Computational Chemistry: Concepts, Theories and Applications

Explore the potential energy surface, linking structure to energy, computed at zero kelvin. Build it by scanning bond lengths and angles on a grid, illustrating dimensionality with diatomic and ozone.

Explore the hypersurface concept by analyzing the potential energy surface for ozone to iso ozone conversion, identifying minima, a transition state, and the reaction coordinate.

Use the Newton-Raphson method to locate stationary points on the potential energy surface, distinguishing local minima from saddle points and transition states via first and second derivatives.

Use the Newton-Raphson geometry optimization to locate stationary points on the potential energy surface, distinguishing local minima from transition states by gradient and Hessian analyses.

Identify stationary points on a complex PES from one reactant to two products. Locate minima, saddle points, and transition states, and distinguish endothermic and exothermic pathways and their activation energies.

Derive normal modes from the Hessian by diagonalization; eigenvectors define mode directions and eigenvalues are force constants. Include translations, rotations, and vibrations; imaginary frequencies indicate transition states under harmonic approximation.

Analyze the given frequency data to identify a true minimum, a transition state with one imaginary frequency, and an inaccurate transition state, using the sign and count of imaginary frequencies.

Explore zero point vibrational energy and its role in reaction profiles, and learn how partition functions describe the distribution of electronic, translational, rotational, and vibrational energies to thermodynamic quantities.

Explain translational, rotational, and vibrational partition functions for an ideal gas, deriving translational and rotational energies as 3/2 RT, and outlining vibrational frequencies in internal energy and entropy.

Explore the postulates of quantum mechanics and ab initio electronic structure methods, focusing on wave functions, Hermitian operators, eigenvalues, and the role of the Schrödinger equation in predicting electronic properties.

Explore the molecular Hamiltonian and Born-Oppenheimer approximation, deriving a four-term electronic Hamiltonian with fixed nuclei. Obtain the electronic energy from the Schrödinger equation and add nuclear repulsion for total energy.

Examine the many body problem in the molecular Hamiltonian and use variational methods with basis-set trial wave functions to minimize energy and bound the ground state energy.

Build trial wavefunctions as linear combinations of basis functions, optimize coefficients to minimize energy using a secular determinant with overlap and resonance integrals, demonstrating gaussian basis expansion.

Define a molecular trial wave function using Hartree product, where electrons occupy orthonormal molecular orbitals under Pauli exclusion and antisymmetry, reflecting indistinguishability and spin alongside spatial coordinates.

Explain how the Slater determinant enforces indistinguishability and antisymmetry in Hartree-Fock theory, linking molecular orbitals as linear combinations of basis functions and variationally optimizing coefficients to include exchange effects.

Explore Hartree-Fock energy by its one-electron, Coulomb, and exchange components, and see how two-electron energy arises and how the Fock operator minimizes energy.

Develop and solve Hartree-Fock equations using the Fock operator to obtain molecular orbitals and the Slater determinant for total energy via self-consistent field iterations.

Explore how post-hartree-fock methods address Hartree-Fock limitations by incorporating dynamic and static correlation through multi-determinant wave functions, configuration interaction, and coupled-cluster methods such as CCSD.

Learn perturbation theory in post-hartree-fock methods, deriving zeroth, first, and second order energy and wave-function corrections from a solvable h0 and a perturbation v.

Apply perturbation theory to chemical problems with the Fock operator h0, link zeroth/first order to Hartree–Fock energy, and derive MP2 as second-order correction, noting electron correlation and non-variational, high cost.

Dive into coupled cluster calculations, using an exponential T operator to include single, double, and perturbative triple excitations for a size-consistent, gold-standard quantum chemical method.

Calculate correlation energy and percent correlation energy for post-Hartree-Fock methods (MP2, CISD, CCD, CCSD, CCSDT) using HF as zero reference for hydrofluoric acid, with CCSDT at 100%.

Capture static correlation with multiple Slater determinants beyond a single resonance form. Use complete active space self-consistent field to optimize orbitals and determinants within an active space.

Learn density functional theory and how the Hohenberg-Kohn theorems link ground-state energy to electron density. Examine Thomas-Fermi theory and its limitations in kinetic energy and electron interactions.

Trace the sources of error in the Thomas-Fermi model, the exchange correction via Hartree-Fock and Slater methods, and how Kohn-Sham DFT uses exchange-correlation functionals to fix self-interaction.

Explore how exchange correlation functionals in density functional theory connect Kohn-Sham orbitals to energy and density, and how approximations shape accurate electronic structure predictions.

Compare empirical versus non-empirical exchange-correlation functionals in density functional theory. Learn LDA, GGA, meta-GGA, and hybrid functionals with examples like BP86, PW91, PBE, and B3LYP.

Explains basis functions and basis sets, shows how molecular orbitals are built from basis functions, compares Slater and gaussian-type orbitals, and discusses contraction and computational implications in Hartree-Fock and DFT.

Explore basis functions and contracted gaussian orbitals, enabling nodal and angular behavior with spherical or Cartesian harmonics. Basis sets are atom-centered collections, from minimal sto3g to larger zeta expansions.

Explain basis set types from single to quadruple zeta, including contracted and primitive Gaussians, split valence, and polarization functions, plus core valence considerations and correlation consistent sets.

We calculate primitive and contracted Gaussian functions for water with the STO3g minimal basis, showing oxygen with 5 contracted and 15 primitive, and hydrogen with 2 contracted and 6 primitive.

Learn how polarization and diffuse functions extend basis sets like 6311 GDP, with oxygen and water as examples, and interpret star notation and heavy vs light atoms.

Identify dispersion-corrected DFT functionals and diffuse basis sets for weak interactions in water and benzene dimers. For coordination complexes, use hybrid or meta-GGA functionals with diffuse and polarization functions.

Learn how plane wave basis sets model periodic condensed-phase systems under periodic boundary conditions. Explore energy cutoffs, g-vectors, and effective core potentials that reduce basis functions and enable relativistic effects.

In this practice problem, compute primitive and contracted Gaussian functions for C2H4 using a 6-31G ADP basis with DP polarization and no diffuse function.