Chemistry 700

1. Chemistry 700 Lectures

2. Resources • Grant and Richards, • Foresman and Frisch, Exploring Chemistry with Electronic Structure Methods (Gaussian Inc., 1996) • Cramer, • Jensen, • Ostlund and Szabo, Modern Quantum Chemistry (McGraw-Hill, 1982)

3. Why is one interested in computational chemistry? • • experiments are expensive, and often indirect. • structure and property prediction is of great value. • We need to know where the electrons are to be able to predict how eg. light will affect them or whether they are ready to create bonds with other atoms/molecules. • Possible applications include: • 1. drug design. • 2. development of new materials.

4. There exist various approximation levels, the major being: • Molecular Dynamics – called also Molecular Mechanics – treats atoms as classical • objects, with interactions described by predetermined potentials, usually fitted • to some analytical functions. Major applications are to perform geometry • optimization and eg. study docking (how a small drug molecule can bind to a • molecular macromolecule). This approach allows to study systems consisting • of thousands of atoms but its quality is limited by the choice of the force • field/potential. • Ab-initio Theory starts from fundamental equations of quantum theory and works • is up from there. Since strict analytical formula exists for energies and other • system properties, many various properties can be computed, including MD • potentials and interaction with light or magnetic field. However, full quantum- • mechanical treatment is expensive and the systems studied are severely limited • in the size to tens or hundreds of atoms. This course is focused mostly on this • method.

5. Schrödinger Equation • H is the quantum mechanical Hamiltonian for the system (an operator containing derivatives) • E is the energy of the system •  is the wavefunction (contains everything we are allowed to know about the system) • ||2 is the probability distribution of the particles

6. Hamiltonian for a Molecule • Kinetic energy of the electrons • Kinetic energy of the nuclei • Electrostatic interaction between the electrons and the nuclei • Electrostatic interaction between the electrons • Electrostatic interaction between the nuclei

7. Solving the Schrödinger Equation • Analytic solutions can be obtained only for very simple systems • Particle in a box, harmonic oscillator, hydrogen atom can be solved exactly • Need to make approximations so that molecules can be treated • Approximations are a trade off between ease of computation and accuracy of the result

8. Expectation Values • for every measurable property, we can construct an operator • repeated measurements will give an average value of the operator • the average value or expectation value of an operator can be calculated by:

9. Variational Theorem • the expectation value of the Hamiltonian is the variational energy • the variational energy is an upper bound to the lowest energy of the system • any approximate wavefunction will yield an energy higher than the ground state energy • parameters in an approximate wavefunction can be varied to minimize the Evar • this yields a better estimate of the ground state energy and a better approximation to the wavefunction

10. Born-Oppenheimer Approximation • The nuclei are much heavier than the electrons and move more slowly than the electrons • In the Born-Oppenheimer approximation, we freeze the nuclear positions, Rnuc, and calculate the electronic wavefunction, el(rel;Rnuc) and energy E(Rnuc) • E(Rnuc) is the potential energy surface of the molecule (i.e. the energy as a function of the geometry) • on this potential energy surface, we can treat the motion of the nuclei classically or quantum mechanically

11. Born-Oppenheimer Approximation • freeze the nuclear positions (nuclear kinetic energy is zero in the electronic Hamiltonian) • calculate the electronic wavefunction and energy • E depends on the nuclear positions through the nuclear-electron attraction and nuclear-nuclear repulsion terms • E = 0 corresponds to all particles at infinite separation

12. Nuclear motion on the Born-Oppenheimer surface • Classical treatment of the nuclei (e,g. classical trajectories) • Quantum treatment of the nuclei (e.g. molecular vibrations)

13. Hartree Approximation • Assume that a many electron wavefunction can be written as a product of one electron functions • If we use the variational energy, solving the many electron Schrödinger equation is reduced to solving a series of one electron Schrödinger equations • each electron interacts with the average distribution of the other electrons

14. Hartree-Fock Approximation • the Pauli principle requires that a wavefunction for electrons must change sign when any two electrons are permuted • the Hartree-product wavefunction must be antisymmetrized • can be done by writing the wavefunction as a determinant

15. Spin Orbitals • each spin orbital I describes the distribution of one electron • in a Hartree-Fock wavefunction, each electron must be in a different spin orbital (or else the determinant is zero) • an electron has both space and spin coordinates • an electron can be alpha spin (, , spin up) or beta spin (, , spin up) • each spatial orbital can be combined with an alpha or beta spin component to form a spin orbital • thus, at most two electrons can be in each spatial orbital

16. Fock Equation • take the Hartree-Fock wavefunction • put it into the variational energy expression • minimize the energy with respect to changes in the orbitals • yields the Fock equation

17. Fock Equation • the Fock operator is an effective one electron Hamiltonian for an orbital  •  is the orbital energy • each orbital  sees the average distribution of all the other electrons • finding a many electron wavefunction is reduced to finding a series of one electron orbitals

18. Fock Operator • kinetic energy operator • nuclear-electron attraction operator

19. Fock Operator • Coulomb operator (electron-electron repulsion) • exchange operator (purely quantum mechanical -arises from the fact that the wavefunction must switch sign when you exchange to electrons)

20. Solving the Fock Equations • obtain an initial guess for all the orbitals i • use the current I to construct a new Fock operator • solve the Fock equations for a new set of I • if the new I are different from the old I, go back to step 2.

21. Hartree-Fock Orbitals • for atoms, the Hartree-Fock orbitals can be computed numerically • the ‘s resemble the shapes of the hydrogen orbitals • s, p, d orbitals • radial part somewhat different, because of interaction with the other electrons (e.g. electrostatic repulsion and exchange interaction with other electrons)

22. Hartree-Fock Orbitals • for homonuclear diatomic molecules, the Hartree-Fock orbitals can also be computed numerically (but with much more difficulty) • the  ‘s resemble the shapes of the H2+ orbitals • , , bonding and anti-bonding orbitals

23. LCAO Approximation • numerical solutions for the Hartree-Fock orbitals only practical for atoms and diatomics • diatomic orbitals resemble linear combinations of atomic orbitals • e.g. sigma bond in H2  1sA + 1sB • for polyatomics, approximate the molecular orbital by a linear combination of atomic orbitals (LCAO)

24. Basis Functions • ’s are called basis functions • usually centered on atoms • can be more general and more flexible than atomic orbitals • larger number of well chosen basis functions yields more accurate approximations to the molecular orbitals

25. Roothaan-Hall Equations • choose a suitable set of basis functions • plug into the variational expression for the energy • find the coefficients for each orbital that minimizes the variational energy

26. Roothaan-Hall Equations • basis set expansion leads to a matrix form of the Fock equations FCi = iSCi • F – Fock matrix • Ci – column vector of the molecular orbital coefficients • I – orbital energy • S – overlap matrix

27. Fock matrix and Overlap matrix • Fock matrix • overlap matrix

28. Intergrals for the Fock matrix • Fock matrix involves one electron integrals of kinetic and nuclear-electron attraction operators and two electron integrals of 1/r • one electron integrals are fairly easy and few in number (only N2) • two electron integrals are much harder and much more numerous (N4)

29. Solving the Roothaan-Hall Equations • choose a basis set • calculate all the one and two electron integrals • obtain an initial guess for all the molecular orbital coefficients Ci • use the current Ci to construct a new Fock matrix • solve FCi = iSCi for a new set of Ci • if the new Ci are different from the old Ci, go back to step 4.

30. Solving the Roothaan-Hall Equations • also known as the self consistent field (SCF) equations, since each orbital depends on all the other orbitals, and they are adjusted until they are all converged • calculating all two electron integrals is a major bottleneck, because they are difficult (6 dimensional integrals) and very numerous (formally N4) • iterative solution may be difficult to converge • formation of the Fock matrix in each cycle is costly, since it involves all N4 two electron integrals

31. Summary • start with the Schrödinger equation • use the variational energy • Born-Oppenheimer approximation • Hartree-Fock approximation • LCAO approximation

32. Ab initio methods 1. The Hartree-Fock method (HF)

33. The Hartree-Fock method We want to solve the electronic Schrödinger equation: For this, we need to make some approximations These will lead to the Hartree-Fock method (which is the simplest ab initio method)

34. The Hartree-Fock method Approximation 1: Decompose Y into a combination of molecular orbitals (MOs) MO: one-electron wavefunction (fn) However, this is not a good wavefunction, as wavefunctions need to be antisymmetric: swapping the coordinates of two electrons should lead to sign change Good wavefunction:

35. The Hartree-Fock method The antisymmetry of the wavefunction can be achieved by constructing the wavefunction as a Slater Determinant: fi is a “spinorbital”: contains also the spin of the electron

36. The Hartree-Fock method Approximation 2: The Hartree-Fock wavefunction consists of a single Slater Determinant (This implies that the electron-electron repulsion is only included as an average effect => the Hartree-Fock method neglects electron correlation)

37. LCAO: Linear Combination of Atomic Orbitals AO or basis function MO expansion coefficients The Hartree-Fock method Approximation 3: The MOs fi are written as linear combinations of pre-defined one-electron functions (basis functions or AOs)

38. How to obtain the optimal coefficients cmi? The Hartree-Fock method • The Hartree-Fock wavefunction is a single Slater Determinant • The MOs in the Slater Determinant are expressed as linear combinations of atomic orbitals • The exact form of the wavefunction depends on the coefficients cmi • The Hartree-Fock method aims to find the optimal wavefunction

39. “The energy calculated from an approximation to the true wavefunction will always be greater than the true energy” The Hartree-Fock method Variation Principle So, just find the coefficients cmi that give the lowest energy! This leads to the Hartree-Fock equations, which can be solved by the Self-Consistent Field (SCF) method

40. The Hartree-Fock method Approximations leading to the Hartree-Fock method • Start with the electronic Schrödinger equation • Decompose Y into a combination of MOs => antisymmetry imposed by using Slater Determinant wavefunctions • Wavefunction consists of a single Slater Determinant • MOs are linear combinations of AOs (which are predefined) • Variation principle to find optimal coefficients

41. The Hartree-Fock method The main weakness of Hartree Fock is that it neglects electron correlation In HF theory: each electron moves in an average field of all the other electrons. Instantaneous electron-electron repulsions are ignored Electron correlation: correlation between the spatial positions of electrons due to Coulomb repulsion - always attractive! Post-HF methods include electron correlation

42. Ab initio methods Post-HF methods Hartree-Fock Møller-Plesset Perturbation theory (MP2, MP3, MP4,…) Multiconfigurational SCF (MCSCF) Coupled Cluster (CCSD, CCSDT, …) Configuration Interaction (CI)