Physical Chemistry 2 nd Edition

1. Chapter 26 Computational Chemistry Physical Chemistry 2nd Edition Thomas Engel, Philip Reid

2. Objectives • Discover the usage of numerical methods. • Discussion is the Hartree-Fock molecular orbital model.

3. Outline • The Promise of Computational Chemistry • Potential Energy Surfaces • Hartree-Fock Molecular Orbital Theory: A Direct Descendant of the Schrödinger Equation • Properties of Limiting Hartree-Fock Models • Theoretical Models and Theoretical Model Chemistry

4. Outline • Moving Beyond Hartree- Fock Theory • Gaussian Basis Sets • Selection of a Theoretical Model • Graphical Models • Conclusion

5. 26.1 The Promise of Computational Chemistry • Sufficient accuracy can be obtained from computational chemistry. • Approximations need to be made to realize equations that can be solved. • No one method of calculation is likely to be ideal for all application. • Hartree-Fock theory leads to ways to improve on it and to a range of practical quantum chemical models.

6. 26.2.1 Potential Energy Surfaces and Geometry • Energy minima give the equilibrium structures of the reactants and products. • Energy maximum defines the transition state. • Reactants, products, and transition states are all stationary points on the potential energy diagram.

7. 26.2.1 Potential Energy Surfaces and Geometry • In the one-dimensional case, 1st derivative of the potential energy with respect to the reaction coordinate is zero: • For many-dimensional case, each independent coordinate, Ri, gives rise to 3N-6 second derivatives:

8. 26.2.1 Potential Energy Surfaces and Geometry • Stationary points where all second derivatives are positive are energy minima: where ζi = normal coordinates • Stationary points where all but one are positive are saddlepoints: where ζi = reaction coordinate

9. 26.2.2 Potential Energy Surfaces and Vibrational Spectra • The vibrational frequency for a diatomic molecule A-B is • k is the force constant which is defined as • And μis the reduced mass.

10. 26.2.3 Potential Energy Surfaces and Thermodynamics • The energy difference between the reactants and products determines the thermodynamics of a reaction. • The ratio is as follow,

11. 26.2.3 Potential Energy Surfaces and Thermodynamics • The energy difference between the reactants and transition state determines the rate of a reaction. • The rate constant is given by the Arrhenius equation and depends on the temperature:

12. 26.3 Hartree-Fock Molecular Orbital Theory: A Direct Descendant of the Schrödinger Equation • 3 approximations need to realize a practical quantum mechanical theory for multielectron Schrödinger equation: • Born-Oppenheimer approximation • Hartree-Fock approximation • Linear combination of atomic orbitals(LCAO) approximation

13. MATHEMATICAL FORMULATION OF THE HARTREE-FOCK METHOD The Hartree-Fock and LCAO approximations, taken together and applied to the electronic Schrödinger equation, lead to a set of matrix equations now known as the Roothaan-Hall equations: where c = unknown molecular orbital coefficients ε = orbital energiesS = overlap matrixF = Fock matrix

14. MATHEMATICAL FORMULATION OF THE HARTREE-FOCK METHOD For Fock matrix, where Hcore = core Hamiltonian Coulomb and exchange elements are given by

15. MATHEMATICAL FORMULATION OF THE HARTREE-FOCK METHOD P is called the density matrix The cost of a calculation rises rapidly with the size of the basis set:

16. 26.4 Properties of Limiting Hartree-Fock Models • For computation, it is expected to have errors in: • Relative energies • Geometries • Vibrational frequencies • Properties such as dipole moments

17. 26.4.1 Reaction Energies • Hartree-Fock models is compare with homolytic bond dissociation energies. • For example in methanol,

18. 26.4.1 Reaction Energies • The poor results seen for homolytic bond dissociation reactions do not necessarily carry over into other types of reactions as long as the total number of electron pairs is maintained.

19. 26.4.2 Equilibrium Geometries • Systematic discrepancies are also noted in comparisons involving limiting Hartree-Fock and experimental. • They are geometries and bond distances. • The reason is that limiting Hartree-Fock bond distances is shorter than experimental values.

20. 26.4.3 Vibrational Frequencies • The error in bond distances for limiting Hartree-Fock models calculated frequencies are larger than experimental frequencies. • The reason is that the Hartree-Fockmodel does not dissociate to the proper limit of two radicals as a bond is stretched.

21. 26.4.4 Dipole Moments • Electric dipole moments are compared, the calculated values are larger than experimental values.

22. 26.5 Theoretical Models and Theoretical Model Chemistry • Limiting Hartree-Fock models do not provide results that are identical to experimental results. • Theoretical model chemistry is a detailed theory starting from the electronic Schrödinger equation and ending with auseful scheme.

23. 26.6 Moving Beyond Hartree-Fock Theory • Improvements will increase the cost of a calculation. • 2 approaches to improve Hartree-Fock theory: • Increases the flexibility by combining it with wave functions corresponding to various excited states. • Introduces an explicit term in the Hamiltonian to account for the interdependence of electron motions.

24. 26.6.1 Configuration Interaction Models • Improvements will increase the cost of a calculation. • 2 approaches to improve Hartree-Fock theory: • Increases the flexibility by combining it with wave functions corresponding to various excited states. • Introduces an explicit term in the Hamiltonian to account for the interdependence of electron motions.

25. 26.6.2 Møller-Plesset Models • Møller-Plesset models are based on Hartree-Fock wave function and ground-state energy E0 as exact solutions. where = small perturbation λ = dimensionless parameter

26. MATHEMATICAL FORMULATION OF MØLLER-PLESSET MODELS Substituting the expansions into the Schrödinger equation and gathering terms in λn yields

27. MATHEMATICAL FORMULATION OF MØLLER-PLESSET MODELS Multiplying each by ψ0 and integrating over all space yields the following expression for the nth-order (MPn) energy:

28. MATHEMATICAL FORMULATION OF MØLLER-PLESSET MODELS In this framework, the Hartree-Fock energy is the sum of the zero- and firstorder Møller-Plesset energies: The first correction, E(2) can be written as follows

29. MATHEMATICAL FORMULATION OF MØLLER-PLESSET MODELS The integrals (ij || ab) over filled (i and j) and empty (a and b) molecular orbitals account for changes in electron–electron interactions as a result of electron promotion, in which the integrals (ij | ab) and (ib | ja) involve molecular orbitals rather than basis functions. The two integrals are related by a simple transformation,

30. 26.6.3 Density Functional Models • Density functional theory is based on the availability of an exact solution for an idealized many-electron problem. • The Hartree-Fock energy may be written as where ET = kinetic energyEV = theelectron–nuclear potential energy EJ = CoulombEK = interaction energy

31. 26.6.3 Density Functional Models • For idealized electron gas problem: where EXC = exchange/correlation energy • Except for ET, all components depend on the total electron density, p(r):

32. MATHEMATICAL FORMULATION OF DENSITY FUNCTIONAL THEORY Within a finite basis set (analogous to the LCAO approximation for Hartree Fock models), the components of the density functional energy, EDFT, can be written as follows:

33. MATHEMATICAL FORMULATION OF DENSITY FUNCTIONAL THEORY Better models result from also fitting the gradient of the density. Minimizing EDFT with respect to the unknown orbital coefficients yields a set of matrix equations, the Kohn-Sham equations, analogous to the Roothaan-Hall equations Here the elements of the Fock matrix are given by

34. MATHEMATICAL FORMULATION OF DENSITY FUNCTIONAL THEORY FXCis the exchange/correlation part, the form of which depends on the particular exchange/correlation functional used. Note that substitution of the Hartree-Fock exchange, K, for FXCyields the Roothaan-Hall equations.

35. 26.6.4 Overview of Quantum Chemical Models • An overview of quantum chemical models.

36. 26.7 Gaussian Basis Sets • LCAO approximation requires the use of a finite number of well-defined functions centered on each atom. • Early numerical calculations use nodeless Slater-type orbitals (STOs), • Ifthe AOs are expanded in terms of Gaussian functions,

37. 26.7.1 Minimal Basis Sets • The minimum number is the number of functions required to hold all the electrons of the atom while still maintaining its overall spherical nature. • This simplest representation or minimal basis set involves a single (1s) function forhydrogen and helium. • In STO-3G basis set, basis functions isexpanded in terms of three Gaussian functions.

38. 26.7.2 Split-Valence Basis Sets • Minimal basis set is bias toward atoms with spherical environments. • Asplit-valence basis set represents core atomic orbitals by one set of functions andvalence atomic orbitals by two sets of functions: • for lithium to neon • for sodium to argon

39. 26.7.3 Polarization Basis Sets • Minimal (or split-valence) basis set functions are centered on atoms rather than between atoms. • The inclusion of polarization functions can be thought about either in terms of hybridorbitals.

40. 26.7.4 Basis Sets Incorporating Diffuse Functions • Calculations involving anions can pose problems as highest energy electrons may only beloosely associated with specific atoms (or pairs of atoms). • In these situations, basis setsmay need to be supplemented by diffuse functions.

41. 26.8 Selection of a Theoretical Model • Hartree-Fock models have proven to be successful in large number of situations and remain a mainstay of computational chemistry. • Correlated models can be divided into 2 categories: • Density functional models • Møller-Plessetmodels • Transitionstategeometry optimizations are more time-consuming than equilibriumgeometry optimizations, due primarily to guess of geometry.

42. 26.8.1 Equilibrium Bond Distances • Hartree-Fock double bond lengths are shorter than experimental distances. • Treatment of electroncorrelation involves the promotion of electrons fromoccupied molecular orbitals to unoccupied molecularorbitals.

43. 26.8.2 Finding Equilibrium Geometries • An equilibrium structure corresponds to the bottom of a well on the overall potential energy surface. • Equilibrium structures that cannot be detected are referred to as reactive intermediates. • Geometry optimization does not guarantee that the final geometry will have a lower energythan any other geometry of the same molecular formula.

44. 26.8.3 Reaction Energies • Reaction energy comparisons are divided into three parts: • Bond dissociation energies • Energies of reactions relating structural isomers • Relative proton affinities.

45. 26.8.4 Energies, Enthalpies, and Gibbs Energies • Quantum chemical calculations account for thermochemistry by combining the energies of reactant and product molecules at 0 K. • Residual energy ofvibration is ignored. • We would need 3 corrections: • Correction of the internal energy for finite temperature. • Correction for zero point vibrational energy. • Corrections of entropy.

46. 26.8.5 Conformational Energy Differences • Hartree-Fock models overestimate differences by large amounts. • Correlated models also typically overestimate energy differences but magnitudes of the errors are much smaller than those seen for Hartree-Fock models.

47. 26.8.6 Determining Molecular Shape • The problem of identifying the lowest energy conformer in simple molecules is when the number of conformational degrees of freedom increases. • Sampling techniques will need to replace systematic procedures for complex molecules, thus Monte Carlo methods is used.

48. 26.8.7 Alternatives to Bond Rotation • Single-bond rotation is the most common mechanism for conformer interconversion. • 2 other processes are known: • Inversionis associated with pyramidal nitrogen or phosphorus and involves a planar transition state. • Pseudorotation is associated with trigonal bipyramidal phosphorus and involves a square-based-pyramidal transition state.

49. 26.8.8 Dipole Moments • Dipole moments from the two Hartree-Fock models are larger than experimental values due to behavior of the limiting Hartree-Fock model. • Recognize that electron promotion from occupied to unoccupied molecular orbitals takes electrons from “where they are” to “where they are not”.

50. 26.8.9 Atomic Charges: Real or Make Believe? • Charge distributions assess overall molecular structure and stability. • Mulliken population analysis can be used to formulate atomic charges.