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atomic units

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allow for a simplification of quantum mechanic equations. Values expressed in atomic units are not affected by refinements in various constants. units include bohr, hartree, and hbar. Set all constants to 1.

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interelectronic repulsion

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term in the schrodinger wavefunction for helium that relies on the positions of both electron one and two. It results in a nonseparable term that makes solving the schrodinger equation exactly impossible

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variational principle

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If an arbitrary wavefunction (depending on parameters called variational parameters) is used to calculate the energy, then the calculated value of energy is never less than the true ground-state energy E0.

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Variational Method

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Variational method is an approximate method of determining the ground-state wavefunction (and energy) of a system (atom or molecule) when solving exactly Schrödinger equation is impossible. It uses the Variational Principle In practice, a trial function that include a number of parameters, called variational parameters, is used to obtain an expression for the energy that will depend on these variational parameters. This energy expression is then minimized with respect to the variational parameters, leading to approximate expression for the wavefunction. Can also do Linear Variational Meyhod, in which the trial function is a linear combination of functions with coefficient variational parameters

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trial function

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Used in the Variational Method, a trial function includes a number of parameters, called variational parameters, and is used to obtain an expression for the energy that will depend on these variational parameters. This energy expression is then minimized with respect to the variational parameters, leading to approximate expression for the wavefunction.

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variational parameters

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parameters used in a trial function to obtain an expression for the energy that will depend on these variational parameters. This energy expression is then minimized with respect to the variational parameters, leading to approximate expression for the wavefunction. Could also be coefficients in linear variational method

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effective nuclear charge

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The effective nuclear charge is the net positive charge experienced by valence electrons. It follows the equation Zeff = Z-S, where S is found by Slater's Rules.

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matrix elements

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integral expressions that result in the linear variational method evaluation of energy. Two examples are the coulomb integral and the overlap integral. If the coluomb integral is Hermitian then, the two matrix elements are equal to each other.

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Coulomb integral

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H, a matrix element, a result from the evaluation of the numerator of the energy expression in the linear variational method. The origin of its name results from noting that the charge in the volume element dr is similar to the coulomb energy of interaction It also appears in MO theory

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resonance integral or overlap integral

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S, a matrix element, a result from the evaluation of the denominator of the energy expression in the linear variational method. It is only significant when two atomic orbitals have a large overlap, dependent on the internuclear separation. It also appears in MO theory

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secular determinant

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The determinant that results from the minimization of the energy with respect to the variational paramters in the linear variational method. It should be equal to zero. Solving the determinant results in a secular equation.

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secular equation

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The equation that results from solving a secular determinant, which is equal to zero. This equation is an ultimate result of the minimization of the energy with respect to variational paramters in the linear variational method. It's small value solution is the variationla approximation for the ground state energy of the trial function.

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Slater orbitals

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defined as: give equation, introduced by John Slater The components of a slater orbital include variational paramter n and zeta, which is arbitrary and not necessarily equal to Zeff, spherical harmonics (Y). The radial portion does not have nodes like hydrogen atomic orbitals do. Snlm is not orthogonal to Sn'lm, so integrals that were once zero must now be included. STO

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Perturbation theory

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It as a method allows for determining the energy and the wavefunction of a real system as a successive set of corrections based on the perturbation which is the difference between the Hamiltonian of the real system and that of an ideal system for which an exact solution is known.

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unperturbed Hamiltonian operator

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the hamiltonian in the schrodinger equation for a system that can be solved exactly. Denoted H^0

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perturbation

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an addition made to H^0 in order to approximate the hamiltonian of a system whose schrodinger equation cannot be solved exactly. Denoted H^(n) where n is nonzero integers.

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spin

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is an intrinsic angular momentum possessed by elementary particles. For example, an electron has a built in spin of 1/2 or -1/2 because IT IS an electron.

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spin quantum numbers

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The spin quantum numbers are s (Spin angular momentum) and m_s (secondary/magnetic spin QN), they are analogous to l and m_l. m_s determines the z component of the electron spin angular momentum. m_s can take on 2s+1 values, for for an electron it can have two values. They can be -s,...,+s Any given elementary particle can only have one value for s, which can be a half integral or an integral.

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spin operator

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The two spin operators are S^2 (total spin angular momentum) and S_z (Z component of spin angular momentum), which are equal to (give equations)

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spin eigenfunctions

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the functions alpha and beta, they are orthonormal to each other (normalized and orthogonal) and act upon spin variable

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spin variable

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Represented by sigma, has no classic analogue, acted upon spin eigenfunctions

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fermion

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Fermions are particles of half-integer spin that are described by antisymmetric wavefunctions (that do change sign upon changing the labels of two particles).

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boson

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Bosons are particles of integer spin. According to the Pauli principle, when the labels of any two identical bosons are exchanged, the total wavefunction retains its sign. In other words, the wavefunction describing a boson is symmetric.

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sixth postulate of QM

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All electron wavefunctions must be antisymmetric under the exchange of any two electrons.

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Pauli exclusion principle

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Two electrons in an atom cannot have the same values for the four quantum numbers n, l, m_l, m_s

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Pauli principle

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When the labels of any two identical fermions are exchanged, the total wavefunction changes sign when the labels of any two identical bosons are exchanged the total wavefunction retains the same sign

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spin orbital

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One electrons wavefunctions, they are normalized

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Slater determinant

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A Slater determinant is the electronic wavefunction written as a determinatal wavefunction to ensure that the wavefunction is antisymmetric.

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determinantal wavefunction

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is a wavefunction represented by a slater determinant to ensure that the wavefunction is asymmetric. The elements of the determinant are orthonormal spin orbitals. Properties of these include wavefunction = 0 if two columns are the same, and psi changes sign if columns are interchanged

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Hartree-Fock/SCF method/approximation

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In the HF method, the wavefunction for the system depends on the effective hamiltonian operator. However, the operator also depends on the wavefunction. This circular logic is solved by using a self-consistant field procedure where a wavefunction is guessed, used to determine effective mean field potential, define an effective 1e hamiltonian operator, then use that to solve for a new wavefunction and repeat until there is no change.

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effective one-electron Hamiltonian operator

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This is the effective hamiltonian operator for an individual electron used in the HF method for He. It represents the the kinetic energy of the electron plus its attraction to the nucleus plus the operator Veff, the effective potential energy or mean field that the electron feels at any point in space.

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Hartree-Fock equation

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relates the effective 1e hamitonian operator and the wavefunciton to the energy.

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Fock operator

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the effective hamiltonian in the HF method when the wavefunction is considered as a spin orbital (expressed with slater determinant)

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Hartree-Fock orbitals

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Self constitant orbitals (the final wavefunction) in the HF orbital when the wavefunction is considered as a spin orbital and a Fock operator is used

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Koopman's theorem

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the ionization potential energy of an electron is approximately the negative magnitude of the electron's orbital energy IP_i = -E_i

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correlation energy

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energy used to correct the inexact HF method. Can be approximated with perturbation theory. CE= E_exact - E_HF

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electron configuration

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reports the subshells or orbitals that are occupied in an atom or ion and gives how many electrons are in which orbital

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Russell-Sounders coupling

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a method of providing more detailed information on the electronic state in an atom in a form of an atomic term symbol

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total angular momentum quantum number

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J, accounts for both orbital angular momentum and spin angular momentum

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total orbital angular momentum quantum number

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L, values represent letters S, P, D, F, ..., is sum of l values

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total spin angular momentum quantum number

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S, use 2S+1 in atomic term symbol to get multipliciy

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atomic term symbol

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gives information on the electronic state of an atom

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spin multiplicity

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result of 2S+1

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Hund's rules

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The state with the largest value of S then L is the most stable If S and L are same for two states, then if the subshell is less than half filled, the smallest value of J is the most stable if the subshell is more than half filled, the state with the largest value of J is most stable

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microstate

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a set of m_l and m_s values for each electron in the atom (all possible permutations)

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fine splitting

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in atomic spectra, the splitting of various n levels into sets of closely lying energy levels can be observed. This is due to spin orbit coupling.

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spin-orbit coupling

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the interaction of the spin magentic moment of an electron with the magnetic field generated by the electric current produced by the electron's own orbital motion.

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fine structure

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increased spectral complexity caused by the spin-orbit coupling

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Born-Oppenheimer approximation

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Born-Oppenheimer approximation is an approximation used to calculate the electronic properties of molecules. Due to larger mass of nuclei compared to electrons, it is considered that nuclei are fixed when solving for electronic wavefunction

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potential energy curve (or surface)

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the treatment of nuclear motion on a curve that includes the electronic energy and the internuclear repulsion energy. It is a result of the Born-Oppenheimer approximation. (draw an example)?

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Molecular Orbital Theory

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a method to describe the bonding properties of molecules in a form of orbitals (or electron wavefunctions) distributed over the whole molecule called molecular orbitals

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molecular orbitals

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a orbital that is distributed over the whole molecule, is the result of combination of atomic orbitals

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linear combination of atomic orbitals (LCAO)

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the combination of atomic orbitals in a linear manner, for example for H2+, Psi_+- = c1x1sA +- c2x1sB

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exchange integral

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The exchange integral is responsible for the existence of the chemical bond. It doesn't have a classical equivalent. For H2+ ion, the expression is:

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bonding orbital

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denoted psi+ or psi_b, a bonding orbital is the constructive interference due to the combination of two atomic orbitals

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antibonding orbital

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denoted psi- or psi_a, a bonding orbital is the destructive interference due to the combination of two atomic orbitals

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Molecular energy level diagram

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a representation of the respective energy levels of atomic and molecular orbitals in a molecule, and the placement of electrons in those orbitals, used in MO theory

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sigma orbital, pi orbital, delta orbital

answer

a sigma bonding orbital is cylindrically symmetric with respect to the internuclear axis a pi bonding orbital has bonding above and below the internuclear axis, but has 1 nodal plane that contains the internuclear axis A delta bond is the result of the faces of four lobes of d orbitals interacting, it has 2 nodal planes that contain the internuclear axis

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u (ungerade) and g (gerade)

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labels detailing the symmetry of the sign under the inversion of the orbital through the midpoint odd, even german

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homonuclear/heteronuclear diatomic

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having the same or different two atoms in a molecule

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bond order

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give equation A high bond order is equivalent to a small internuclear distance and a high bond energy, and vice versa A molecule with 0 bond order is unstable and does not exist

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photoelectron spectroscopy

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the measurement of the energies of the electrons ejected by radiation incident on gaseous molecules This method provides evidence of the existence of molecular orbitals

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molecular term symbol

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designates the electronic state of a molecule (the symmetry properties of molecular wavefunctions)

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hybrid orbitals

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the concept of hybrid orbitals was introduced to interpret molecular shape, and why C was tetravalent Hybrid orbitals are the result of mixing between atomic orbitals of an atom Hybrid orbitals in valence bond theory include sp, sp2, sp3, sp3d, sp3d2, etc

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s and p character of hybrid orbitals

answer

are the terms associated with the s and p orbitals respectively in the combination of s and p orbital wavefunctions A hybrid orbital that is x%s character and (100-x)% p character can be expressed as an sp^100-x/x hybrid orbital

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Walsh correlation diagrams

answer

a walsh correlation diagram is a plot of the energy of a molecular orbital as a function of change in molecular geometry

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pi-electron approximation

answer

The pi electrons are moving in an fixed, effective electrostatic potential due to the electrons in the sigma framework and can be independently treated from the sigma electrons.

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