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.