space isotropy
6.14 Uniform passage of time
6.15 Hamilton’s equations of motion
6.16 Invariance under canonical
transformations
6.17 Time reversibility in
Hamiltonian theory
6.18 Hamilton-Jacobi theory
6.19 Illustrating with the harmonic
oscillator
6.20 Contact between quantum and
classical mechanics
6.21 Poisson’s brackets
6.22 Classical time propagator
II BASICS OF QUANTUM MECHANICS
7 Wave-particle duality of matter
7.1 Young’s experiment
7.2 Interference of waves
7.3 Photo-electron experiment
7.4 Compton’s experiment
7.5 Davisson-Germer’s experiment
7.6 De Broglie’s hypothesis
7.7 Bohr’s complementary principle
8 Quantization of the energy
8.1 Planck’s energy equation
8.2 Blackbody radiation experiment
8.3 Rayleigh-Jeans law
8.4 Wien’s displacement law
8.5 Ultraviolet catastrophe
8.6 Planck’s law
8.7 Franck-Hertz experiment
8.8 Heisenberg’s uncertainty
principle
8.9 Appendix: Planck’s radiation
intensity law
9 Quantization of the angular
momentum
9.1 Orbital angular momentum and
spin
9.2 Characterizing a particle with
spin
9.3 Stern-Gerlach experiment
9.4 Wave-particle duality and spin
of a particle
9.5 Fermions and bosons
9.6 Pauli’s exclusion principle and
Hund’s rule
9.7 Appendix: magnetic moment
9.7.1 Electric current in a circular
loop
9.7.2 Magnetic g factor
9.7.3 Magnetic energy and magnetic
work
9.7.4 Zeeman effect
9.7.5 Electron spin
9.7.6 Paschen-Back effect
9.7.7 Applications of the spin
resonance technique
10 Postulates of quantum mechanics
10.1 Reformulating the conceptual
world
10.2 Postulates of quantum mechanics
10.2.1 First postulate
10.2.2 Second postulate
10.2.3 Third postulate
10.2.4 Fourth postulate
10.2.5 Fifth postulate
10.2.6 Sixth postulate
10.3 Stationary states
10.4 Superposition principle of
quantum states
10.5 Bohr’s correspondence principle
10.6 Selection rules
10.7 Pauli’s principle in the
electronic wave function
10.8 Wave function of the electrons
in a molecule
10.9 Variational principle of the
energy
10.10 Appendix: proposing the wave
equation for matter waves
10.11 Appendix: expansion of a determinantal
wave function
III FIRST-PRINCIPLES MOLECULAR
DYNAMICS
11 Dynamics of electrons and nuclei
11.1 The electronic and nuclear
dynamics are coupled
11.2 The molecular Hamiltonian
11.3 Approximating the total wave
function 20611.4 The time-dependent self-consistent field equations
12 Classical limit of the nuclear
motion
12.1 Polar form of the nuclear wave
equation
12.2 Continuity and Hamilton-Jacobi
equations
12.3 Conditions to describe the nuclear
particles classically
12.4 Simplification of the nuclear
potential
12.5 Parameterizing the potential
function
12.6 Total energy of the molecular
system
12.7 Establishing the accuracy of
atomic forces
12.8 Diffusion from the continuity
equation
12.9 Diffusion equation and particle
flux
12.10 Expansion of the electronic
wave equation
12.11 Expansion of the Newtonian
equation of the nuclei
12.12 Appendix: the Bohm’s quantum
potential
IV CLASSICAL MOLECULAR DYNAMICS
13 Classical molecular dynamics
13.1 Model interaction potentials
13.2 Forcefields
13.3 Atom types
13.4 The united atom
13.5 Bond elongation and compression
13.6 Combination rules
13.7 Bond angle vibration
13.8 Plane bending
13.9 Angle inversion
13.10 Torsional motion
13.11 Electrostatic interaction
13.12 Van der Waals forces
13.13 Interaction potential
functions of water
13.14 Polarizability of atoms
13.15 External fields and potentials
13.16 Parameterization of forcefields
13.17 Model potentials of
non-biological systems
13.18 Sutton-Chen potential function
13.19 Gupta potential function
13.20 Tersoff potential function
13.21 Appendix: harmonic model of
the dispersion energy
14 Extended systems
14.1 Fixed and flexible boundaries
14.2 Periodic boundary conditions
14.3 The P BC system is an open
system
14.4 Electrostatics in the P BC approach
14.5 Ewald sum approach
14.6 Using the Poisson equation
14.7 Short-range interactions
14.8 Dealing with the electrostatic
self-interaction
14.9 Long-range interactions
14.10 Ewald electrostatic energy
14.11 Smooth particle mesh Ewald
approach
14.12 Shifted potentials and forces
V TIME EVOLUTION OPERATORS
15 Integrating the equations of
motion
15.1 The Liouville operator as a
time propagator
15.2 Discretizing the time
propagator
15.3 Evolving positions and momenta
15.4 Simplified time integrators
15.5 Leapfrog algorithm
15.6 Verlet algorithm
15.7 Bond constraints
Ruben Santamaria is originally from California, United States. He studied physics at UNAM and earned his doctorate degree in Molecular Physics from the University of Oxford. He did postdoctoral studies in Molecular Biophysics at Northwestern University in Chicago. He has vast experience as a researcher and works at the Physics Institute of the University of Mexico (UNAM). He has given talks at various national and international institutions and taught courses in his specialty. His research focuses mainly on developing methodologies in atomic and molecular physics, and proposing new processes with applications to nanostructured systems and molecular biophysics, using the latest technological tools in computing and artificial intelligence.
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