1Electronic Structure Calculations Introduction and Motivation a) To calculate any property of a molecule, the molecule’s wavefunction is needed. b) The wavefunction function is found by solving the Schrödinger equation using the Hamiltonian of the molecule. c) The Hamiltonian of the molecule includes 1) Kinetic energy terms for all the electrons 2) Coulomb attraction terms between each pairing of an electron and a nucleus 3) Coulomb repulsion terms between each pairing of electrons 4) Coulomb exchange terms (related to the antisymmetry requirement of the wavefunction. 5) Coulomb repulsion terms between nuclei d) To find the wavefunction a) Assume that the molecule orbitals that comprise the wavefunction can be written as an unknown Linear Combination of Atomic Orbitals (LCAO). b) Use the variation principle to find the best LCAO that minimizes the energy. Molecular Properties Electronic Energy Definition: The electronic energy is the energy lost when all of the particles (electrons and nuclei) assemble from infinity. - These energies are considerably greater than the “normal” chemical energies of several hundred kJ/mol. - The calculations are done with the nuclei fixed in a single arrangement. - These calculations are also known as single point energy calculations. Optimal Molecular Geometry - Changing the position of the nuclei will change Coulombic energy and therefore change the arrangement of the electrons. - To find the optimized molecular geometry, the nuclei are shifted multiple times to find which nuclear arrangement will yield the lowest electronic energy. - An optimized geometry should give and bond angles and distance that are close to experimental values.2 Optimal Conformations - Sometimes geometry optimization calculations produce molecular geometries that yield a “local minimum” rather than a “global minimum”. - Conformer searches move the nuclei into different conformations to find the one that yields the lowest energy. - To find an accurate geometry of large molecules and ring molecules, conformer searching is usually necessary. Dipole Moments/Magnetic Properties - Once a wavefunction is found, molecular properties can be found be calculating expectation values. - Calculating dipole moments (and other multipole moments) are valuable in the study of intermolecular forces and certain types of spectroscopy. - Magnetic properties such as NMR shielding tensors can be calculated (used to assign peaks in an NMR spectrum). Population Analyses - With a population analysis, one can calculate the charges of atoms within the molecule by calculating the number of electrons within each of the atomic orbitals used to make the molecule orbital. - Clever application of population analyses can be used to describe the bonding within a molecule based on the overlap of atomic orbitals. One application would be examine the hybridization of an atom involved in a chemical bond. Electron Densities/Electrostatic Potentials/Molecular Orbitals - The wavefunction can be used to find the electron density at any point (therefore, every point) within a molecule. These calculations can be used to visualize the total electron cloud of the molecule. - Another useful property is the electrostatic potential, that is, the voltage that surrounds a molecule. - The electrostatic potential is useful since it shows the tendency of how an outside charge will be attracted to the molecule. (Negative charge is attracted to positive potential and positive charge is attracted to negative potential.) - The total wavefunction is product of molecular orbitals. One can break apart the wavefunction and visualize the individual orbitals.3Electronic Transition Energies - Most calculations yield two types of molecular orbitals a) Occupied orbitals – orbitals that are occupied by a pair of electrons in the molecule’s ground state b) Virtual orbitals – orbitals higher in energy than the occupied orbitals where electron may go when the molecule is in an excited state. - By taking the difference between the energies of the individual orbitals, one calculate transition energies. - Also important are transition intensities (how likely is the transition to occur). With the total electronic wavefunction and the vibrational wavefunctions, one can calculated electronic transition intensities. Vibrational Transition Energies - By moving the nuclei slightly, a vibration’s potential energy curve can be calculated. - From the potential energy curve, the vibration’s force constant can be found. - Thus vibrational energies and frequencies can be found. Thermochemical Properties - Quantum mechanics connects to thermodynamics through the study of statistical thermodynamics. - By taking a special sum of quantum mechanical energies, one can calculate any thermodynamic quantities such as the enthalpy or entropy. - Thus after finding electronic, vibrational, rotational energies, thermodynamic quantities can be found. Transition States - We can find transition states (activated complexes) for reactions by searching for vibrations with negative force constants. Such a situation corresponds to a potential energy that is approximately quadratic but is concave down rather than concave up. - Vibrations with negative force constant yield vibrational imaginary frequencies. Thus looking at a vibration with imaginary frequencies should yield information about the transition state.4Basis Sets Introduction: A Problem To find a wavefunction for a molecule, we attempt to find an appropriate Linear Combination of Atomic Orbitals (LCAO). To do the linear combination, we need atomic orbitals. The question arises: What atomic orbitals shall we use? The atomic orbitals of the hydrogenic atom seem like a sensible choice. However, those orbitals were found without considering any electron repulsion or electron exchange. Solution: Slater orbitals A solution can be found by using orbitals that use a couple of adjustable parameters, an effective nuclear charge, Z′, and an effective principal quantum number, n*. ()()()0Zrn1nalmr, , Nr e Y ,∗∗⎛⎞′−⎜⎟⎜⎟−⎝⎠ψθφ= θφ Orbitals of this form are called Slater orbitals. The Z′ and n* are a way of accounting for the shielding of the nuclear charge from the other electrons in the atom.