Research can be carried out by acquiring knowledge in one or more of the following areas.
Depending on the area of research specialization knowledge has to be acquired in two or three advanced areas.
In this connection, previous co-workers have specialized and have presented their knowledge in group seminars. Here are some seminar titles of the past years:
The course is structured into five parts:
The main part of the practical work consists of working with computers of different type. All parts of the practical work are compulsory:
Definition of the term ab initio, goals; advantages; size-extensivity, approximations involved; limitations, classification; difference between empirical, semiempirical, and nonempirical methods; What is calculated? Comparison with experimental measurements; acronyms; units; conventions; history.
Unitary vector space, Hilbert space, basis vectors; scalar product, operators, dyadic product, projection operator, Hermitian operator, turn-over rule, unitary operator, eigenvalue problem, Hamilton operator; Born-Oppenheimer approximation, single determinant wave-function, what is an orbital? spin orbitals and space orbitals; Variational principle, Hartree product; antisymmetry principle, Slater determinant; matrix elements for Slater determinants; Slater-Condon rules, exchange and Coulomb operator, Fock operator, Hartree-Fock equations; canonical form, orbital energy, separation of spin, energy for closed-shell system, restricted HF (RHF), LCAO approach; basis functions, metric of a non-orthogonal basis, overlap integrals, Fock matrix, Roothaan-Hall equations, energy expressions, density matrix.
Flow chart for ab initio calculations; input; choice of the coordinates; Cartesian and internal coordinates, geometry optimization and the right choice of coordinates, z-matrix formalism; dummy atoms; puckering coordinates; determination of symmetry, number of independent internal coordinates, molecular framework group, what ab initio programs are available? How to get them; how to use them?
Building-block principle, H-type functions and H-type orbitals; Slater-type functions and Slater-type orbitals; the exponent zeta; diffuse and compact basis functions, Slater rules; Gaussian-type functions and Gaussian-type orbitals, cusp problem, energy of the H atom; LCGTF, difference between STFs and GTFs, cartesian Gaussians; advantages and disadvantages; first order and second order GTF, the index l, Gaussian lobe functions.
Notation; minimal zeta basis; double zeta basis; choice of the exponent, split valence basis, extended basis sets; isotropic limit, HF limit, augmented basis sets; hidden variables; floating functions; bond functions; polarization functions (p, d, f, g, h); radial and angular polarization, notation for augmented basis sets, weight, size, and position of a basis function, uncontracted and contracted basis sets; construction of contracted basis sets; contraction criteria; segmented and generalized contraction, the scaling theorem; notation; Huzinaga-Dunning basis sets; Pople minimal basis sets; shell constraints; STO-NG; split valence basis sets, augmented split valence basis sets; addition of diffuse functions; even-tempered basis sets, selection of a basis set, Pople's recipe, basis sets for special molecular properties, Dunning basis sets.
Single bar and double bar integrals, physical and chemical notation of integrals; number of integrals; shell structure; one-electron integrals; overlap integrals, Cartesian Gaussian functions, spherical Gaussians, transformation from cartesian to spherical Gaussians, angular shell, contaminants, Gaussian product theorem, Laplace transform of r12-1, incomplete Gamma functions, shift of angular momentum, differentiation of Gaussian functions, recurrence relationships, Cartesian Hermite Gaussian-type functions, translational invariance, Gaussian Quadrature, overlap integrals, kinetic energy integrals, nucleus-electron attraction integrals, two-electron repulsion integrals (ERIs), [ss|ss] ERI, prescreening of ERIs, McMurchie-Davidson scheme, Dupuis-King-Rys scheme, Rys polynomials, Pople-Hehre scheme, exponent sharing, early contraction, late contraction, axes rotation, PRISM algorithm, contraction and scaling, choice diagrams, Obara-Saika scheme, Resolution of the identity (RI) method, canonical ordering, sequential and random storing, batch processing, packing and unpacking of ERI labels, synchronous/asynchronous IO, buffering of ERIs.
Conventional Roothaan-Hall SCF, the trial and error method, iterative solution of the Roothaan-Hall equations, initial guess, diagonalization of the core hamiltonian, extended Hückel type guess, INDO and MNDO guess, guess from atomic densities, basis set expansion, solution of the pseudo-eigenvalue problem, canonical orthonormalization, spectral form of S, Schmidt orthonormalization, symmetric orthonormalization, density matrices, projector idempotency constraint, Jacobi diagonalization, Givens-Householder diagonalization, stationary state conditions (with regard to orbitals and basis functions), unitary transformation of MOs, mixing of occupied and virtual orbitals, orbital rotation, energy gradient with regard to expansion coefficients, Brillouin theorem, construction of the Fock matrix, permutational symmetry of ERIs, supermatrix formalism, Raffenetti ordering, timings for construction of the F matrix.
Convergence criteria for SCF, convergence problems, oscillations, state switching, divergence, counteractions, convergence acceleration, state loyalty, univariate search methods, Fock matrix partitioning, pseudocanonical orbitals, l-dependent form of F, mixing coefficient, energy gradient with regard to l, Bessel equation, overlap of spinorbitals, Camp-King method, unitary transformation by an exponential matrix, diagonalization of a rectangular matrix, orbital rotation, extrapolation methods, Hartree damping, dynamic damping, 3-point extrapolation, Aitken d2 method, 4-point extrapolation, level shifting, starting and termination strategies, Pulay's DIIS, linear dependence of changes in F and P, errors in P and stationary state conditions, ADEM-DIOS, QC-SCF, linear convergence, quadratic convergence, orbital mixing expressed by a CI formalism, Newton-Raphson formulation of the SCF problem.
Different open shell cases, generalized Brillouin theorem, generalized HF equations, coupling terms, generalized Roothaan-Hall equations, partitioned HF (PHF); ROHF according to Roothaan; the McWeeny method for ROHF; unrestricted HF (UHF); Pople-Nesbet equations; properties of the UHF energy; dissociation problem by RHF and UHF; UHF wave function; the expectation value of S2; spin contamination; spin projection methods; UHF electron and spin densities.
Complex GHF, real GHF, complex UHF, real UHF, complex RHF, real RHF, form of general spinorbitals, possible constraints, internal and external stability, stability test, form of the Hessian, symmetry dilemma of UHF, singlet instability, non-singlet instabilities, complex HF, the O2 molecule and its 1Dg state, complex orbitals vs. real orbitals, complex Fock matrix, complex HF equations, eigenvalues for the complex problem, flow chart diagram.
M4-myth, number of ERIS for large molecules, storage problem, recalculation of ERIS; prescreening of two-electron integrals; batch processing, recurrence formula for the Fock matrix; cost for an ERI in dependence of l, selective storage of integrals; minimization of errors; changes in the number of ERIs per iteration step
CI wave function, properties of CI methods, full CI, truncated CI, tape driven and integral driven CI, UGA, SGA
Model space and orthogonal space, projection operators.