YouTube Video Lessons

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Differential Equations: These are equations involving derivatives of a function. In this chapter, ordinary differential equations with one independent variable are discussed, as well as linear and nonlinear differential equations 1. Boundary Conditions: These are conditions that specify the value of a function or its derivatives at specific points. They are used to determine the constants in the general solution of a differential equation 1. Linear Differential Equation: A type of differential equation where the dependent variable and its derivatives appear to the first power and are not multiplied together 1. Nonlinear Differential Equation: A differential equation that cannot be written in the form of a linear differential equation. It involves terms where the dependent variable or its derivatives appear to a power other than one or are multiplied together 1. Homogeneous Differential Equation: A linear differential equation where the function on the right-hand side is zero 1. Inhomogeneous Differential Equation: A linear differential equation where the function on the right-hand side is not zero 1. Schrödinger Equation: A fundamental equation in quantum mechanics that describes how the quantum state of a physical system changes over time. The time-independent Schrödinger equation is used in this chapter to solve for the stationary-state wave functions and energy levels of a particle in a one-dimensional box 1. Wave Function (ψ): A mathematical function that describes the quantum state of a particle. The square of the wave function's magnitude gives the probability density of finding the particle at a given position 1. Quantum Number (n): A number that quantizes the energy levels of a particle in a box. Each quantum number corresponds to a different wave function and energy state 1. Nodes: Points where the wave function is zero. The number of nodes increases with the quantum number 1. Ground State: The lowest energy state of a particle in a box. It corresponds to the quantum number n=1 1. Excited States: Energy states higher than the ground state. They correspond to quantum numbers n=2, 3, etc 1. Bohr Correspondence Principle: A principle stating that the predictions of quantum mechanics converge to those of classical mechanics as the quantum numbers become very large 1. Orthogonality: A property of wave functions where the integral of the product of two different wave functions over all space is zero 1. Normalization: The process of adjusting the wave function so that the total probability of finding the particle is one 1. Tunneling: A quantum mechanical phenomenon where a particle can pass through a potential energy barrier that it classically should not be able to pass 1.

Quantum Mechanical Operators

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Operator: A rule that transforms a given function into another function. Operators are fundamental in quantum mechanics for describing physical quantities and their interactions 1. Energy Operator: An operator that, when applied to the wave function, returns the wave function multiplied by an allowed value of the energy 1. Differentiation Operator: An operator that differentiates a function with respect to a variable, often denoted with a circumflex (e.g., ( \hat{D}_n )) 1. Sum and Difference of Operators: The sum of two operators ( \hat{A}_n ) and ( \hat{B}_n ) applied to a function ( f(x) ) is defined as ( (\hat{A}_n + \hat{B}_n)f(x) = \hat{A}_n f(x) + \hat{B}_n f(x) ). The difference is similarly defined 1. Product of Operators: The product of two operators ( \hat{A}_n ) and ( \hat{B}_n ) applied to a function ( f(x) ) is defined as ( \hat{A}_n \hat{B}_n f(x) = \hat{A}_n (\hat{B}_n f(x)) ) 1. Operator Algebra: A set of rules and operations for manipulating operators, including the associative law of multiplication and commutative properties 1. Commutator: The commutator of two operators ( \hat{A}_n ) and ( \hat{B}_n ) is defined as ( [\hat{A}_n, \hat{B}_n] = \hat{A}_n \hat{B}_n - \hat{B}_n \hat{A}_n ). If the commutator is zero, the operators are said to commute 1. Eigenfunction: A function ( f(x) ) that, when operated on by a linear operator ( \hat{A}_n ), results in the function being multiplied by a constant ( k ). This constant is called the eigenvalue 1. Eigenvalue: The constant ( k ) that results from operating on an eigenfunction with a linear operator. It represents a characteristic value associated with the operator 1. Hamiltonian Operator: The operator corresponding to the total energy of a system, often used in the Schrödinger equation to find the energy eigenvalues and eigenfunctions 1. Linear Operator: An operator ( \hat{A}_n ) that satisfies the properties ( \hat{A}_n (f(x) + g(x)) = \hat{A}_n f(x) + \hat{A}_n g(x) ) and ( \hat{A}_n (cf(x)) = c\hat{A}_n f(x) ), where ( f ) and ( g ) are functions and ( c ) is a constant 1. Laplacian Operator: An operator denoted by ( \nabla^2 ) (del squared), which is the sum of the second partial derivatives with respect to each spatial coordinate 1. Expectation Value: The average value of a physical property ( B ) for a system in state ( \psi ), calculated as ( \langle B \rangle = \int \psi^* \hat{B} \psi , d\tau ), where ( \hat{B} ) is the operator corresponding to ( B ) 1. Normalization: The process of adjusting the wave function ( \psi ) so that the total probability of finding the particle is 1. This ensures that ( \int |\psi|^2 , d\tau = 1 ) 1. Quadratically Integrable: A function ( \psi ) is quadratically integrable if ( \int |\psi|^2 , d\tau ) is finite. This property is necessary for the function to be normalized 1.

The Harmonic Oscillator

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Harmonic Oscillator: A system in which a particle experiences a restoring force proportional to its displacement from equilibrium. It is a fundamental model for understanding molecular vibrations 1. Schrödinger Equation: A key equation in quantum mechanics that describes how the quantum state of a physical system changes over time. In this chapter, it is solved for the harmonic oscillator 1. Power-Series Method: A mathematical technique used to solve differential equations by expressing the solution as an infinite sum of terms 1. Force Constant (k): A parameter that measures the stiffness of the bond in a molecule. It is the proportionality constant in the force equation ( F = -kx ) 1. Vibration Frequency (ν): The frequency at which a molecule vibrates. For a harmonic oscillator, it is given by ( \nu = \frac{1}{2\pi} \sqrt{\frac{k}{m}} ) 1. Zero-Point Energy: The lowest possible energy that a quantum mechanical system may have. For a harmonic oscillator, it is ( \frac{1}{2} h \nu ) 1. Eigenvalues and Eigenfunctions: Solutions to the Schrödinger equation that describe the allowed energy levels and corresponding wave functions of a quantum system 1. Recursion Relation: A relation that defines each term of a sequence as a function of preceding terms. It is used in the power-series method to solve the Schrödinger equation 1. Hermite Polynomials: A set of orthogonal polynomials that arise in the solution of the Schrödinger equation for the harmonic oscillator 1. Classically Forbidden Region: Regions where the potential energy exceeds the total energy of the system, making it impossible for a classical particle to be found there 1. Reduced Mass (m): A hypothetical mass used in the analysis of two-body problems, defined as ( m = \frac{m_1 m_2}{m_1 + m_2} ) 1. Anharmonicity: The deviation of a system from the ideal harmonic oscillator model, leading to non-equally spaced energy levels 1. Boltzmann Distribution Law: A statistical law that describes the distribution of particles among various energy states in thermal equilibrium 1. Wavenumber (ν̅): The number of wavelengths per unit distance, often used in spectroscopy. It is defined as ( \nu̅ = \frac{1}{λ} = \frac{ν}{c} ) 1.

Angular Momentum

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 Angular Momentum: A measure of the amount of rotation a particle has, taking into account its mass, shape, and speed. In quantum mechanics, it is quantized and has discrete values. Eigenfunction: A function that is associated with a particular operator in quantum mechanics, where the operator acting on the function yields the function multiplied by a constant (the eigenvalue). Eigenvalue: The constant value obtained when an operator acts on its eigenfunction. It represents measurable quantities in quantum mechanics. Commutator: An operation used to determine whether two operators can be simultaneously measured. It is defined as [A, B] = AB - BA. Simultaneous Eigenfunctions: Functions that are eigenfunctions of two or more operators at the same time, indicating that the corresponding physical quantities can be simultaneously measured. Uncertainty Principle: A fundamental concept in quantum mechanics stating that certain pairs of physical properties, like position and momentum, cannot be simultaneously measured with arbitrary precision. Standard Deviation: A measure of the spread or dispersion of a set of values. In quantum mechanics, it represents the uncertainty in a measured property. Orbital Angular Momentum: The component of angular momentum associated with the motion of a particle through space, distinct from spin angular momentum. Spin Angular Momentum: An intrinsic form of angular momentum carried by particles, independent of their motion through space. Spherical Coordinates: A coordinate system used to describe the position of a point in space using three values: the radial distance, the polar angle, and the azimuthal angle. Ladder Operators: Operators used to raise or lower the eigenvalue of another operator, often used in the context of angular momentum. Normalization: The process of adjusting the magnitude of a function so that its total probability density equals one. Degeneracy: The condition where two or more eigenfunctions correspond to the same eigenvalue, indicating multiple states with the same energy. Associated Legendre Functions: Special functions used in the solution of angular momentum problems in quantum mechanics. Spherical Harmonics: Functions that describe the angular part of the wavefunction in spherical coordinates, often used in the context of angular momentum.

Theorems in Quantum Mechanics

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Schrödinger Equation: A fundamental equation in quantum mechanics that describes how the quantum state of a physical system changes over time. Hamiltonian: An operator corresponding to the total energy of the system, including both kinetic and potential energies. Variation Method: An approximation method used to find the ground state energy of a quantum system by minimizing the energy expectation value. Perturbation Theory: A method used to find an approximate solution to a problem by starting from the exact solution of a related, simpler problem and adding corrections. Bracket Notation: A notation introduced by Dirac, used to represent the inner product of two functions in quantum mechanics. Matrix Element: The integral of an operator sandwiched between two functions, representing the transition amplitude between states. Hermitian Operator: An operator that is equal to its own conjugate transpose, ensuring that its eigenvalues are real numbers. Eigenvalues: The possible outcomes of a measurement of a physical quantity, represented by the operator. Eigenfunctions: The functions that correspond to the eigenvalues of an operator, representing the state of the system. Orthogonality: A property of functions where their inner product is zero, indicating that they are independent. Completeness: A property of a set of functions where any well-behaved function can be expanded as a linear combination of the set. Parity Operator: An operator that replaces each Cartesian coordinate with its negative, used to determine the symmetry properties of wave functions. Commuting Operators: Operators that can be applied in any order without changing the result, indicating that the corresponding physical quantities can be simultaneously measured. Uncertainty Principle: A principle stating that certain pairs of physical properties, like position and momentum, cannot be simultaneously measured with arbitrary precision. Superposition: The principle that a quantum system can exist in multiple states at once, and the overall state is a combination of these states. Reduction of the Wave Function: The process by which a quantum system's state becomes definite upon measurement, also known as wave function collapse. Dirac Delta Function: A function that is zero everywhere except at a single point, where it is infinitely high and integrates to one, used to represent point particles. Heaviside Step Function: A function that is zero for negative arguments and one for positive arguments, used to represent sudden changes. Time-Dependent Schrödinger Equation: An equation describing how the quantum state of a system evolves over time. Stationary State: A quantum state with all observables independent of time, typically an eigenstate of the Hamiltonian.

The Variation Method

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Variation Method: An approximation technique used to estimate the ground-state energy of a system without solving the Schrödinger equation 1. Variation Theorem: A theorem stating that for any normalized, well-behaved function that satisfies the boundary conditions of a system, the expectation value of the Hamiltonian is an upper bound to the ground-state energy1. Hamiltonian Operator (H n): The operator corresponding to the total energy of the system, including both kinetic and potential energies1. Ground-State Energy (E 1): The lowest energy eigenvalue of the Hamiltonian operator for a given system1. Trial Variation Function (f): A well-behaved function used in the variation method to approximate the ground-state energy1. Variational Integral: The integral of the product of the trial variation function, the Hamiltonian operator, and the trial variation function, divided by the integral of the square of the trial variation function1. Normalization Constant (N): A constant used to ensure that the trial variation function is normalized1. Eigenfunctions (c k_): The stationary-state wave functions that are solutions to the Schrödinger equation for a given Hamiltonian1. Eigenvalues (E k_): The energy values corresponding to the eigenfunctions of the Hamiltonian operator1. Orthonormal Set: A set of functions that are both orthogonal and normalized1. Kronecker Delta (d k j_): A function that is 1 if the indices are equal and 0 otherwise1. Parabolic Function: A function of the form f = x_1_l - _x_2 for a particle in a one-dimensional box1. Harmonic Oscillator: A system in which the potential energy is proportional to the square of the displacement from equilibrium1. Gaussian Elimination: A method for solving systems of linear equations by transforming the system's matrix into an upper triangular form1. Gauss–Jordan Elimination: An extension of Gaussian elimination that reduces the matrix to row echelon form1. Linear Variation Function: A linear combination of linearly independent functions used in the variation method1. Overlap Integral (S jk_): The integral of the product of two basis functions1. Secular Equation: An algebraic equation derived from the variation method that determines the approximate energies of the system1. Matrix Diagonalization: The process of finding the eigenvalues and eigenvectors of a matrix1. Hermitian Matrix: A matrix that is equal to its conjugate transpose1. Orthogonal Matrix: A matrix whose inverse is equal to its transpose1. Unitary Matrix: A matrix whose inverse is equal to its conjugate transpose1. Eigenvector: A non-zero vector that changes by only a scalar factor when a linear transformation is applied1. Characteristic Equation: An equation that determines the eigenvalues of a matrix1. Symmetric Matrix: A matrix that is equal to its transpose1. Diagonal Matrix: A matrix in which the entries outside the main diagonal are all zero1. Tridiagonal Matrix: A matrix that has non-zero elements only on the main diagonal and the diagonals immediately above and below it1. QR Method: An algorithm for finding the eigenvalues and eigenvectors of a matrix1. Cyclic Jacobi Method: An iterative method for diagonalizing a symmetric matrix1. Gaussian Variational Function: A trial function of the form e -cx 2 used in the variation method1. Block-Diagonal Form: A matrix form where the matrix is divided into smaller square matrices along the diagonal1. Normalization Condition: The condition that the integral of the square of the trial variation function is equal to 11. Expectation Value: The average value of a physical quantity in a given quantum state1. Schmidt Orthogonalization: A method for orthogonalizing a set of functions1. Symmetric Orthogonalization: A method for orthogonalizing a set of functions using the overlap matrix1. Rayleigh-Ritz Theorem: A theorem that provides an upper bound to the ground-state energy using the variation method1. Numerov Method: A numerical method for solving differential equations1. Particle-in-a-Box: A model system in quantum mechanics where a particle is confined to a one-dimensional box with infinite potential walls1. Quartic Oscillator: A system in which the potential energy is proportional to the fourth power of the displacement from equilibrium1. Double-Well Potential: A potential energy function with two minima separated by a barrier1. Harmonic Oscillator Basis Functions: The eigenfunctions of the harmonic oscillator Hamiltonian1. Particle-in-a-Box Basis Functions: The eigenfunctions of the particle-in-a-box Hamiltonian1. Radial Equation: The part of the Schrödinger equation that depends only on the radial coordinate in spherical coordinates1. Hydrogen Atom: A model system in quantum mechanics consisting of a single electron orbiting a proton1. Eigenvalue Problem: The problem of finding the eigenvalues and eigenvectors of a matrix or operator1. Matrix Algebra: The branch of mathematics that deals with matrices and their operations1. Linear Transformation: A transformation that preserves the operations of addition and scalar multiplication1. Unit Matrix: A square matrix with ones on the main diagonal and zeros elsewhere1. Inverse Matrix: A matrix that, when multiplied by the original matrix, yields the unit matrix1. Characteristic Polynomial: The polynomial obtained from the characteristic equation of a matrix1.

Perturbation Theory

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Perturbation Theory: A quantum-mechanical approximation method used to find an approximate solution to a problem by starting from the exact solution of a related, simpler problem 1. Hamiltonian Operator: An operator corresponding to the total energy of the system, including both kinetic and potential energies 1. Schrödinger Equation: A fundamental equation in quantum mechanics that describes how the quantum state of a physical system changes over time 1. Eigenfunctions and Eigenvalues: Solutions to the Schrödinger equation where eigenfunctions represent the possible states of the system and eigenvalues represent the corresponding energy levels 1. Unperturbed System: A system whose Hamiltonian is exactly solvable and serves as the starting point for perturbation theory 1. Perturbed System: A system whose Hamiltonian is slightly different from the unperturbed system, making it more complex to solve 1. Perturbation: The difference between the Hamiltonians of the perturbed and unperturbed systems 1. Nondegenerate Perturbation Theory: Perturbation theory applied to energy levels that are not degenerate (i.e., each energy level corresponds to a unique state) 1. Degenerate Perturbation Theory: Perturbation theory applied to energy levels that are degenerate (i.e., multiple states share the same energy level) 1. First-Order Energy Correction: The initial correction to the energy of a system due to perturbation 1. Second-Order Energy Correction: The subsequent correction to the energy of a system, taking into account the first-order correction 1. Intermediate Normalization: A simplification method where the perturbed wave function is required to satisfy a specific normalization condition 1. Configuration Interaction: The mixing of different configurations in the wave function due to perturbation 1. Variation-Perturbation Method: A method that combines variational principles and perturbation theory to estimate higher-order energy corrections 1. Coulomb Integral: An integral representing the electrostatic energy of repulsion between two charge distributions 1. Exchange Integral: An integral representing the interaction between two electrons when their positions are exchanged 1. Transition Dipole Moment: A measure of the probability of a transition between two states due to interaction with electromagnetic radiation 1. Selection Rules: Rules that determine the allowed transitions between quantum states based on the change in quantum numbers 1. Time-Dependent Perturbation Theory: Perturbation theory applied to systems exposed to time-dependent perturbations, such as electromagnetic radiation 1. Stimulated Emission: The process by which an atom or molecule emits a photon when exposed to radiation, causing a transition to a lower energy state 1. Spontaneous Emission: The process by which an atom or molecule emits a photon without external stimulation, causing a transition to a lower energy state 1. Absorption: The process by which an atom or molecule absorbs a photon, causing a transition to a higher energy state 1. Secular Equation: An algebraic equation used to find the energy corrections in degenerate perturbation theory 1. Hermitian Operator: An operator whose eigenvalues are real and whose eigenfunctions form a complete orthonormal set 1. Orthonormality: The property of eigenfunctions being orthogonal and normalized 1.

Electron Spin Theorem

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Electron Spin: A fundamental property of electrons that gives rise to their intrinsic angular momentum. It is not a classical effect and cannot be visualized as a physical rotation1. Spin Angular Momentum: The intrinsic angular momentum of a particle, distinct from orbital angular momentum. It is represented by operators ( S_x ), ( S_y ), ( S_z ), and ( S^2 )1. Spin Quantum Number (s): A quantum number that describes the intrinsic spin of a particle. For electrons, ( s = \frac{1}{2} )1. Spin Eigenfunctions: Functions that describe the spin state of a particle. For electrons, the eigenfunctions are denoted by ( \alpha ) and ( \beta ), corresponding to spin up and spin down states1. Spin–Statistics Theorem: A fundamental principle in quantum mechanics stating that particles with half-integer spin (fermions) must have antisymmetric wave functions, while particles with integer spin (bosons) must have symmetric wave functions1. Fermions: Particles with half-integer spin that obey the Pauli exclusion principle, meaning no two fermions can occupy the same quantum state1. Bosons: Particles with integer spin that do not obey the Pauli exclusion principle and can occupy the same quantum state1. Pauli Exclusion Principle: A principle stating that no two electrons can occupy the same spin-orbital in an atom, a consequence of the antisymmetry requirement for fermions1. Spin Magnetic Moment: The magnetic moment associated with the spin of a particle. For electrons, it is proportional to the spin angular momentum1. Nuclear Magnetic Resonance (NMR): A spectroscopic technique that observes transitions between nuclear spin energy levels in an applied magnetic field1. Spin–Spin Coupling: An interaction between the spins of adjacent nuclei that affects the magnetic field experienced by each nucleus, leading to splitting of NMR lines1. Slater Determinant: A mathematical expression used to construct antisymmetric wave functions for a system of electrons, ensuring that the wave function changes sign upon interchange of any two electrons1. Ladder Operators: Operators used in quantum mechanics to raise or lower the eigenvalue of the spin angular momentum component ( S_z ). They are denoted by ( S_+ ) and ( S_- )1.

Many Electrons Atoms

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Hartree–Fock Self-Consistent-Field Method: A computational approach used to find approximate wave functions for many-electron atoms. It involves iteratively solving the Schrödinger equation by assuming each electron moves in the field created by the nucleus and a hypothetical charge cloud formed by other electrons 1. Hamiltonian Operator: An operator representing the total energy of a system, including kinetic and potential energy. For an n-electron atom, it includes terms for electron-nucleus attractions, electron-electron repulsions, and kinetic energy of electrons 1. Central-Field Approximation: An approximation where the effective potential acting on an electron in an atom is assumed to depend only on the distance from the nucleus, simplifying calculations by averaging over angular coordinates 1. Slater Determinant: A mathematical expression used to describe the wave function of a multi-electron system, ensuring the antisymmetry required by the Pauli exclusion principle. It is a determinant of spin-orbitals 1. Spin-Orbit Interaction: A relativistic effect where the electron's spin interacts with its orbital motion, leading to energy level splitting. It is proportional to the dot product of the electron's spin and orbital angular momentum 1. Coulomb Integral: An integral representing the electrostatic interaction energy between two electrons in different orbitals. It is part of the Hartree–Fock energy calculation 1. Exchange Integral: An integral representing the exchange interaction energy between two electrons with the same spin in different orbitals. It arises due to the antisymmetry of the wave function 1. Configuration Interaction (CI): A method to improve the accuracy of wave functions by considering contributions from multiple electron configurations. It involves expressing the wave function as a linear combination of configuration state functions 1. Electron Correlation: The interaction between electrons that is not accounted for in the Hartree–Fock method. It includes instantaneous interactions that cause electrons to avoid each other, leading to a more accurate description of the system 1. Term Symbol: A notation used to describe the quantum state of an atom, including its total electronic spin and orbital angular momentum. It is written as 2S+1LJ, where S is the spin multiplicity, L is the orbital angular momentum, and J is the total angular momentum 1.

Molecular Symmetry

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Symmetry Elements: Geometrical entities (point, line, or plane) with respect to which a symmetry operation is carried out. Examples include axes of rotation, planes of reflection, and centers of inversion 1. Symmetry Operations: Transformations of a body such that the final position is physically indistinguishable from the initial position, and the distances between all pairs of points in the body are preserved. Examples include rotations, reflections, and inversions 1. n-Fold Axis of Symmetry (C_n): An axis about which a molecule can be rotated by 360/n degrees to give a configuration that is physically indistinguishable from the original position. The order of the axis is denoted by n 1. Plane of Symmetry (σ): A plane that divides a molecule into two mirror-image halves. Reflection through this plane gives a configuration that is physically indistinguishable from the original one 1. Center of Symmetry (i): A point in a molecule such that inversion through this point gives a configuration that is physically indistinguishable from the original one 1. Rotation-Reflection Axis of Symmetry (S_n): An axis about which a molecule can be rotated by 360/n degrees followed by reflection in a plane perpendicular to the axis to give a configuration that is physically indistinguishable from the original one 1. Dipole Moment: A measure of the separation of positive and negative charges in a molecule. Symmetry considerations can determine whether a molecule has a dipole moment and along which axis it lies 1. Optical Activity: The ability of certain molecules to rotate the plane of polarization of plane-polarized light. Molecules that are not superimposable on their mirror images may be optically active 1. Symmetry Point Groups: Classifications of molecules based on their symmetry elements. Examples include groups with no C_n axis, groups with a single C_n axis, and groups with multiple C_n axes 1. Group Theory: A mathematical framework used to describe the symmetry operations of molecules. It involves the study of groups, which are sets of elements with a rule for combining them that satisfies certain requirements 1.

Electronic Structures of Diatomic Molecules

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Born–Oppenheimer Approximation: An approximation in molecular quantum mechanics that assumes nuclei are fixed while electrons move, simplifying the molecular Hamiltonian operator 1. Molecular Hamiltonian: The operator representing the total energy of a molecule, including kinetic energy of nuclei and electrons, and potential energy from repulsions and attractions between nuclei and electrons 1. Schrödinger Equation: A fundamental equation in quantum mechanics used to find the wave functions and energies of a molecule 1. Electronic Schrödinger Equation: The Schrödinger equation for electronic motion, obtained by omitting nuclear kinetic energy terms 1. Purely Electronic Hamiltonian: The Hamiltonian operator for electronic motion, excluding nuclear kinetic energy terms 1. Internuclear Repulsion: The potential energy term representing the repulsion between nuclei in a molecule 1. Equilibrium Internuclear Distance: The internuclear separation at which the potential energy of a molecule is minimized 1. Equilibrium Dissociation Energy: The energy difference between the potential energy at infinite internuclear separation and at the equilibrium internuclear distance 1. Zero-Point Energy: The lowest possible energy that a quantum mechanical system may have, even at absolute zero temperature 1. Harmonic Oscillator Approximation: An approximation that treats the vibration of a diatomic molecule as a harmonic oscillator, useful for calculating vibrational energy levels 1. Vibrational Anharmonicity: The deviation of a molecule's vibrational energy levels from those predicted by the harmonic oscillator approximation 1. Centrifugal Distortion: The distortion of a molecule due to rotational motion, affecting its vibrational energy levels 1. Morse Function: A potential energy function used to approximate the potential energy of a diatomic molecule, accounting for anharmonicity 1. Atomic Units: A system of units used in quantum chemistry where fundamental constants like the electron mass, charge, and Planck's constant are set to 1 1. Hartree: The atomic unit of energy, equivalent to the energy of the ground state of the hydrogen atom 1. Bohr Radius: The atomic unit of length, representing the average distance between the proton and electron in a hydrogen atom 1. Hydrogen Molecule Ion (H₂⁺): The simplest diatomic molecule, consisting of two protons and one electron, used as a model for studying molecular electronic structure 1. Confocal Elliptic Coordinates: A coordinate system used to solve the Schrödinger equation for the hydrogen molecule ion 1. LCAO-MO (Linear Combination of Atomic Orbitals - Molecular Orbital): A method for constructing molecular orbitals by combining atomic orbitals 1. Bonding and Antibonding Orbitals: Molecular orbitals formed from the combination of atomic orbitals, where bonding orbitals lower the energy and antibonding orbitals raise the energy of the molecule 1.

Theorems of Molecular Quantum Mechanics

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Electron Probability Density: This term refers to the likelihood of finding an electron in a specific region of space. It is derived from the wave function of a molecule and is crucial for understanding the spatial distribution of electrons in molecular systems 1. Dipole Moment: The dipole moment of a molecule is a measure of the separation of positive and negative charges within the molecule. It is calculated from the wave function and is important for understanding the molecule's interaction with electric fields 1. Hartree–Fock Method: This is a computational method used to determine the wave function and energy of a quantum many-body system in a stationary state. It approximates the wave function as a single Slater determinant of spin-orbitals 1. Virial Theorem: This theorem relates the average kinetic energy and potential energy of a system in a bound state. It is useful for understanding the stability and bonding of molecules 1. Hellmann–Feynman Theorem: This theorem states that the force on a nucleus in a molecule can be calculated as the sum of the electrostatic forces exerted by the other nuclei and the electron charge density. It simplifies the calculation of forces in molecular systems 1. Molecular Orbital (MO): A molecular orbital is a region in a molecule where there is a high probability of finding an electron. MOs are formed by the combination of atomic orbitals and are used to describe the electronic structure of molecules 1. Slater Determinant: This is a mathematical expression used to describe the wave function of a multi-electron system in a way that satisfies the Pauli exclusion principle. It ensures that the wave function changes sign when any two electrons are exchanged 1. Coulomb Integral: This integral represents the electrostatic interaction between electrons in different orbitals. It is a key component in the calculation of molecular energies using the Hartree–Fock method 1. Exchange Integral: This integral accounts for the exchange interaction between electrons due to their indistinguishability and the Pauli exclusion principle. It is also a key component in the Hartree–Fock method 1. Roothaan Equations: These are a set of linear equations derived from the Hartree–Fock method, used to determine the coefficients of the molecular orbitals in terms of a chosen basis set 1.