Statistical Mechanics and Multi-Scale Simulation Methods ChBE 591-009

1. Statistical Mechanics and Multi-Scale Simulation MethodsChBE 591-009 Prof. C. Heath Turner Lecture 05 • Some materials adapted from Prof. Keith E. Gubbins: http://gubbins.ncsu.edu • Some materials adapted from Prof. David Kofke: http://www.cbe.buffalo.edu/kofke.htm

2. Open-Shell Systems Spin-restricted HF (RHF)= uses same spatial functions for both the a and b spins. Spin-unrestricted HF (UHF)= uses separate spatial functions for the a and b spins, resulting in two Fock matrices: ** electron spin everywhere is NO LONGER zero. A spin density can now be defined: • When is UHF needed? • open-shell systems • radicals • molecules undergoing dissociation • some ground-state molecules

3. Open-Shell Systems Spin-unrestricted HF (UHF)- visualization

4. Advanced Ab Initio Methods • Configuration Interaction • Møller-Plesset perturbation theory • Coupled-Cluster Theory • Gaussian-2/Gaussian-3 • ONIOM

5. Advanced Ab Initio Methods • Electron Correlation • HF: e- move in an average potential. Dynamics of e- neighbors (called ‘dynamical correlation’) are neglected. • Fock operator = 1e- operator  1e- MOs  wave function (Slater determinant) • Ecorr = E – EHF • Example: H2  H + H, w/o e- correlation e- are predicted to spend equal time on both nuclei, even at infinite H-H separation. • wavefunction is a single determinant – sometimes too rigid… • Example: Degenerate Orbitals • Alternate approach: construct the wavefunction using multiple determinants (ci = weight factors, subject to normalization)

6. Beyond HF: Configuration Interaction The CI method employs a wavefunction which is constructed of a linear combination of the HF wavefunction with excited determinants The expansion coefficients are then selected so that they variationally minimize the expectation value of the electronic energy with respect to the CI wavefunction where the wavefunction is restricted by the normalization condition

7. Configuration Interaction for Singles If we imagine the case of H2 we can implement CI for singles by mixing in the excited state configurations. In molecules of high symmetry only the configurations of appropriate symmetry can “mix” in this way. CI that includes singles only is appropriate for improving transition energies, but does not help ground state properties.

8. Multielectron Configuration Interaction • The CI expansion is variational and, if the expansion is complete (Full CI), gives the exact correlation energy (within the basis set approximation). • The number of determinants in Full CI grows exponentially with the system size, making the method impractical for all but the smallest systems. For this reason the CI expansion is usually truncated at some order, for example CISD, where only singly and doubly excited determinants are considered. • Brillouin's Theorem states that singly excited determinants do not mix with the HF determinant. Therefore CISD is the cheapest worthwhile form of CI, yet this method scales as O(N6) where N is the size of the system. • CI is NOT size consistent: (EA+EB)CI≠ (EA)CI+(EB)CI • QCISD = size-consistent CISD theory

9. Perturbation Theory Møller-Plesset perturbation theory (MPn) ** Used to account for e- correlation: Create a more tractable operator. Using exact eigenfunctions and eigenvalues of simplified operator, estimate the eigenfunctions and eigenvalues of the more complete operator. Complete operator = H (Hamiltonian): Correction term to the approximate Hamiltonian = difference b/t true Hamiltonian (1st term) and Fock operator (2nd term): The wave function and corresponding energy can be found:

10. Perturbation Theory Møller-Plesset perturbation theory (MPn) The various energy values can be calculated if the wavefunctions are known: ** The higher order wave functions developed by promoting electrons into the virtual orbitals taken from the HF calculation. MP2 = double excitations MP3 = little improvement over MP2 MPn = size independent, not variational

11. Other Methods… Coupled-cluster (CC) theory – a perturbation theory of e- correlation with an excited configuration that is “coupled” to the reference configuration. CCD = include double excitations CCSD = single + double excitations CCSD(T) = includes triples (approximately) Gaussian-2/Gaussian-3 (G2/G3) = composite methods using various levels of theory to compute thermochemistry ONIOM = model critical parts of the system with various levels of theory

12. Density Functional Theory Increased in popularity within last 2 decades. Given a known e- density  form the H operator  solve the Schrödinger Eq.  determine the wavefunctions and energy eigenvalues. HF – the wavefunction is essentially uninterpretable, lack of intuition. Hamiltonian depends ONLY on the positions and atomic number of the nuclei and the number of e-. HF – optimize the e- wavefunction DFT – optimize the e- density

13. Definitions Function: a prescription which maps one or more numbers to another number: y = f(x) = x2. Operator: a prescription which maps a function onto another function: Functional: A functional takes a function as input and gives a number as output. An example is: F[f(x)] = y Here f(x) is a function and y is a number. An example is the functional to integrate x from –¥ to¥.